Multiplicateurs sur les espaces de Banach de fonctions sur un groupe localement compact abélien Violeta Petkova To cite this version: Violeta Petkova. Multiplicateurs sur les espaces de Banach de fonctions sur un groupe localement compact abélien. Mathématiques [math]. Université Sciences et Technologies - Bordeaux I, 2005. Français. �tel-00011714� HAL Id: tel-00011714 https://tel.archives-ouvertes.fr/tel-00011714 Submitted on 2 Mar 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. #! 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WÂzR^ ¶ ¥ ¥ !2$ ± «$% ³§y L2 (R+) °$ Ð¥ !2 ¸¬®|¦´ µqQ¦´¥ $«$³´ ¬Å !#"$»{! ³¿"#¥ δ "Ð¥ ! ³§ ¥ R+ ³ δδ "Ð¥ ! ƶ+ ! « " ³§!¬$ ¥ ¼±Ä ¦§Ä» " ³´« "¼ ¬Å !" ³´" ³´² ¥y R+ ² $ ³ ± !"L¦§ ³§·"¼$O ¥ ³´²Ä± !"¼ D \¨µ´¨ δ(x + y) + (x) = sup ess δf < +∞, pour x ≥ 0, y≥0 \¨µ + (x) = sup ess δf y≥0 ·"^ ! L2δ (R+ ) Yl ³´" n δ(y) δ(y) < +∞, pour x < 0 δ(y − x) + := f mesurable sur R | Z +∞ 2 0 2 o |f (x)| δ(x) dx < +∞ . P + : L1loc (R) −→ L1loc (R+ ) ¦ $ ±Ä"¼ ¥ L !y³ ± L¦Á±¶+ ¬®¥ ¦§Ð P +f (x) = f (x), µ µ x ≥ 0 " P +f (x) = 0 » ¥ µXL!1 ¼±Ä" $¥ T | ! D ¥ L2 (R+) $W"S± ¦§ $ ±Ä"¼ ¥ O É-³§ ! $\Â\R^ ¶ ³ x<0 δ P + S−a T Sa f = T f, ∀a ∈ R+ , ∀f ∈ L2δ (R+ ), ³§Ê ! Ŧ ¼±Ä" $¥ ! ³g ¥y L1 (R) ± S f (x) = f (x − a), p.p. µbAH! ! Ä"¼ a loc Í ¦ Ç $ ± « H $ B ¼ Ä ± " $¥ SÉ ´ ³ $! W \  L R E ¶ ¥ ·"B!Æ!y "¼ C ∞(R+) ¦ÍÇ$ ± «$ 2 Wδ Lδ (R+ ) B¶+! « " ³§!y «$¦§± C ∞ %® ¥ "O±Ä! R+ µ Uz¦j "L ³§ !)«$!y!¥+« ¶\µc #¥ ¥ " ¥y" T ∈ W » ³´¦·Èl³§W"¼D¥ ! D ³´ " ³§y¥y"¼³´! µ "¼$¦´¦§ #¥ 1 T \¨µ T f = P + (µ ∗ f ), pour f ∈ C ∞ (R+ ). Sa T c FS ¦§¥ »³§¦·Èl³§W"¼D¥ ! D¶+! «·"¼³§ ! h ∈ L∞(R) »± ¦§ ¦§ ¸¬®|¦´ T » " $¦´¦§#¥ W¨µÇ© T f = P + F −1 (hfˆ), pour f ∈ L2 (R+ ). A!T$Ç$W"yÈl®«$ ¬Å¬ ¬Å³´$L $«·"¼³¿¶Q &Ê $! ¼± ¦´³§ \¨µ O·" \¨µ© ¥ T ∈ W »| δ ² $ ³ \¨µ ¯ ·µC{ y¦§°$¬ ·"¾± ³¿"® ³§¬³§¦Á±Ä³§ ®± ¥ ¦§° ¬Å Ʀ ÃÈy³´ " $! « ) ¥ ¸¬®|¦´ δ ¥y!¬&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y_ ¥ Q¥ !Ð$ ± «$L% ³§ L2 (R) » #¥ ³·"¾± ³¿"Q± ¥ ³¥ ! ¦´°$¬¥y² $ "$µ FH± ! H¦§± $¬³§° ± " ³§I ¥2?K½± ³´" »! ¥ ω^" ¼± ³¿"¼! L± | Z« ® ! ³´$ ¦´°$¬ µ A!¢± y ¦§¦§ ³´ ) ¥ R " ¥y"¼º± y ¦§³´«ª±Ä" ³§!G¬$ ¥ ¼±Ä ¦§Ä» " ³§«·"¼ ¬Å !#" ³¿"¼³¿² ¥y R ²$ ³ ±Ä!#"L¦§±®«$! y³´"¼³´!) ¥y³´²Ä± !"¼Ð W¨µ ¯ ω(x + t) < +∞, pour tout t ∈ R. ω̃(t) := sup ess Yl³´" q_¥ x∈R ω(x) ¦ ±Ä«$² « "¼ ³§ ¦E $B¶+! «·"¼³´! L2 (R) ω !Ty ! ³¿"O ¥ a ∈ R, Z R f ¬Å ¥ ± ¦´$O ¥ R "¼$¦´¦§ ¥y |f (x)|2 ω(x)2 dx < +∞. L2ω (R) ¦ÍÇ $ ¼±Ë"¼$¥yB "¼¼±Ä! ¦§±Ä"¼³´! Sa,ω ± B¦Á±®¶+ ¬®¥ ¦´® (Sa,ω f )(x) = f (x − a) p.p. A!1± W¨µÀ kSa,ω k = sup ess x∈R ω(x + a) ω(x) "ÆÊ « % ¦Íǽ¸ " ½ °$ W¨ µ ¯Ã»}¦ ¼±Ä" $¥ S ":| ! ĵAH!G!y "¼ M ¦ ! ¬® ¦´ $:¬®¥ ¦¿"¼³ ¦§³´«ª±Ä" $¥ Å ¥ L2 (R) »«Ä " Âz%ËÂzy³§ ~¦a,ωÍÇ$! $¬® ¦´ $) ¼±Ä" $¥ 1ω | ! $ » #¥ ³ «$¬¬®¥y" $!"±ª² « S ¥ Lω "¼¥y" a ∈ R µjCQ± ¦§Ê °$ M ¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ B ¥ L2 (R) $ "L ³´$!:«$ ! !#¥ a,ω » H ³ M ∈ M » ³§¦ÃÈl³§ " h ∈ L∞(R) "¼ ¦ ¥y W¨µ d M f = hfˆ, ∀f ∈ L2 (R). Cb±&¶+ ! « " ³§! h $W"L± $¦§ H¦§S ¸¬®|¦§Hy M µ^ ¥ Ê ! $ ± ¦§³´ ! K«$ ¥ ¦¿"¾±Ä" ¥yK"¼¥l" $ ± «$B% ³´ $µÄ^ ¥ ! ¥y ¬¬$³´! ³§ } ¥ ¦¿"¾±Ä" Q± !± ¦´Ê¥ _ ± ! ¦´K«ª±Ä ³§ «$ "ªµ qg¦´¥ $«$³´ ¬Å !#"$»Yl½ ³§ ¦§ B¬!"¼ ± ! #¥ " ¥y"L¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ @ ¥ lσ2 (Z) n := v = (vn )n∈Z | X 2 2 |vn | σ (n) < +∞ $ "^± «$³´D%¥ !yD¶+ ! « " ³§!:½ ¦´¬ ½yS| ! D ¥ $ ³§Ê! ± !" ¦´1 ½ ³´¶ " ¥ ¥yÐ" ° $« $¬¬Å !" ± S n n∈Z o 1 z ∈C| < |z| < ρ(S) , ρ(S −1 ) l2 (Z) µ@CE1«ª±ÄÅ 1 = ρ(S) D>Qσ "¼ ¦´± ! µ{ρ(S FH± −1!yI) ¦§± ¬Å³´°$ o ¬® ¦´1!j ±ª² §³ "¼:"¼ ± ³´" ± " ³§®y¥ K? ½ ± ³´" l» ! ¥ ® "¼± ¦´³§ ! I¥ ! $ ¥y¦´"¾±Ë"®± ! ± ¦§Ê ¥ % W¨µ ¥yI" ¥y" ¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ ¥ L2 (R) » ¥ "¼ ¥ D¦§$ ³´ ω ² $ ³ ± !"I ¥ ¦§ ¬Å !" ¦ ½¸ "¼½ ° W¨µ ¯Ãµ{>Q!º" $!± !"I«$ ¬ " ωÐ $ ³§¬³§¦§³¿"¼¥ [email protected]$!"¼ ¦§$K ¼±Ä" $¥ B HÉ-³§ ! $\Â\RL ¶E "O¦´$¬&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y $»³´¦ "O!±Ë"¼¥ $¦ Å«$! $« " ¥ #¥ "¼¥l"& $ ±Ä"¼ ¥ D ÅÉ-³´$! WÂzR^ ¶± ¬ "I¥ ! !#"¼±Ä"¼³´!5± ! ±Ë ¦´Ê¥ &% W¨ µ© õUz¦b$W" ³´$!Z« ! !#¥ #¥ &"¼ ¥y"H $ ±Ä"¼ ¥ Sy É-³§ ! $\Â\RL ¶g ¥y L2 (R+) " ·"¼ ¬³§! ± P +M » M $W"¥ !¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ S ¥ L2 (R) I·"®±Ä! ®«$Å«ª± \¨µ Q·"W¨µÇ© !#"@ }« ! #¥ $!y«$$K³§¬¬Å ³Á±Ä" $[email protected] $ ¥ ¦´"¼±Ä"¼g ¥ g¦§$g¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ $µ L¥ D!±¹² ! ± ¥2 ·"¼ ¬³§! H ³" ¥y" $ ±Ä"¼ ¥ É-³§ ! $\Â\RL ¶@ ¥ L2(R+) " ¦§±& "¼ ³§«·"¼³§ !)% L2 (R+) j¥ !:¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ Q ¥ L2 (R) » ω "B¥ ! ³´ K ¥δ R ³§ ! «¾½ ³§ ³ »" $¦ #¥ ω|δ = δ µj^$± ! ¬³§! S¦§$S¬Å·"¼½ $ωS ² $¦§ $ ¥ H¦´$H¬®¥y¦´"¼³ ¦§³´«ª±Ë " $¥ } ¥ }¦´$g$ ± «R$% ³§ »± ± "¼ $±ª² $« ¥y$¦#¥ K¬ly³ «$±Ä"¼³´! Q%¦Á± ³´"¼¥ ±Ä"¼³´!Ð $ $ ±Ä"¼ ¥ g BÉ-³´$! WÂ\RL ¶\»Ä! ¥ g!" $ ¬³§ O $ ¥ O¦§ ¦§° ¬Å ¥yL¦ !$Ç "¾±Ä³´" ³´! ³´" ³Á± ¦´$¬$!" " ¼± ³¿"¼$ µqQ¦§¥ $« ³§ ¬Å !"^ !1 y"¼³´$!"O¦§ $ ¥y¦´"¾±Ë"^ ¥ ³´²Ä± !"$µ 6¨µ´¨ δ !"! $# R+ T ∈ W # ! - a ∈ J 0/ (T f ) ∈ L2 (R+) ,# f ∈ C ∞δ %'(R&"+( ). 1)+* ,# δ. a . c 2 3 1 4 ! 5 6 8 9 7 6 : ; >= ∞ 2)+* ,# νa ∈ L (R) (<( a ∈ Jδ ( + 3)>? @ ! ! ( . d (T f )a = P + F −1 (νa (f )a ), pour f ∈ Cc∞ (R+ ). rδ− < rδ+ . 2 134!5 6A796: ( ◦ ν ∈ H ∞ (Ω δ ) - = B ,# 5 (<( ◦ a ∈ Jδ ν(x + ia) = νa (x), p.p sur R+ kνk∞ ≤ cδ kT k. Uz« ³b !1¥l"¼³§¦´³§ H¦§ B! "¾±Ë"¼³§ ! B ¥ ³¿²Ä± !"¼$D (f )a (x) = f (x)e−ax , p.p., ∀f ∈ L2δ (R+ ), ∀a ∈ R, 1 1 + (n) n , r − = lim δ + (−n)− n , f rδ+ = lim δf δ Jδ := n→+∞ [ln rδ− , ln rδ+ ], n→+∞ Ωδ := {z ∈ C | Im z ∈ Jδ }, Z 2 + (u)du. cδ = exp 2 ln δf 1 L¥ Æ! ¥ Æ ¬Å¬$Å$! ¥ ³´" Z Z ² $! ³´)± ¥ ¦´°$¬T ~¦ÍÇ·Èl³§W"¼$!y«$Z ¥ W¸l¬&¦´SÇ¥ !)¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ Q ¥ K¥y!)$ ± «$Hys± !±Ä«¾½) ^¶+!y« "¼³´! } !y³§$K! ! ¦´¥ ¥ R »{¬Å± ³§ ¥ ¥y!xÊ ¥ C_?KN µ_?K ¦´°$¬ "¾±Ä!#" |ª± ¥ « ¥ x ¦§¥ DÊ ! $ ± ¦ #¥ ¦§ ¦´°$¬®"¼¼±Ä³´"¼ ± ! I¦§?K½± ³´"¼ l»{! ¥yI!y¥ I ¬¬Å ± ! «¾½ $ ¥ D¦´«ª± ¥1Ê ¥ Z µ|Yl ³´" S : CZ 3 (x(n))n∈Z −→ (x(n − 1))n∈Z . C HF ± ! ¦´D?K½± ³´"¼ Hy»l! ¥ K«$! ³§ !yB¦§$K$ ± «$ B Ds± !±Ä«¾½ ± !"L¦§$ ³§·"¼$O ¥ ´³ ²Ä± !"¼ D R ¨ ÅCQ ! ¬® ¦´ »¦Á±&¶+! «·"¼³´! n∈Z $ "L« !"¼³§!#¥ D RH BA!T± F (Z) ± !y C µ E S(E) ⊂ E ¥ y$Æ ¥ ³´" $ !y³§$Å ¥ Z E H ¥ ³´" $B ¥y Z »l² $ ³  "Æy$! ± ! E " ¥ Å" ¥y" pn : x −→ x(n) S −1 (E) ⊂ E µ SR AH!ű ψ (E) ⊂ E, ∀z ∈ T " sup kψ k < +∞ » Ц $ ±Ä"¼ ¥ z z ± B¦Á±&¶+ ¬®¥ ¦´ ψz (a)(n) = a(n)z n ∀n ∈ Z.z∈T ψz $W"g !y³ FH± ! L¦Á±® ¥ ³´" I!y¥ L $ ³§Ê! ! ± spec(S) ¦§ « "¼ y ¦ $ ±Ä"¼ ¥ S % ¬ ± ³´! $!y 1± ! E µQ^¥yб l« ³§! ®%" ¥y"¬&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y ¥ E ¥ ! ƶ+ ! « " ³§!5 W F (Z) ◦ $!" ³§$¦´¦§ ¬Å !#"T| ! ¥ spec(S) " #¥ ³ $W" ¦§¥ T½ ¦§ ¬Å ½ º ¥ spec(S) »L ³ ◦ µq_¥yL" ¥y" k ∈ Z »! ¥ S± ¦§!y e ¦§±Ð ¥ ³´" D ¥ Z »|y!"L" ¥ L¦§ L« l spec(S) 6= ∅ «$³§ !"¼O !"L!#¥ ¦§L%¦ÍÇ·Èl "¼³´!: e (k) #¥ ³E$ "L Êk± ¦{%:¨µAH!T± ¥ " ¥y" M ∈ M(E) » k W¨µ M a = a ∗ M (e0 ), ∀a ∈ F (Z). A!2! Ä"¼ M c ¦§±Æ ¥ ³¿"¼ M (e0 ) µbAH! ¥y" ± l« ³§$D%:«¾½± #¥ ¬®¥ ¦¿"¼³ ¦§³´«ª±Ä" $¥ L¥ ! Ð ³´y Cb±Ä¥ !#"O¶+ ¬Å ¦§¦´ M f !y³§ ± B¦§±¶+ ¬®¥ ¦´ Nv"¼¥y" f(z) = M X c(n)z n , ∀z ∈ C. M » ! $¥y"^± ¥y ³E±Ä «$³´$B¥ !y $ ³´H¶+ ¬$¦´¦§H ! ± !" a∈E ã(z) = n∈Z X a(n)z n , ∀z ∈ C. Cb± ³§ " W ¨µ K !#" ¼±ÄΧ! I± ¥: ! L¶+ ¬$¦ n∈Z ga(z) = M f(z)ã(z), ∀z ∈ C. M ^ " ! C = {z ∈ C | |z| = r} » ¥ r > 0 µL¥ L $¬!"¼ ! L± !y^¦´&?K½ ± ³´" Ŧ´ "¼½ ° ¬År ¥ ³´²Ä± !"ªµ 5¨ µ < E 0 !5 /,: , / /,: ,"! <5 ! ! $# Z # B/ ! ! ! ) ) ? ! ! ! ! # / - ,# ! S S −1 ! # ( ! ! $# E % ( ( % n o ! / ! 1 spec(S) = ρ(S ) ≤ |z| ≤ ρ(S) % 1) < M ∈ M(E) * ,# r > 0 5 = C ⊂ spec(S) M f ∈ L∞ (Cr ) / 2) % ( r . f(z)| ≤ kM k . % % ! ,# Cr % |M ◦ ρ(S) > 1 M f ! ( # "! ,# spec(S). 3) . ρ(S ) N^³§! ³_! ¥ SÊ ! $ ± ¦§³´ !y^¦´ $ ¥y¦´"¾±Ë"S ®Yl½ ³´$¦§y µ^¥ S± ¦§¦´! L± ¥ ³_ $¬!"¼ $ ±Ä! ¦§&?K½± ³¿"¼ #¥ ³ E "¥ ! ±Ä«$ sK± !± «¾½Z² $ ³ ± !"¦´$H« ! ³¿"¼³§ ! R ¨ û RH S·" RS ^¦§®JB½y$ °$¬1¨ µ $ " ² ±Ä¦Á± ¦´ µFH± ! D« «ª± ! ¥ ²$ !y #¥ ³ S " | !y® " S −1 !j " ± S ! »|!2± ρ(S −1) = +∞ µFS®¬ $¬& ³ S −1 "D| !y& " S !Ç$W" ± g ! » !ű ρ(S) = +∞ µ#^ ¥ g $¬Å± ¥y$ ! ¥yL ³¦§ @ ¼±Ä" $¥ S " S −1 ! K !" ± _ ! Q ¥ E »Ä¦´KJB½ ° ¬ÅS¨µ L! " ±Ä²Ä± ¦Á± y¦§g$!Ê ! $ ± ¦ÍµÄC{$Q¬ " ½ ly$ ¬ ¦´¹¸$ ± ! ^¦§ «ª± Hy$H¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ B ² !"I± ¥ ³Q%Ʊ «$³´$S ± ! S¦´?K½± ³´" %&" ¥y"O $ ±Ä"¼ ¥ Jbl ¦§³¿"¼Ì^¥ !) ¸¬®|¦§Äµ FH±Ä! B¦§D?K½ ± ³´" H©y»y! ¥yB«$! ³§ !yB¦Á± ³´"¼¥ ±Ä"¼³´!TÊ $! ¼± ¦´& G $ "S¥ !ZÊ ¥ ®Cb?KN&µjAH!~! Ä"¼ Gb ¦´ Ê ¥ y¥± ¦b G µ|q_¥ »y ³¿" S ¦ $ ±Ä"¼ ¥ B ! ³E ¥ L1 (G) ± x∈G −1 −1 x qb¥ b »!:! χ∈G loc " Sx f (y) = f (y − x), p.p. Γχ ¦ÍÇ $ ±Ä"¼ ¥ L1loc (G) 3 f −→ χf. L¥ ±Ä¦§¦§ ! E«$! ³§ _ $ ± « $ }s± ! ± «¾½ R ¨ ! ± ⊂ E ⊂ L1loc (G) ! C c(G) µ E RHq_ ¥ "¼¥y" µ K⊂G RS Bq_ ¥ B"¼¥y" E ¼±Ä" ³§ ¶ ±Ä³§¼±Ä!#"_¦§ _« ! ³¿"¼³´! E ¥ ³¿²Ä± !"¼$} »_¦§$& $¥lȳ§! « ¦§¥ ³§!y® "¼± !"«$!" ³§!#¥ $ »Q " » " x ∈ G Sx (E) ⊂ E b » Γχ (E) ⊂ E χ∈G " supx∈K kSx k < +∞ supχ∈Gb kΓχ k < +∞. Cc (G) $ " » ¥ "¼¥y"D« ¬ ± « " C{$^½¸ " ½ °$ $H« ³¿Âzy$ ¥ S !#"^ " & ¥ ÊÊ $ ± O¦´$L«$! y³´"¼³´! $S ± ! S ¦´D«ª± Z »@¬Å± ³§Ð! $¬Å± ! T³§« ³ #¥ 1"¼¥l"¼$ ¦§$Ð"¼ ± ! ¦Á±Ä" ³§! Ð ³§ !#" ! $ ¥ E Kµ A! ± y ¦§¦§¬&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y} ¥ E " ¥y"O $ ±Ä"¼ ¥ B| ! M : E −→ E "¼$¦ #¥ Sx M = M Sx , ∀x ∈ G. Cg± ¦´Ê°$ ¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ D ¥ E $¼±! Ä"¼$ M(E) µYl³´" Ge ¦§ÅÊ ¥ $¬W ½ ³§ ¬Å «$ !#" ³§!#¥ O G ± ! C∗ ·"B ³´" Gf+ ¦§HÊ ¥ H L¬ ½ ³§ ¬Å «$ !#" ³§!#¥ B G ± ! R+ A!:¬®¥y! ³´" Ge ¦Á±&" ¦§Ê ³§L D¦Á±&« !²$ Ê$!y«$&¥y! ³´¶+ ¬Å^ ¥ B" ¥y"B« ¬ ± « "$µ A!)²Ä±Ð $«¾½y$ «¾½y$S¥y!T ¥ WÂ$! $¬®y¦§ Gf Ge "¼ ¦ #¥ ¥yO" ¥y" M ∈ M(E) " ¥ "¼¥y" f ∈ C (G) ¦§±Ð¶+ ! « " ³§! (M f )θ−1 ±Ey ± " ³§ ! ! % L2(G) » ¥ O"¼¥y" θ ∈ Gf µj?K$« ³ c E ¬ " " ¼±&yH ! ³§} H¬Å± ! ³§° ^!±Ä"¼¥y $¦´¦§L¦Á±"¼ ± ! ¶+ ¬$L ]¥ ³§$}Ê$!y$¼±Ä¦§³§ $Sy ¥ Gf µj>Q!~" $!± !"S«$¬ "¼ $S± Ê ¥ ¬Å !"¼Sy b " Í»¥ !Z«$± ! ³´±Ä"S!±Ä"¼¥y $¦ Mf $ "L¦ ! ¬®E ¦§ Z n o f e GE = θ ∈ G | f (x)θ −1 (x)dx ≤ kMf k, ∀f ∈ Cc (G) , Mf ∈ M(E) "L¦ÍÇ $ ±Ä"¼ ¥ B G D«$!² ¦§¥y" ³§! E 3 g −→ f ∗ g. ^ " ! A(E) ¦Í± ¦§Ê °$ ƶ+$ ¬$1$!yÊ$! y $ ± Ц§ Ð ¼±Ä" $¥ M »@±ª² « f ∈ C (G) " B(E) ¦Í± ¦§Ê° ®¶+ ¬$$! Ê $! $ ± D¦§ D" ¼± !y ¦Á±Ë"¼³§ ! $µEFH±Ä! ¦´fÅ?K½± ³¿"¼ Щ »{!yc ¥ ²$ !y #¥ )¦ÍÇ$! $¬®y¦§ Gf "³´ ¬ ½ %T¦ÍÇ$! $¬® ¦´ ! !²l³´ y$®«$± ¼± «·"¼° $y ¦Í± ¦§Ê° A(E) µA! E Uz¦{$ "O¶ ± « ³§¦´H D ¬ ± #¥ $ #¥ Yl³´" U ¥ !1¥y² "^ Cp $ "L ³¿"¼D± ! ± ¦´¸#"¼³ #¥ ¥ f+ = {|θ|, θ ∈ G fE }. G E f+ G. fE = G b G E µX^! ¶+! «·"¼³´! U e Π : U 3 λ −→ Π(λ) ∈ G » ³ ¥ B" ¥y" x∈G »¦§±&¶+! «·"¼³§ ! U 3 λ −→ Π(λ)(x) ∈ C $ "@±Ä!± ¦´¸#" ³#¥ K ¥ U µAH!! "¼ d ¦§±S¬Å ¥ B ³§ «$ ° "¼B ¥ Gf+ µ^¥yy" $! [email protected]¦´K $ ¥y¦´"¾±Ë" E ¥ ³¿² ±Ä!#"$µ ¨ µ < E !5 /,: , / /,: # B/ ! ! ! ) ) ( . ) % θ ∈ Gf * ,# 5 f ∈ C (G) (M f )θ−1 ∈ L2(G). * ! ! i) M ∈ M(E) c . $# 5 δ ∈ Gf+ ,# # ! = E % 5 χ ∈ Gb . E \ g M f (δχ) = (M f )δ −1 (χ). ( 2134!- 6 8796: fE , d ⊗ m) hM ∈ L ∞ ( G - > = (<( ^ (M f ) = hM f˜, ∀f ∈ Cc (G) ! 6+: ! / 56> / /!>, khM k∞ ≤ CkM k, C M% < , 6 9 7 6 : / / p < f ii) U # C % Π : U −→ GE ( 6 796:; H ∈ L∞(G, 5 = 5 ∞ b H (U )) (<( ,# λ ∈ U $# M,Π b. χ∈G g ˜ M f Π(λ)χ = HM,Π (χ)(λ)f Π(λ)χ , ∀f ∈ Cc (G). HF ±Ä! ~¦´º?K½± ¿³ "¼ 2© »B!y¥ :²$ !y #¥ $¬Å± #¥ !y^±Ä¥ ³ # ¥ Yl³ G f+ G E = 2 134!- # % ! = ( 5 $ "T¦§ ÊÄÂz« !²·Èlº "1«$ ¬ ±Ä« "ªµOL¥ fE ⊂ {θ ∈ G, e |θ −1 (x)| ≤ ρ(Sx ), ∀x ∈ G}. G $ "L¥ !1Ê ¥ ³§ «$ ·"L¥:¥ !TÊ ¥ D«$¬ ± «·"O! ¥ B² $ ! #¥ fE = {θ ∈ G, e |θ −1 (x)| ≤ ρ(Sx ), ∀x ∈ G}. G L¥ «$ ! $«·"¼¥ ! #¥ )¦Á±~«$± ¼±Ä« "¼ ³´¼±Ä" ³§! $« $ !"¼:$W"² ±Ä¦Á± ¦´ ¥ &" ¥y"Ê ¥ Cb?KN&µFH±Ä! }¦´$Q"¼ ³§Q«¾½ ± ³´" }! ¥ @± ¹Èl³´¬Å[email protected]¥y!Ŭ®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ ¥ g¦Á±S"¼ ¦´Ê³´ ¶+W"¼} $_ ¼±Ä" $¥ ±Ä_¥ !yK ¥y³´"¼@¥ ± _¥y! K ¥ ³´" }Ê ! $ ± ¦§³´ }Ç $ ±Ä"¼ ¥ _ K« !²Ä ¦´¥y"¼³´! ±ÄQ $Q¶+! «·"¼³´! « !"¼³§!#¥ @%D ¥ "g« ¬ ± « "$µ#P~± ³´¦§ Q¬Å·"¼½ Q$¬ ¦§¹¸ $ ! ¥ ³¿"¼ ± ! L¦§«$± SÇ¥ !TÊ ¥ ICb?KN Ê $! ¼± ¦E !#"L"¼ °$L ³$ $!"¼ S I«$ ¦§¦´$O¥y"¼³´¦§³´ $ ±Ä! D¦§?K½ ± ³´" lµj>Q!ºj·"D± ! ¦´Ð?K½± ³´"¼ Ðl»! ¥ ! ¥ ² ! j¥ !y¬Å·"¼½ « ! " ¥ «·"¼³¿² »±Ä¦§ #¥ ± ! B¦´D?K½± ³´"¼ H© »l! ¥ B¥y" ³§¦´³§ ! K¦´$«$± ¼± «·"¼° $O O± ¦§Ê °$ $ ·" B(E) "D ³§·"¼$D I± ¦§Ê° D s± !± «¾½º "I! ¥ ¶ ± ³§ ! ¦Á±Ä Ê ¬Å !#" A(E) ¥ ± Ê S¦Í±ËÈl³§¬^y¥1«¾½y³¿È»y« #¥ ³ $ ¬Å·"B$! ±Ä "¼³´«$¥ ¦´³§ } ^"¼¼±Ä³´"¼ K¦´$Ê ¥ $BCb?KN ! ! σ «$¬ ± «·"¼ µ ?RLNHqQUzJOM^> egfWh ®( e ykQilh ^h d\hSh e L2δ (R+ ) c HF ±Ä! _« }«¾½ ± ³´" g!I·È @¦§ b $ ¥ ¦´"¼±Ä"¼E ¬Å!" $E±Ä! # " µËA! ± $¦´¦§ ¼± ³§y1 ¥ R "¼¥y" º± ¦´³§«ª±Ë"¼³§ !<¬Å ¥ ± ¦§Ä»B " ³§«·"¼ ¬Å !#" ³¿"¼³¿² ¥y R ² $ ³ ± !"T¦Á± « ! ³¿"¼³§ !1 ¥ ³´²Ä± !" lµ´¨ ω(x + t) ω̃(t) := sup ess < +∞, pour tout t ∈ R. Yl ³´" x∈R ω(x) ¦ÍÇ$ ± «$² $« " ³´$¦{ L¶+ ! « " ³§! L2 (R) ω AH!1¬®¥ ! ³¿" Z f ¬$ ¥ ¼± y¦§$O ¥ R "¼ ¦§¦´$ #¥ |f (x)|2 ω(x)2 dx < +∞. yI¦§±® " ¥ «·"¼¥ &RL³§¦§|$W"¼³´$! ! H± l« ³§$I± ¥ ly¥ ³´"O «ª± ¦§± ³§ L2ω (R) R < f, g >:=< f, g >ω = Z f (x)g(x)ω(x)2 dx, Ð " #¥ Ц ± « C ∞(R) y$¶+! «·"¼³§ ! RЫ$¦§± C ∞ ·"I%) ¥ W" « ¬ ± « "D " ! I± !y L2 (R) µq_¥y c a ∈ R, !1 !y³´"O ¥y L2 (R) ¦ÍÇ $¼±Ë"¼$¥y "¼ ± ! ¦§±Ä"¼³´! S a,ω ±ÄL¦§±&¶+ ¬&¥ ¦§ω® (Sa,ω f )(x) = f (x − a) µ µ|AH!~ω± lµ ω(x + a) kS k = sup ess a,ω x∈R ω(x) ·"OÊ « D%®¦Íǽ¸ " ½ °$ Ílµ¿¨ ·»l¦ÍÇ $¼±Ë"¼$¥y S $W"O ! ĵAH!:! "¼ M ¦ ! ¬® ¦§S $ ¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ { ¥ L2 (R) »Ï« Ç$W" Âz%ËÂz ³´ g¦ÍÇ$!ya,ω $¬& ¦§@ $b $ ±Ä"¼ ¥ M ∈ω B(L2 (R)) »#¥ ³ « ¬Å¬&¥y"¼ !#" ±ª² « S ω ¥ "¼¥l" a ∈ R » B(X) $ ³§Ê!±Ä!#"I¦ ! ¬® ¦§Ð ω ¼±Ä" $¥ | !y$S ¥ L¥ !~ ± a,ω « & s± ! ± «¾½ X µCQ± ¦§Ê °$ M $^¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ K ¥ L2 (R) $ " ³´$!1«$!y!¥y Í» H ³ M ∈ M » ³§¦ÃÈl³§ " h ∈ L∞(R) "¼$¦ #¥ lµ d M f = hfˆ, ∀f ∈ L2 (R), y$ ³´Ê!± !"¦§±" ¼± ! W¶+ ¬$ ]¥y ³§ ^j¥ !y&¶+! «·"¼³§ ! f ∈ L2 (R) µCE±Å¶+! «·"¼³§ ! h $ " fˆ ± y ¦§$S¦§L ¸¬®|¦§^ M µyML «$³ #¥ $¬$!"ª»l¦Á±¶+ ¬&¥ ¦§ Ílµ @± l« ³§H%I"¼¥y" ^¶+!y« "¼³´! ¥ !<¬&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y ¥ L2 (R) µKFS$) $ ¥ ¦´"¼±Ä"¼)±Ä!± ¦§ Ê¥ $Å ¥ ¦ÍÇ·Èl³§W"¼$!y«$ ¥xW¸¬® ¦§Ç¥ !º¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ S !" « ! !#¥ I± ! D¦´«ª± D ³´ «$ "$µEqg¦´¥ «$³´ $¬$!"$» Yl½ ³§ ¦§ B¬!"¼ ± ! #¥ " ¥y"L¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ @ ¥ h ∈ L∞ (R) n o X lσ2 (Z) := v = (vn )n∈Z | |vn |2 σ 2 (n) < +∞ $ "^± «$³´D%¥ !yD¶+ ! « " ³§!:½ ¦´¬ ½yS| ! D ¥ n∈Z 1 < |z| < ρ(S)}, ρ(S −1 ) "} ¥ l2 (Z) µlC{^«$± K 1 = ρ(S) !yS ¬® ¦´H±ª²³´}·"¼L"¼¼±Ä³´"¼ #¥ S>QW"¼$σ ¦§® ± ! µFHρ(S ± ! −1H)¦´ «$± H« !"¼³§!#¥Zy$H·"¼¥ ± ¶+!y ³§ {z ∈ C | $ ³§Ê!±Ä!#"}¦§L ½y³´¶ "¼ ° S $«$ ¬Å¬$!" ± ¥Æ ¸¬®|¦´^Ç¥ !Ƭ&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y ¥ L2 (R) !"K "¼L¶ ± ³´" $ ¥ ω ¥ ! ³§ ± W"¼³´«$¥ ¦´³§$ yÈl «·¶\µ S·" ·µK?K $!y± !"ª»}¦ÍÇ·Èωl³§W"¼$!y«$~y¥G ¸¬®|¦§1Ç¥ !G $ ±Ä"¼ ¥ y Mω $¬& ¦§ ! ±Ä±ª²³´^·"¼®·"¼¥ ³´$ ¥yS¥ ! ³´ #¥ ¦§«$ ! #¥ µA!~ ·"¾± y¦§³§ ¥ !6 $ ¥ ¦´"¼±Ä"б ! ± ¦§Ê ¥ Æ% lµ ¥ &"¼ ¥y"¬®¥ ¦¿"¼³ ¦§³´«ª±Ä" $¥ D ¥y L2 (R) » ¥ ®" ¥ ®¦´$ ω ²$ ³ ±Ä!#" ¥ ¦§ ¬Å !"&¦ ½¸ Ä"¼½ ° lµ¿¨ ·µE>"¾±Ä!#" !y! $ ¦§$I ¥ ¦¿"¾ω±Ä" y @·" »E³§¦ $ "L!±Ë"¼¥ $¦{ D«$ ! ³´ $ $O¦Á±®±Ä! S Aω = {z ∈ C | ln Rω− ≤ Im z ≤ ln Rω+ }, 1 1 Rω+ = lim ω̃(x) x = lim k(S1,ω )n k n = ρ(S1,ω ) " x→+∞ n→+∞ 1 Yl³´" M ∈ Mω µqb¥ x→+∞ a∈R " 1 1 Rω− = lim ω̃(−x)− x = lim k(S−1,ω )n k− n = » !1 ! ³´"O¦§±¶+ ! « " ³§! n→+∞ f ∈ L2ω (R) ρ(S−1,ω ) (f )a : t −→ eat f (t). A!:²Ä±¬!"¼ $ #¥ ¥ a ∈ I := [ln R−, ln R+] " ± ± W"¼³´$!"L% L2 (R) " #¥³´¦·Èl³´ω "¼ ν ∈ Lω∞(R) " ω$¦ #¥ a " f ∈ Cc∞ (R) . »¦Á±®¶+! «·"¼³§ ! d \ (M f )a (x) = νa (x)(f )a (x) p.p. kνa k∞ ≤ Cω kT k, Cω = exp ?K$«$³{ "^¥ ! DÊ ! $ ± ¦§³´¼±Ä" ³§!Æy³§ « " % Z 1 2 2 ln ω̃(y)dy . ¥T $ ¥ ¦´"¼±Ä" Í lµ K« ! «$ !±Ä!#" Mω Rω− ≤ 1 ≤ Rω+ . ! − 12 M ³ (M f )a FS ¦´¥ $» #¥±Ä! Rω− < Rω+ »¦§±&¶+! «·"¼³§ ! ν : z = a + ix −→ νa (x) "L½ ¦´¬Å ½ L ¥y A◦ µ L¥ B ! ! O³´«$³|«$$W"¾± ³´! $B¬ " ½ $B " Í»l¬ ± ³´ ¦´Å«$± &« !"¼³§!#¥ $!" ω) Å $ ³§$¥y $& ³ «$¥y¦´"¼ ¥ ¦´$¬$!"¾±Ä³§ ± & ± " ± ¥x«ª± ³´ « "$µ>Q!< "$»K ³ ω $W")¥ ! ³§y #¥ $¦´«$! ¥y »³´¦L!j " ± ¥8"¼ ¥y") ²³´ $!" #¥ » ¥y M ∈ M , a ∈ I ·" f ∈ C ∞(R) µ^ ¥ H"¼ ± ³´" ! S ! ¬Å³´$ \ (M f )a ∈ S(R)0 ¦´1«$± Ð $Ð $ ±Ä"¼ ¥ Ð 1«$ !#² ω¦§¥l"¼³§ ! ±ª² ω$«1¥ ! )¶+! c«·"¼³§ ! y C ∞(R) µQJ_ ¥y" ±Ä » ±Ä! ^¦§± ± " ³§ D«$I«¾½± ³¿"¼ Ä»! ¥ O! ¥y^ ± ¬Å ! ! L± ¥1«ª±Ä^Ç¥ c ! ³´ O«$!" ³§!#¥»#¥ ³ ² $ ³ D $ ³§·"¼$B ¥ ¦´$¬$!"¾±Ä³§ $µyFH± ! O¦Á± ± " ³§l»y!: ¬Å!" #¥ ¥ K" ¥y" »_³´¦@ÃÈl³§ " ¥ !yÆ ¥ ³¿"¼ (φ ) ⊂ C ∞(R) "¼ ¦§¦§ #¥ M "¦§±1¦´³§¬³´"¼ ¥ ¦Á± M ∈ Mω c " ¦§Ê³´I¶+ "¼& $D $ ±Ä"¼ ¥ ¦n §n∈N ± ¥y³´"¼ (M »j M : f −→ f ∗ φ $W"D¦§ φ )n∈N ¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ } ¥ L2 (R) ± «$³§I% φ µFS ¦§¥y$» ! ¥ ^y³§ φ ! BÇ¥ !~« !"¼ ¦§n ¥ O¦Á± ! ¬y M µ{qQ¥ ³§ ω»{± ! D¦§±: $«·"¼³§ !ºn ©y»{Ê « Ð% ¦§¥ ³§$¥y ¦§ ¬Å¬$H" $«¾½ !y³ #¥ $ »E! ¥ ¬!"¼ ! #¥ φ n n n b |φ(α)| ≤ kTφ k, ∀φ ∈ Cc∞ (R), ∀α ∈ Aω . L¥ O $y¥ ³§ ! B « $O ¥ ¦¿"¾±Ä" $» %¦§±& « "¼³´!T¯l»y¦´S"¼½y$ °$¬«$! « $ !± !"L¦§$K¬®¥ ¦´" ³ ¦´³¿Â «$±Ä"¼ ¥ L ¥ L2 (R) $!y! «$I«$³ Âz ¥ $µ ω xlµ´¨< ω ! ! ,# R ! < M ∈ M . ω 0 / 2 ∞ 1 ) (M f )a ∈ L213(R) . 6 $79# f6:∈ C c (R) a ∈ Iω- % = 4 ! 2) * $# a ∈ Iω . ( νa ∈ L∞ (R) (<( d \ (M f )a (x) = νa (x)(f )a (x), ∀f ∈ Cc∞ (R), p.p. ? ( ! . 0/ kνa k∞ ≤ Cω kM k, ∀a ∈ Iω % R− < R+ 2134!- 68796: ν ∈ H∞(A◦ ) 5 = 3) ω (<( ω ω. ( 4) d M f = ν fˆ, ∀f ∈ Cc∞ (R), \ d M f (x + ia) = (M f )a (x) ,# a ∈ Iω . f ∈ Cc∞ (R) % spec(Sω ) = {z ∈ C | Rω− ≤ |z| ≤ Rω+ } % ◦ FH±Ä! S « &«¾½± ³´"¼ !y¥ H±Ä¦§¦§ ! B " ¥ ³§ S±Ä¥ ³_¥ ! §¦ ° ¬ÅI± $Ì& ³§¬³§¦§± ³§ µyAH!~ ³´¼± #¥ δ $W"Å¥y! ³§ ¥ R+ = [0, ∞[ ³ δ "Å¥ ! )¶+! «·¼" ³´!6¬Å ¥ ± ¦§Ä»Q " ³§«·"¼ ¬Å !#" ³¿"¼³¿² ¥y R+ ²$ ³ ±Ä!#"L¦´$ ³´ " $O ¥ ³¿²Ä± !"¼$D lµ© 0 < sup essy≥0 δ(x + y) < +∞, pour x ≥ 0, δ(y) Ílµ ¯ 0 < sup essy≥0 δ(y) , pour < +∞ x < 0. δ(y − x) ^ " I $«·"¼³¿¶Q$W"S $¬!"¼ $L¥ !T" ½ $ °$¬&y $ !"¾±Ä" ³§! ¥ L¦§ ^ ¼±Ä" $¥ É-³§$!y$WÂzR^ ¶ ¥ O¦ÍÇ$ ± «$ L2δ (R+ ) n + := f mesurable sur R | Z +∞ o |f (x)|2 δ(x)2 dx < +∞ . Uz¦b "S«$¦Á±Ä³§ # ¥ &¦ ± « L2 (R+) $W"H¥ !Z ±Ä«$®0 &R^³´¦§|$ "L ³b !Z¦§I¬®¥ ! ³¿"L &¦§±Ð¶+ ¬Å $#¥ ³§¦´³§! $± ³§ H ! ³§ ± B 䦧±®¶+ ¬®¥y¦§ < f, g >:=< f, g >δ = A!: ! ¿³ "ª» ¥ » a≥0 Z R+ f (x)g(x)δ(x)2 dx, ∀f ∈ L2δ (R+ ), ∀g ∈ L2δ (R+ ). Ua,δ : L2δ (R+ ) −→ L2δ (R+ ) ± B¦Á±&¶+ ¬®¥ ¦´ Ua,δ f (x) = f (x − a), p.p pour x ≥ a, Ua,δ f (x) = 0, pour 0 ≤ x < a. A!: ! ¿³ "ª» ¥ ± B¦Á±&¶+ ¬®¥ ¦´ q_ ! " a>0 » Va,δ : L2δ (R+ ) −→ L2δ (R+ ) Va,δ f (x) = f (x + a), p.p. pour x ≥ 0. + (a) = kU k, ∀a ≥ 0 δf a,δ + (a) = kV δf −a,δ k, ∀a < 0. V¥ ± ! ³§¦! ¸± ± }y^ ³§ #¥ L L«$!l¶+¥ ³´!»! ¥y} «$ ³´ !y U + $ µ µ ·" U $ µ V B± ¥:¦§³´$¥: U + $ µ V ·µAH!1 a!y³´" Va,δ 1 1 ± @¦Á±H¶+ ¬&¥ ¦§D P + : L1loc (R) −→ L1loc (R+ ) P + f (x) = f (x), µ µ x≥0 ·" P + f (x) = 0 » ¥y P − : L1loc (R) −→ L1loc (R− ) Va g± ¥Å¦´³§ ¥y x<0 Ua,δ µ#·"K!Å ! ³´" ± B¦Á±&¶+ ¬®¥ ´¦ P −f (x) = f (x), µ µ x ≤ 0 " P −f (x) = 0 » ¥ x > 0 µ < l µ¿¨ 0 # / 5 ,# T ∈ B(L2 (R+)) [email protected]/ # / 5 $# , 6 # ( δ > 7"! Va T Ua f = T f, ∀a ∈ R+ , ∀f ∈ L2δ (R+ ). AH![!y ¼" W ¦ÍÇ$ ± « ~ $ ±Ä"¼ ¥ 1 2É-³§$!y$WÂzR^ ¶ ¥ L2 (R+) "T![! Ä"¼ δ δ % ¥ " ± ! $ " ∞ + ¦ ±Ä«$Z ¶+! «·"¼³´! y ∞ + µ}CE~«ª± Cc (R ) Cc (R) R δ = 1 ³´$!:«$ ! !#¥ +« ¶\µ·µ >g!1 "$» ¥ "¼ ¥y" T ∈ W » ³´¦j·Èl³´ "¼D¥ !y ³§W"¼ ³´ ¥y" ³§! µ " $¦§¦´ 1 T #¥ lµ À T f = P + (µT ∗ f ), pour f ∈ Cc∞ (R+ ). FS ¦´¥ $» ³´¦·Èl³´ "¼D¥ !yD¶+¥y! « " ³§! h ∈ L∞(R) »± $¦´$¦§W¸¬® ¦§ T » "¼ ¦§¦§ ¥y lµ T f = P + F −1 (hfˆ), pour f ∈ L2 (R+ ). L¥ I± ¦§¦´! HÊ $! ¼± ¦´³§ H¦§ ¥ ¦¿"¾±Ä" lµÀ ^ " lµ ¥ T ∈ W » δ ² $ ³ ͵´¨ õ >Q!:"¼$! ± !"^« ¬ "¼D L ³§¬³§¦´³´"¼¥y [email protected]$!"¼ ¦´$B $ ±Ä"¼ ¥ O É-³´$! δWÂ\RL ¶_ "O¦´$B¬®¥ ¦¿"¼³  ¦´³§«$±Ä"¼ ¥ »³§¦ "I!±Ä" ¥ ¦Q «$ ! $«·"¼¥ $ ¥yÐ" ¥y"D $ ±Ä"¼ ¥ D É-³§ ! $\Â\R^ ¶}± ¬ " ¥ !yS !#"¼±Ä"¼³´!)± ! ± ¦§Ê ¥ L% ͵ õJ_ ¥y" $ ±Ä"¼ ¥ @ LÉ-³´$! WÂzR^ ¶ ¥ L2 (R+) " ·"¼ ¬³§! ±Ä P +M »{ M "I¥ !2¬®¥ ¦¿"¼³ ¦§³´«ª±Ä" $¥ L ¥ L2 (R) S "I± !y«$Ыª± lµ À Q·"Ílµ !#"@ }« ! #¥ $!y«$$K³§¬¬Å ³Á±Ä" $[email protected] $ ¥ ¦´"¼±Ä"¼g ¥ g¦§$g¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ $µ q_± H«$ !#" Ä»!~! & ± ³´" ± S ³b" ¥y"S ¼±Ä" $¥ ^ É-³´$! WÂzR^ ¶ ¥ L2 (R+) $W"¦Á± W " ³§«·"¼³´!% L2 (R+) j¥ !Ь®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ E ¥ L2 (R) » ω $ "g¥y! ³´ δ¥ R y³§$!&«¾½ ³§ ³Í» " $¦ #¥ ω| δ = δ µ^$± ! ¬³§!y¦§ L¬ " ½ $Oω ·²$¦´ $ ¥ O¦§ B¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ } ¥ ¦´$ ± « $LR % ³§ ¥y² !" ·"¼ D± ± " $ O±¹² $« #¥ ¦ #¥ $O¬ly³ «$±Ä"¼³´! }%®¦Á± ³´" ¥±Ä"¼³´! $¼±Ë"¼$¥y ® É-³§ ! $\Â\R^ ¶\µL¥ ®y "¾±Ä³§¦§¦´$ ! D¦§ & "¼± $® ¥5 ± ³§ ! ! ¬Å !" #¥ ³ ³ j° !#"$µ^¥yL y"¼ ! ! L¦§ $ ¥y¦´"¾±Ë"O«$³¿Â $ ¥y$µ qb !y + 1 1 + (n) n , r − = lim δ + (−n)− n , f rδ+ = lim δf δ Jδ := n→+∞ [ln rδ− , ln rδ+ ], n→+∞ Ωδ := {z ∈ C | Im z ∈ Jδ }, Z 2 + (u)du. 2 ln δf cδ = exp xlµ δ !"!1$# R+ T ∈ W # ! - a ∈ J 0/ (T f ) ∈ L2 (R+) ,# f ∈ C ∞δ %'(R&"+( ). 1)+* ,# . δ. a c 2 3 1 4 ! 5 6 8 9 7 6 : ; >= ∞ 2)+* ,# a ∈ Jδ ( νa ∈ L (R) (<( 3)>? @ ! ! ( . d (T f )a = P + F −1 (νa (f )a ), pour f ∈ Cc∞ (R+ ). rδ− < rδ+ . 2 134!5 6A796: ( ◦ ν ∈ H ∞ (Ω δ ) ν(x + ia) = νa (x), p.p sur R+ kνk∞ ≤ cδ kT k. 5C - = B ,# 5 (<( ◦ a ∈ Jδ ` {Yl³¿" Uz¦{$ "L« ¦Á± ³´ #¥ "¼¥y" L2ω (R) ` S^¥ L± ¦´¦§!y ! !y$ #¥ $¦ ¥y$L·Èl$¬ ¦§ O ³´ B ¥y R µ ` a Ec Ec ω ² $ ³ Í lµ¿¨ û ` QYl ³´" a∈R ω(x) = 1 x<0 " ω(x) = ex » ³§! !jµ NS¦´ B!1± kSa,ω k = ea , ∀a ∈ R. Rω+ = e ·" Rω− = 1 » ¥yS"¼¥l" µUz«$³ » !1± x #ω(x) ¥ #¥ = eR+ = R− = e µ x ∈ R ω ω µ|A!: $¬Å± ` Yl³´" » ¥ µ{A!2± Rω+ > Rω− µ kSa,ω k = ea < +∞ »¥ » ¥ ^" ¥y" x ∈ R µAH!2± kS k = +∞ » ¥ ^" ¥y" a,ω @ µ E C 2 ± + ¶ y ! « ¼ " ´ ³ ! ± Å Á ¦ ± $ « ! ´ ³ " § ³ ! ͵´¨ õ@?K $! ± !"ª»K ± ! ¦ÍÇ$ ± «$ a 6= 0 "¼¥yH¦§ H¬®¥ ¿¦ "¼³ ¦§³§«$±Ä"¼ ¥ B !""¼ ³´²³Á± ¥Èj»|³ÍµÇ µ|$ʱ ¥lÈZ% ¥ ! « ! "¼± !"¼®¶+³´S¦ ³´ $!l L2ω (R) "¼³´" «·¶\µ ·µ ω(x) = ex ! 1²$ ³ ω 2 c a c LFH± ! @«$·" " ^ $« " ³§!j»!Ŭ!"¼ ¥y^"¼ ¥y" ³´ ω "}$!ж ± ³´" #¥ ³´²Ä± ¦´$!"K%&¥ ! ³§y ω # ¥ ³²$ ³ S H|! ! ³´ " $KyH Ê¥ ¦Á±Ä ³´" µ#FH±Ä! «$S ¥y"ª» ! ¥ Å $!y! ³´«$³O Æ0± Ê¥ ¬$!"¼ ² $¦´ $ ±Ä sK ¥ ¦´³§! ʺ "ÆP~± ¦´¦§³§±¹²³´! ± ! q_ ! ͵ γ(x) = ln(ω(x)) p.p. "L ³¿" b(t) := sup essx∈R |γ(x + t) − γ(x)|, ∀t ∈ R. F& ± ° Ílµ¿¨ ûb ! ± b(t) < +∞ » ¥ D"¼¥y" t ∈ R µ_CE±1¶+!y« "¼³´! b " ± ³§ Å·"® ¥ W ± ³¿"¼³´² µF^ ¦§¥ »$¦§¦´H$W"L¬Å ¥ ± §¦ ĵ?K¬¬Å¦ ¥y! ³§!)y$L$! $¬® ¦´$ $ " R »¦Á±Ð¬Å ¥ ¥y En = {t ∈ R | b(t) ≤ n} En $ "^ " ³´« "¼ ¬Å !" ³¿"¼³¿² ¥ n E = {t ∈ R | b(t) ≤ M } ± Ì Ê ± ! µYl ³´" M >0 "¼ ¦ " )¬$ ¥y )! !6!#¥ ¦´¦§ µVH¥ ³´" "¼Æ% $ ¥ ³´ E »! $¥y" ¥ ¥y1¦§±~¬$ ¥y ) $W" !y³§ µbAH! g(x) = χ ∗ χ (x) » ¥ " ¥y" x ∈ R »{ χ $ " ¦Á±Æ¶+!y« "¼³´! E E «$± ¼± «·"¼ ³§W"¼³ ¥yIy E µC{ ¥ EW"^y −E $ W L " $ « ! ¼ " # ! ~ ¥ ± y ! g E1 = {t ∈ R | t = t1 − t2 , t1 ∈ E, t2 ∈ E}. EC ±1¶+ ! « " ³§! g $W"®«$ !#" ³§!#¥ Å«ª± I« Ç$W"¦Á±:«$!² ¦§ Ç¥ ! ¶+! «·"¼³§ ! L1 (R) ±ª² «Æ¥ ! ¶+!y« "¼³´!G L∞ (R) µ?K ¬Å¬ g(0) ") ʱ ¦H%x¦Á± ¬$ ¥ Zy E » #¥ ³S ": " ³´« "¼ ¬Å !" ³¿"¼³¿² » g "K "¼ ³§«·"¼$¬$!" ³´" ³´² L ¥ K¥y!Ų ³§ ³´!± ÊL 0 ·" E1 «$ !#" ³§ !#"}¥ !Ƴ´!"¼$W²Ä± ¦§¦´ ¥l²$W"S!y!1²³´ µ?K ¬Å¬ b $W"L ¥ \Â\± y ³´" ³´²Ä» !T± ¥y"¼¥y"L« ¬ ± « " K R lµ sup b(x) < +∞. qb !y Yl ³´" AH!T± x∈K AH!yÈl M0 = supx∈[−1,1] b(x). Ma " $¦ #¥ µAH!:²Ä±Ð¬!"¼ $ #¥ sup essx∈[−a,a] |γ(x)| < +∞. µ|Yl³¿" ¥ "¼ ¥y" |b(t)| ≤ Ma t ∈ [−2a, 2a] n o Ja = (x, t) ∈ [−a, a] × [−2a, 2a] | |γ(x + t) − γ(x)| > Ma . "L± °$O¦§H" ½ $ °$¬D I]¥ ³§!y³j!1y" ³§$!" χ dx dt = 0 J a −a R 2a R a −2a a>0 Z a Z 2a χJa dt dx = 0. ?K ¦Á±³§¬ §¦ ³ #¥ #¥ ¥ #¥ D"¼¥y" x ∈ [−a, a] » !T± |γ(x + t) − γ(x)| ≤ M , ¥ a $¥y "¼ ¥y" t ∈ [−2a, 2a]. AH! lÈy x ∈ [−a, a] ¥yB¦§#¥ ¦{!T± −a −2a 0 ¥ $ #¥ D"¼ ¥y" |γ(x0 + t) − γ(x0 )| ≤ Ma t ∈ [−2a, 2a] ·"L!T y"¼³´$!" |γ(x0 + t)| ≤ Ma + |γ(x0 )|, ¥ $#¥ Æ"¼¥l" t ∈ [−2a, 2a]. NS³´! ³ »E!6± |γ(z)| ≤ |γ(x0)| + Ma , ¥y $ #¥ " ¥y" z ∈ [−a, a]. ?K ¦Á±Ð³§¬ ¦§³ #¥ #¥ γ $W"S¦´l«$± §¦ ¬Å !#"³§!"¼ ʼ± y¦§ µ|AH! ¥y"^ !y³§O¥ ! ³§y ω0 » ± O¦Á±&¶+ ¬®¥ ¦´® lµ ω0 (x) = exp Z 1 2 − 12 γ(x + u) du , ∀x ∈ R. EC ³§ ω "Q« !"¼³´!¥jµAH! >g!1 "$» 0 ¥ "¼¥l" x ∈ R !~± γ0 (x) = ln(ω0 (x)) γ0 (x) = " γ0 0 (x) = γ FS §¦ ¥ »!1± «ª± Z γ0 (x + t) − γ0 (x) = γ0 Z 1 +x 2 µÄCE±^¶+! «·"¼³´! γ0 $W"Q¦§³ "¼«¾½ ³¿"¼Ì ³§$!y! µ γ(t) dt − 21 +x 1 +x −γ x− p.p. 2 2 1 x+t γ0 0 (u) du, pour tout x ∈ R, pour tout t ∈ R, $ "^± y ¦´¥ ¬Å !"O«$!"¼³´!#¥ µ|?K¬¬ x sup essx∈R |γ0 0 (x)| = b(1) ≤ M0 , !:y"¼³´$!" Ílµ¿¨¹ FS ¦§¥ »!1± |γ0 (x + t) − γ0 (x)| ≤ M0 |t|, ∀x ∈ R, ∀t ∈ R. ω˜0 (y) = sup exp (γ0 (x + y) − γ0 (x)) ≤ eM0 |y| , ∀y ∈ R, x∈R ± ° l µ¿¨¹ ·µyNS³§!y ³Í» ¥y"¼¥y" Ílµ¿¨¨ L" ¦§ ³´ Ílµ¿¨Ï K «$¬ ± «·"O R !T±) sup ω˜0 (y) < +∞ ² $ ³ D¦Á± ³´ " y∈K ω0 FS ¦§¥ »¦§ ³§y lim sup ω˜0 (y) = 1. n→+∞ ω0 "L#¥ ³¿² ±Ä¦§$!"^± ¥ ³§ ω µ >g!1 "$»!T± ω0 (x) = exp ω(x) FSD¬ ¬ÅÄ» !1± 1 |y|≤ n ≤ exp Z Z 1 2 − 12 1 2 − 12 γ(x + u) − γ(x)du M0 du = eM0 p.p. ω(x) ≤ eM0 p.p. ω0 (x) qb !y 0 (x) βω = sup essx∈R ωω(x) µAH!~± βω = exp Z 1 2 − 12 sup essx∈R (γ(x + u) − γ(x)) du = exp AH!1 $¬Å± ¥y #¥ Z 1 2 ln ω̃(u) du. − 12 ω(x) = exp sup essx∈R ω0 (x) ·"^ !~± Z 1 2 − 12 ln ω̃(−u)du = βω βω−1 ω(x) ≤ ω0 (x) ≤ βω ω(x) p.p. ?K¬¬L¦´ ³´ ω " #¥ ³´²Ä± ¦´$!"}% ¥ ! ³´ g«$!" ³§!#¥» ω ² $ ³ L¦Á± ³´ "¼L ¥y³´²Ä± !"¼I lµ´¨ª 0 < inf essy∈K ω(y) ≤ sup essy∈K ω(y) < +∞, pour tout K compact de R. FS ¦´¥ $» j ± °$ ͵´¨¨ ·» !:l"¼³§ !"D lµ´¨$© sup ω̃(y) < +∞, pour tout K compact de R. QC Ç#¥ ´³ ²Ä± ¦´$! « Q$!"¼ !1! " y∈K ω kT kBω := ·" ω0 ³´¬ §¦ ³#¥ #¥ sup f ∈L2ω (R), L2ω (R) = L2ω0 (R) T ∈ Bω = B(L2ω (R)) kT f kω kT f kω0 et kT kBω0 := sup . f 6=0 kf kω f ∈L2ω (R), f 6=0 kf kω0 Yl³j± ¥ « ¥ ! H« !y¶+¥ ³§!Æ!j " ³§ ¦´ »¦Á±I! ¬Sy T ¼±®! " $ kT k. AH!Æ $¬Å± #¥ #¥ βω −2 kT kBω ≤ kT kBω0 ≤ βω 2 kT kBω . ?K ¦Á±Ð³§¬ §¦ ³ ¥y ·"^¦´$O± !y $ µ$q_¥y Aω " Rω+ = Rω+0 , Rω− = Rω−0 A ω0 ±Ä «$³´$ S± ¥ È ³§ ω " ω0 !"L$ʱĦ§$ µ bq ¥ g $¬!"¼ $Q¦ ÃÈy³´ " $! « B ¥Ð ¸¬®|¦§BÇ¥ !Ð $ ±Ä"¼ ¥ Q $W"¼D DÊ$!y$¼±Ä¦§³´" ¥ y $ #¥ ω "L«$!" ³§!#¥µ Mω ! ¥y"g ! «B ± ! » `_`bc a ` a Ec bc ^ ¥ D± ¦§¦´! ^¬ ± ³´!"¼·Â !± !"B± y ªÈl³§¬$g¥ !Ƭ&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y L2 (R) ± @ ¥ ! S ¥ ³´" SÇ $ ±Ä"¼ ¥ } S« !²¦´¥l "¼³§ !:±¹² $«I $O¶+! «·"¼³´! B C ∞(R). qb¥ ωK ⊂ R «$¬ ± «·"O! c L2ω (R) ∞ CK (R) = {f ∈ Cc∞ (R) | supp f ⊂ K}. Yl³´" H 1 (R) = {f ∈ L2 (R) | f 0 ∈ L2 (R)}, ¦Á±® ³´² $I "¼± !"^«$± ¦§« ¥ ¦§ D± ¥: ! ^y$L ³§W"¼ ³´ ¥y" ³§! µYl ³´" ω ∗ (x) = " ω ¥ ! ³´ B ¥y R µAH! 1 , ∀x ∈ R ω(−x) [f, g] := [f, g]ω = Z f (x)g(−x)dx, ¥y f ∈ L2ω (R) " g ∈ L2ω (R) µC{$ $¥È ¦´$¬¬Å ¥ ³´²Ä± !" !"®«$ ! !#¥ «·¶\µ û ¬ ± ³´Q! ¥ K± ¦§¦´! Q ! ! [email protected]¦§ ¥ $¥y² $«$± }!y¥ @¥y"¼³´¦§³§ $ ! Q¥ ¦¿"¼$ ³§ ¥ ¬Å !#"}¦§ g¬ ¬Å ± Ê¥y¬Å !#" $µ :lµ¿¨ < ω > ! ! ,# R M ∈ M f ∈ C ∞(R) # ! M (f 0 ) ω % &"( c ! / # >, M (f ) / ! ! , !> 4!% # ; ! ( % c Yl³¿" (hn)n∈N ¥ ! H ¥ ³´" S $$¦´¦§ ¥y³j«$ !#² $ ÊH² O&·"B ³¿" f ± ! C ∞(R). c NS¦§ B!1± R ∗ (S−hn f )(x) − f (x) − f 0 (x) ≤ 2kf 0 k∞ , ∀n ∈ N. hn q± O«$ !#² $ Ê$! « & ¬Å³´! $Ä»y!1y" ³§$!" lim "L«$ ¦Á±$!"¼ ±ÄÎÁ!y n→+∞ lim S−hn f − f − f0 hn lim n→+∞ Z +∞ =0 M (S−hn f ) − M f − M (f 0 ) hn ?K¬Å¬DP$W"^¥y!T¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ » n→+∞ ω ω = 0. 2 (M f )(x + hn ) − (M f )(x) − (M (f 0 ))(x) ω(x)2 dx = 0 hn « !² Êв M (f 0 ) ± ! L2 (R) µ{A!2 !x ¥ ³¿" ¥y¦§±Æ $ ³´² $ ± ¥: $!y^ ^y³§ " ³´ ¥y"¼³´! " M (f 0 ). loc Mf Ƶ A< ω !8! ,# R * ,# 5 M ∈ M 2134!5 6 4! # ω. ( % 5 = ∞ µM # # . (<( M f = µM ∗ f . $# f ∈ Cc (R) % " −∞ (S−hn M )f −M f hn ! c ^ ¥ ! ! D¦§& «¾½ ¬ ±)¥y" ³§¦§³´ ± DR ¬Å± ! ·« ¶\µ ^ ± ! ¦§&«ª± Ç¥ !x¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ ^ ¥ L2(R) µbYl ³´" B ¦Á±)|¥ ¦§¥y! ³´" R µEY³´" g ¥ ! ж+! « " ³§!y « ¦Á± C ∞ »E$ʱ ¦§Å%º¨ ± ¥2²³´ ³§! ± ÊÐ Æ: "®%1 ¥ " ± !y B µ_Yl³ f ∈ C ∞(R) »E!± µF^! « gM f "^$ʱ ¦´ p.p. %¥y! ¶+!y« "¼³´!:«$!" ³§!#¥ I "^! c $¥l"S ! ³´ gM f ∈ H 1 (R) « ¬Å¬¦§±²Ä± ¦´$¥ Qy gM f $!Åyµ#Yl³¿" µ ¦Í± ¦§³§«$±Ä"¼³´!& ¥ C ∞(R) !y³§ ± (M f )(0) M c » L ¥ ^ ± § ¦ ´ ¦ ! B Å ¬ ! " $ # ¥ &¦ ± ¦§³´«ª±Ä" ³§! µ ˜ ˜ hµM , f i = (M f )(0) f (x) = f (−x). "D¥ ! & ³§W"¼ ³´ ¥y" ³§!µ|AH!Z²Ä±Æ± y ¦§³ ¥y$S¦´&¦´$¬¬ÅI ÐYl ¦´ ² «·¶\µ Í» µ´¨ À ^% gM fM˜µ L¥ ^±¹² ! 0 ˜ ˜ ˜ ˜ |(M f )(0)| = |(gM f )(0)| ≤ C0 kgM f kL2 (B) + k(gM f ) kL2 (B) , C0 > 0 ! D ! ± O |(M f˜)(0)| ≤ C Z ·"S±ª² $«I¥ ! ± ¥y"¼ «$!y "¾±Ä!#" f µ?K ¦Á±¬Å !#" #¥Ç³§¦ÃÈl³§ " D¥ ! D«$!y "¾±Ä!#" |(M f˜)(x)| dx 2 21 + Z "¼ ¦§¦§ ¥yÅ C 12 0 2 ˜ |(M f ) (x)| dx !1y"¼³´$!"ª»yÊ « I% l µ¿¨¹ S C̃ > 0 |x|≤1 |x|≤1 |(M f˜)(0)| ≤ C̃ kM f˜kω + kM (f˜0 )kω ≤ C̃kM k kf˜kω + kf˜0 kω . Yl ³´" ¥y!T« ¬ ± « "O R µ|Yl³ !T±: 0 ˜ ˜ ˜ |(M f )(0)| ≤ C(K) kf k∞ + kf k∞ , y! ! #¥ ³´ " ³§y¥y"¼³´!µAH!T±) f ∈ Cc∞ (R) C(K) K "& ! «Å¦Í± ¦´³§«$±Ä"¼³´! µM : f −→ (M f˜)(0) $W"®¥ !y (M f )(y) = S−y (M f )(0) = (M S−y )(f )(0) AH!1«$! « ¦§¥y" #¥ =< µM,x , f (y − x) >, ∀f ∈ Cc∞ (R), ∀y ∈ R. M f = µM ∗ f, ∀f ∈ Cc∞ (R). AH!2 ³§ ± #¥ M ∈ M $ "I%Æ ¥ "D« ¬ ± « "¥ ± ! 2¦Á±Æ ³´ " ³§y¥y"¼³´! µ ı «$³´$Ð% ω M $ ^ " % ¥ L " $ « ¬ ± · « ª " µ M < 8µ < M ∈ M ! # : " /,: / # /+! >, M % (<( ( / 796: µ̂ 8 ! $# C / # ω ( M C{D ¸¬®|¦§y St $W" µ̂M (s) = hµM , e−isx i. µ|Yl³ φ ∈ C ∞(R) »¦§W¸l¬&¦´ δbt (s) = e−its c Mφ $W" φ̂ µ AH!1 $ ¥y³´"O ¥TC{$¬¬ÅIlµ ®¦§ $ ¥y¦´"¾±Ë" $« "¼± ³§! ¬Å !"^y³§$!:« ! !#¥ ¥ ³¿²Ä± !"ªµ ! 6< 2lµ´¨ < ω ! ! ,# R / # M ! : / ; % ( ω % c qb¥ M ∈ B(L2 (R)) »!:! " M ∗ ͦ Ç $ ±Ä"¼ ¥ B ¥ L2 (R) "¼ ¦ ¥y ω ω ∗ q_¥ [M f, h] = [f, M ∗ h], ∀f ∈ L2ω (R), ∀h ∈ L2ω∗ (R). » M ∈ Mω f " h ∈ Cc∞ (R) (µM ∗ f ) ∗ h = f ∗ (µM ∗ h). ?K$¦Á± !#" ¼±ÄΧ! " »!1± [M f, h] = ((µM ∗ f ) ∗ h)(0) = (f ∗ (µM ∗ h))(0) = [f, M h] M f = M ∗f » ¥ " ¥y" f ∈ Cc∞ (R) µ|Y³´" U ∈ Mω µyNS¦§ B!T± [U M f, h] = [M f, U ∗ h] = [M ∗ f, U h] = [f, M U h], ∀f ∈ Cc∞ (R), ∀h ∈ Cc∞ (R). SF ! « (M U )h = (U M )∗ h = (U M )h, ∀h ∈ C ∞(R) ·" c "¼¥y" U ± ! M µ MU = UM ¥ H"¼¥l" M " ω < µ´¨ ω 0 ! ! $ # R M ∈ M ! 2134!5 6 ω %&"( # ( ! < 5 (Y ) , : / - ,# ! ! # : " /,:8- = n n∈N ( ( (<( lim kYn f − M f kω = 0, ∀f ∈ L2ω (R) n→+∞ kYn k ≤ kM k, ∀n ∈ N. c Oq_ ¥ ·" t ∈ R » ! (V f )(x) = f (x)e−itx » ¥ Ð"¼¥l" µUz¦K $ ¥ ¦´" ¥" ½ $ °$¬Æ Æ« !² Ê ! «$: ¬³§t!y$ #¥ (V ) $W"¥ !5Ê ¥ x ∈ R ¶+ " $¬$!"D«$!" ³§!#¥xj $ ±Ä"¼ ¥ D ¥ L2 (R) µbAH! yÈl M ∈ M µ{Ylt t∈R ³´" T ¦ ± y ¦§³´«ª±Ä" ³§! ω ω !y³§ ± B¦§±&¶+ ¬®¥y¦§® NS¦§ f ∈ L2ω (R) T : R −→ V−t ◦ M ◦ Vt ∈ B(L2ω (R)) T (t) ∈ Mω » ¥y"¼¥y" t∈R µ>Q!Tj·"ª» T (t)(Sa f )(x) = M (f (s − a)e−its )(x)eitx = M (f (s − a)e−it(s−a) e−ita )(x)eitx = M (f (s − a)e−it(s−a) )(x)eit(x−a) = Sa (T (t)f )(x), ∀a ∈ R, ∀t ∈ R, ∀x ∈ R. !! FS ¦§¥y$» kT (t)k = kM k » ¥ B" ¥y" t ∈ R " T (0) = M µ|CQ ± y ¦§³´«ª±Ä" ³§! t −→ M ◦ V "g«$ !#" ³§!#¥ ¥ _¦Á±^" ¦§Ê ³§Q¶+ " K Q $¼±Ë"¼$¥y $µ F&± ¥y"¼ ± W"ª» V "Q¥ !y³´"¾±Ä³§ ¥ t " ¥y" t ·" T "^«$!"¼³´!#¥ ¥yB¦Á±&"¼ ¦´Ê³´^¶+W"¼D $O $ ±Ä"¼ ¥ µq_t ¥y n ∈ N » ³§ !#" η gn (η) := 1 − χ[−n,n](η), ∀η ∈ R n ·" 1 − cos(nx) , ∀x ∈ R. πx2 n AH!± γb (η) = g (η), ∀η ∈ R, ∀n ∈ N. CE±Å ¥ ³´" (γ ) $ "S¥ " n " «Ä " Âz%ËÂzny³§ γ n "L $¦§¦´ ³¿"¼³´² » kγ k 1 = 1 ¥ " n¥yn∈N n n L Z lim γn (x)dx = 0, ∀a > 0. γn (x) = AH! qb¥ n→+∞ Yn := (T n∈N " |x|≥a N^¦§ ¥y ∗ γn )(0). f ∈ L2ω (R) ! ¥ ³¿"¼& $Ê¥y¦Á± ³§¼±Ä!#" »y!T y"¼³´$!" lim kYn f − M f kω = 0. n→+∞ f ∈ L2ω (R) = ≤ Z Z » !~± +∞ −∞ kYn f k2ω = k(T ∗ γn (0))f k2ω Z +∞ 2 (T (y)f )(x)γn (−y)dy ω(x)2 dx −∞ +∞ Z +∞ |(T (y)f )(x)|γn(−y)dy 2 ω(x)2 dx. Q> !T± ¦´³#¥± !"B¦ ³´! $ʱÄ−∞ ¦§³´" ^y&Ñ−∞! Z ! %¦Á±®¬$ ¥ γ (y)dy "L%¦§±&¶+! « " ³§!)« !²ÃÈy ·"^ !T¥l"¼³§¦´³§ ± !"B¦§H" ½ $ °$¬ I]¥ §³ !y³Í»y !Tl"¼³§ !" n kYn f k2ω ≤ Z ≤ Z +∞ −∞ Z +∞ −∞ |(T (y)f )(x)|2 γn (−y)ω(x)2 dxdy +∞ −∞ kT (y)k 2 kf k2ω = kM k 2 x2 γn (y)dy ≤ kf k2ω , Z +∞ −∞ kM k2 kf k2ω γn (y)dy ∀n ∈ N, ∀f ∈ L2ω (R). AH!« ! «$¦´¥y" #¥ M $W"®¦Á±1¦§³´¬Å³¿"¼ ¥ I¦Á±:"¼ ¦§ ʳ§®¶+W"¼ $& ¼±Ä" $¥ & ŦÁ±1 ¥ ³¿"¼ ·"#¥ kY k ≤ kM k, ∀n ∈ N µÏ^ ¥ ±Ä¦§¦§ ! ¬Å± ³§!"¼ !± !"b! ¥ E³´!#" $ $ $%L¦§±Oy³§W (Yn )n∈N " ³§y¥y"¼³´!®± «$³´$B%n Y µYl ³´" f ∈ C ∞(R) "Q ³´" n ∈ N µ>Q! !± !"Q¦ ± Ê¥ ¬$!"_ ¦Á± n $¥y² S ¥C{$¬¬Llµ´¨ »!ЬŠ!#" #c ¥ L¦Á±H ³¿²$O M (f˜gn) ± ¥Ð ! @ $g ³´ "¼ ³§ ¥l"¼³§ ! " M (f˜g )0 µY³´" g ∈ C ∞(R) ¥y! K¶+! «·"¼³´!®$ʱ ¦§K%&¨B± ¥®² ³§ ³§!± Ê K 0 "g%S ¥ " n c ! ± ! B µNS¦´ A! Ð gM (f˜gn ) ∈ H 1 (R) "H! ¥y" ! ³´ (M (f˜gn ))(0) = (gM (f˜gn ))(0). hµM gn , f i = (M (f˜gn ))(0). !1y" ³§ !#" gM (f˜gn ) 0 |(M (f˜gn ))(0)| ≤ C0 kM (f˜gn )kL2 (B) + k M (f˜gn ) kL2 (B) , >g!T± ¦§³#¥±Ä!#"¦§¦§ ¬Å¬H Yl ¦´ ²Æ% C "O¥ ! « ! W"¾± !"¼³´! ! ± !"¼y f µ q_± ¦´H¬ ¬Å^¼± ´³ !y! $¬$!"ª» #¥ ± ! B¦§± 0 y¥ ²I ¥TCE ¬Å¬Ilµ l» !:¬!"¼ ¥j³´¦{·Èl³§W"¼ C > 0 "¼ ¦§¦´#¥ Å q_¥ 0 ˜ ˜ ˜ |(M (f gn ))(0)| ≤ C kf gn kω + k(f gn ) kω . ∞ f ∈ CK (R) »y!T y"¼³´$!"D |(M (f˜gn ))(0)| ≤ C(K) kf˜k∞ + kf˜0 k∞ , C(K) !y1y $! ¥y1y¥ « ¬ ± « " K ⊂ R µQA! «$ ! $« ¦´¥y" #¥ µ g $W" ³§ ! ¥y! ³§W"¼ ³´ ¥y" ³§!<Ç 6¨µBUz¦D$W"~« ¦Á± ³´#¥ µ g $W"Z%6 ¥ y W"1«$¬ T ± n«·"ªµO>È ³´¬Å!y ¬ ± ³´!"¼$! ± !" Y !1¶+ ! « " ³§!) µ g µ^ ¥ LM±¹² n! n M n ((T ∗ γn )(0)f )(y) = Z = = Z +∞ −∞ = Z Z +∞ (T (−s)f )(y)γn(s)ds −∞ +∞ −∞ M (V−s f )(y)e−isy γn (s)ds hµM,x , f (y − x)eis(y−x) ie−isy γn (s)ds +∞ −∞ hµM,x , f (y − x)e−isx iγn (s)ds = hµM,x , f (y − x) Z +∞ γn (s)e−isx dxi −∞ = hµM,x , f (y − x)gn (x)i = h(µM gn )x , f (y − x)i, ∀f ∈ Cc∞ (R). A!:« ! «$¦´¥y" #¥ Yn f = µM gn ∗ f, ∀n ∈ N, ∀f ∈ Cc∞ (R). q_¥ φ ∈ C ∞(R) »|!Z ! ³¿" M : f −→ φ ∗ f » #¥ ³_$ "H¦§&¬&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥yB ¥ L2 (R) ± l« ³§Å% φ µc AH! ! Ä"¼ C ∞(R) ¦ÍÇφ$ ± «$Å $ ¶+! « " ³§!y Å«$¦§± C ∞ ¥ R %T ¥ ω W" ± ! O¦§« ¬ ± « " K µ|AH!:K ¬ ±Ä #¥ #¥ ¥y φ ∈ C ∞(R) » g ∈ L2 (R) »¦Á±&¶+! «·"¼³´! K R 3 x −→ φ(x)Sx g ∈ L2ω (R) ! ω "^¥ ! ³´¶+ ¬$¬$!"B«$ !#" ³§!#¥ D ¥y Z " R kφ(x)Sx gkdx ≤ kφk∞ kgk sup kSx km(K) < +∞. AH!-«$!y«$¦§¥l"¥y R φ(x)S gdx $W":¥ ! 2³§!"¼ ʼ±Ä¦§Z 2s}l«¾½ !y$:«$!² $ Ê $!"¼ ¥ )¦Á± " ¦§Ê³´K¶+W"¼B }K $ ±Ä"¼ x¥ «·¶\µ »?K½± ³¿"¼ L õ#^¥y}±ª² ! $»#¦§±¶+ ¬®¥ ¦´ ¥y³´²Ä± !"¼ Z lµ´¨¹ ¯ Mφ = φ(x)Sx dx. R >Q! j·"ª» ³´" K ¥ ! ¥ WÂ$ ± «$L«$ ¬ ±Ä« "@ R µA!ű M (C ∞(R)) ⊂ C ∞ " (R) φ K K+supp(φ) ¦§±& "¼ ³§«·"¼³§ ! R φ(x)S dx % C ∞(R) $¥l" " H« ! ³´ $ $H« ¬¬ÅL¥ ! H³´!"¼$Ê ¼± ¦´^ G sK«¾½ ! ¥y C ∞(R) %I² ±Ä¦§x$¥y @± ! K C ∞ µy?K¬Å¬O¦§ K³§!" $Ê ± ¦§ } SsK«¾½ ! $ K K+supp(φ) (R) « ¬Å¬&¥y"¼ !#"^±ª² « ¦´$¶+ ¬Å B¦§³´! ª± ³´ «$!" ³§!#¥ $ »!:y"¼³´$!"ª» ¥ g ∈ C ∞(R) » K x∈K Mφ g(x) = (φ ∗ g)(x) = = Z Z R φ(y)g(x − y)dy = Z c φ(y)(Sy g)(x)dy supp(φ) φ(y)Sy g (x), ∀x ∈ R ·"}¦§± $! ³´" Ly C ∞(R) ± !y $!" ¼±ÄΧ! O¦Á±H¶+ ¬&¥ ¦§ +© µÀ ·µAH!Ð ¬ ± #¥ ¥y ¥ c ¦ÍÇ $ ±Ä"¼ ¥ D«$ !#² ¦§¥l"¼³§ !:±¹² $« φ $W"^± ¥ ³{¥ !1¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ g ¥ L2 (R) µ φ ∈ Lω̃1 (R) ω 6 < lµ "< ω ! ! ,# R < M ∈ M . # ! 2134!- 6 % ( ω &"( ! <- (ψ ) ⊂ C ∞(R) 5 = (<( n n∈N c supp(φ) L2ω (R) lim Mψn = M / ! ! , /5 97 # ->, !+ # / 5 $ # ! @5 = , # - ( ( . (<( n→+∞ n ∈ N. / kMψn k ≤ kn kM k, kn = sup|y|≤ 1 kSy k. c q_¥y} ¬Å !#" @¦§± ³´"¼³´!»Ä³§¦ ¥ Ð"K L¬!"¼ #¥ L" ¥y"@¬®¥ ¿¦ "¼³ ¦§³´«ª±Ë n " $¥ _%B ¥ W"« ¬ ± « "E "_¦Á±B¦§³´¬Å³¿"¼_± ¥ $! b @¦Á±"¼ ¦´Ê³´_¶+W"¼Q _ $ ±Ä"¼ ¥ bÇ¥ ! ¥ ³´" (M ) ⊂ M » ψ ∈ C ∞(R) » ¥yO" ¥y" n ∈ N µ|Yl³¿" M ∈ M % ¥ " « ¬ ± « "}ψ "Kn∈N ³¿" (θ ) ω ¥y! Sn¥ ³´" ^ c$Ê¥ ¦§± ³´¼± !" O" $¦´¦§ #¥ ¥ g"¼¥y" n ≥ 1ω¦§±D¶+ ! « " ³§! n n∈N ^ " $ § ¦ ´ ¦ » ´ ³ ¼ " ³¿²Ä» ± ³´ "^% ¥ "L± ! [− 1 , 1 ] µNS¦´ » ¥ f ∈ L2 (R) !T± θn ω n n n lim kθn ∗ f − f kω = 0. qb¥ n ∈ N » ! M f = M (θ ∗ f ), ¥ "¼¥y" f ∈ L2 (R) µCE±Z ¥y³´"¼ « !²$ Ê ² $ M ¥ B ¦Á±®n "¼ ¦´Ê³´^¶n+W"¼ L ¼±Ä" $¥ L·" Mω = M » n ψ n→+∞ n ψn = µM ∗ θn ∈ Cc∞ (R). !2C (Mn )n∈N A!1± »«ª±Ä Mn f = θ n ∗ M f kMn f k2ω = ≤ ≤ Z +∞ Z Z $ "L¥ !y ±Ä¦§Ê° « ¬Å¬&¥y"¾±Ä" ³´² H " Mω Z +∞ Z −∞ +∞ Z −∞ +∞ −∞ 2 (M f )(x − y)θn (y)dy ω(x)2 dx +∞ −∞ |(M f )(x − y)|θn (y)dy 2 ω(x)2 dx +∞ |(M f )(x − y)|2 θn (y)ω(x)2 dydx, ∀n ∈ N, ∀f ∈ L2ω (R), ± ° T¦ ³´! −∞ $ʱĦ§³´" −∞ Z Ñ $! $!r± ¦§³ #¥ $º% ¦§±5¬Å ¥ x ± ³´¦§³¿"¼ ¶+! « " ³§!)« !²ÃÈy x2 µ>Q!T¥y" ³§¦´³§¼±Ä!#"K¦§H" ½ $ °$¬D I]¥y ³§! ³ »y!)" ¥y² Å ≤ kMn f k2ω ≤ Z sup kSy k 2 kM f k2ω , ∀n ∈ N, ∀f ∈ L2ω (R). q± O«$ ! ¥y$!"ª»|!:y" ³§$!" 1 |y|≤ n θn (y)dy "T¦Á± +∞ −∞ kSy k2 kM f k2ω θn (y)dy kMn k ≤ kn kM k, pour tout n ∈ N. +` a bc ¥y ¥ ω ¥ ! ³§ B ¥ Ílµ¿¨¹À q_¥ ω R Mφ «$!" ³§!#¥2 DFH±Ä! }« " "¼S $«·"¼³§ !Å! ¥ K ¬Å !#" !y b |φ(α)| ≤ kMφ k, ∀φ ∈ Cc∞ (R), ∀α ∈ Aω . «$!" ³§!#¥» ! + Rω,1 = lim x→+∞ + Rω,2 = lim A!: $¬Å± #¥ #¥ x→+∞ ω(y) − x1 ω(x + y) x1 − , Rω,1 = lim sup , sup x→+∞ y≥0 ω(x + y) ω(y) y≥0 sup + Rω,1 y≤0 ω(y) x1 ω(−x + y) − x1 − , Rω,2 . = lim sup x→+∞ y≤0 ω(−x + y) ω(y) − ≥ Rω,1 " + − Rω,2 ≥ Rω,2 ·"^!y¥ O ! ³§ ! − + Iω,1 := [ln Rω,1 , ln Rω,1 ], Aω,1 := {z ∈ C | Im z ∈ Iω,1 }, q_¥ O·"¾± ¦´³§ − + Iω,2 := [ln Rω,2 , ln Rω,2 ], Aω,2 := {z ∈ C | Im z ∈ Iω,2 }. (2.16) »y! ¥ S±ª² ! O ³´!Ty §¦ ¥ ³§$¥y B¦§ ¬Å¬$ µ ! lµ< δ ! ! ,# N 5 = . ( lim sup p→+∞ n≥0 ! &"( # δ(n) 1p ≥ 1. δ(n + p) δ(n + 1)2 = 0. n→+∞ δ(0)2 + ... + δ(n)2 inf lim bq ¥ g¦Á± $¥l²L ¥CE ¬Å¬OlµH! $¥y"g L ± " $Q±Ä¥y$ ¥y"gyB¦Á±H ¬Å!y "¼ ±Ä"¼³´! ¥TC{$¬¬ÅDlµ´¨µy ͵ ! : < ! Axlµ¿¨ < lµ´¨ lµ´¨ + Rω,1 + Rω,2 lµ´¨ª = lim p→+∞ sup supt∈[0, 1] ω̃(t) « #¥ ³{ !#" ¼±ÄΧ! D¦§ B$ʱ ¦´³´" $K² ¥ ¦§¥y$$µ lµÇ© < − Bω,1 n ω ≤ ω(x) ≤ sup ω̃(t)ω(n), t∈[0, 1] ! : < ! $ # := z ∈ C | − ln Rω,1 ! < n→+∞ α,k ω lµ ¨ R ≤ Im z et lim ! 5 2134!5 6+! <- − , ( &": ( # < ,! # ! / α- ∈! Bω,1 lµ Ä i) kf k = 1, ω(−n) p1 ω(−p − n) − p1 − sup , Rω,2 = lim sup , p→+∞ n∈N ω(−n) n∈N ω(−p − n) ω(n) / < ! ω(n) − p1 ω(n + p) p1 , Rω− = lim sup . p→+∞ n∈Z ω(n + p) p→+∞ n∈Z ω(n) !1± x = n + t, n ∈ Z ·" t ∈ [0, 1] ·" x∈R Rω+ = lim >Q!~j·"ª» ¥ ! / ! ,# R %'&"( # ω ( ( ω(n) − p1 ω(n + p) p1 − , Rω,1 = lim sup , = lim sup p→+∞ n∈N ω(n + p) p→+∞ n∈N ω(n) n X e −2k Im z (fα,k )k∈N ⊂ L2ω (R) . # ∀k ∈ N. ! o ω(k) = +∞ . k=0 ii) lim kSfα,k − e−iα fα,k kω = 0. k→+∞ 2 B/ !, 1 ( c {AH!yÈl ·" − α ∈ Bω,1 ∈ ]0, 21 [ gn = A!1±: kgn k2ω n X µ|AH! » λ = e−iα f = χ[−,] ·" λ−p−1 Sp f . p=0 = = Z n X R n X λ −p−1 p=0 |λ| −2p−2 Z 2 n X 2 f (x − p) ω(x) dx = p=0 2 2 f (x) ω(x + p) dx = P~± ³§!" $!± ! p=0 "$»!)² ±¬!"¼ $ #¥ − k Sgn − =k n+1 X p=1 λgn k2ω "L«$ ¦Á±³§¬ §¦ ³ #¥ −2n−2 Z λ −p−1 p=0 n X |λ| Z −2p−2 +p −+p Z f (x − p)2 ω(x)2 dx ω(x + p)2 dx. − kSgn − λgn k2ω = 0. kgn k2ω n X =k λ−p Sp f − = |λ| n X p=0 lim inf n→+∞ A!1±: |λ| −2p−2 Sp+1 f − n X p=0 λ−p Sp f k2ω λ−p Sp f k2ω = k λ−n−1 Sn+1 f − f k2ω p=0 2 ω(x + n + 1) dx + − Z ω(x)2 dx − R R |λ|−2n−2 − ω(x + n + 1)2 dx + − ω(x)2 dx k Sgn − λgn k2ω R Pn = . −2p−2 ω(x + p)2 dx k gn k2ω p=0 |λ| − A!: ! ³¿"^¦´ ³§ NS¦§ B!1±) A!: $¬Å± #¥ #¥ "L³§¦·Èl³§W"¼ σ ± B¦§±®¶+ ¬®¥y¦§ σ(p) = 2 ω(x + p) dx − 12 , ∀p ∈ Z. k Sgn − λgn k2ω |λ|−2n−2 σ(n + 1)2 + σ(0)2 Pn = . −2p−2 σ(p)2 k gn k2ω p=0 |λ| ω(p) ≤ ω̃(−x)ω(x + p), ∀p ∈ Z, ∀x ∈ [−, ] ω(p + x) ≥ C>0 Z "¼ ¦§¦´#¥ ω(p) sups∈[−,] ω̃(s) , ∀p ∈ Z, ∀x ∈ [−, ] ω(p + x) ≥ Cω(p), ∀p ∈ Z, ∀x ∈ [−, ] ! µj?K ¦Á±³§¬ ´¦ ³#¥ ·"^y! « σ(p) = Z 2 ω(x + p) dx 12 ≥ "^¥ ! D«$ ! "¼± !"¼D $¦§¦´ ³´" ³´² µ FSI¬ $¬ » − K Z C 02 ω(p)2 dx Kω(p) ≤ σ(p) ≤ K 0 ω(p) 12 = Kω(p), ∀p ∈ Z, "L³´¦·Èl³§W"¼ sup 12 = K 0 ω(p), ∀p ∈ Z, $!"¼ ±ÄÎÁ!y σ(n) ω(n) ≥ M sup , ∀p ∈ Z, σ(n + p) n∈N ω(n + p) $ "L¥ !y « ! "¼± ! n∈N "¼D $¦´¦§ ³¿"¼³´² µ N^³§! ³Í» lim p→+∞ |λ|p sup 1 σ(n) p1 ω(n) p1 |λ| ≥ |λ| lim M p sup ≥ − ≥ 1, p→+∞ σ(n + p) Rω,1 n∈N ω(n + p) ± °$ Í lµ¿¨ õAH! ¥y"^± ¦´³#¥ $B¦´DC{$¬¬ÅIlµÐ± ¥ ³§ n∈N Pn K? ¬¬ lim "I«$ ¬Å¬¦§ ³´ −2p ω(p)2 = +∞ n→+∞ p=0 |λ| ¦´$!"¼ »!~± lim n→+∞ ]³´!± ¦´$¬$!"ª»l!T« ! « ¦§¥y" #¥ n X |λ|−2p σ(p)2 = +∞ p=0 σ(0) = 0. −2p σ(p)2 p=0 |λ| lim Pn n→+∞ lim inf n→+∞ k Sgn − λgn k2ω = 0. k gn k2ω ! "L!:y" ³§ !#"I |λ|−p σ(p) |λ|−2(n+1) σ(n + 1)2 lim inf n→+∞ Pn = 0. −2p σ(p)2 p=0 |λ| ·" "¼ ¦§¦´#¥ Ð C0 > 0 σ(n) Kω(n) ≥ 0 , ∀n ∈ Z, ∀p ∈ Z. σ(n + p) K ω(n + p) L¥ ^±¹² ! M − $W"^¥y! D«$! W"¾± !" $$¦´¦§− ³¿"¼³¿² µ CQdz§! ʱ ¦§³¿"¼ C 2 ω(p)2 dx ω(p + x) ≤ C 0 ω(p), ∀p ∈ Z σ(p) ≤ 0 ω(p + x) ≤ ω̃(x)ω(p), ∀x ∈ [−, ] ·"^y! « Z ω " σ !"I#¥ ³´²Ä±Ë Uz¦B·Èl³§W"¼: ! «:¥ !yT ¥ W ¥ ³¿"¼: $¥lÈ)«$! y³´"¼³´! lim k→+∞ ( kggnnkω )n∈N » #¥Ç! ²Ä±! " $ α,k,p c bAH! yÈl k→+∞ 2 2 |f (x)| ω(x) dx = " !$! ¥ ³´" #¥ ¥y R Vp ω kS 1 hα,k,p − e−i p hα,k,p kω = 0. p µYl ³´" ¦§ ³§ ρ(x) = √1 ω( x ), ∀x ∈ R µAH!~y ! ³¿" p p » ¥ f ∈ L2 (R) ¬®¥ ¦´ (V f )(x) = f(x) − α ∈ Bω,1 ρ ± {¦§±B¶+ Vp : L2ω (R) −→ L2ρ (R) Z # ! , # 5 α ii) lim ¦Í± ¦´³§«ª±Ë"¼³§ ! " x ∈ R µAH!T±) p ∈ N∗ %"&"( tlµ ¯ < ω ! : ;< ! ,# R ! < 2134!5 6 ! <5 (h ) ⊂ L2 (R) - = − α ∈ Bω,1 α,k,p k∈N . ( (<( ω Ílµ i) k h k = 1, ∀k ∈ N. Ílµ »² $ ³ ± !"¦§ k Sfα,k − e−iα fα,k kω = 0, k fα,k kω = 1, ∀k ∈ N. (fα,k )k∈N Z p ω p Z 1 y 2 y 2 dy = |(Vp f )(y)|2ρ(y)2 dy ω f p p p R "Q¥y! ³§R ¬ "¼ ³§@ ¥ $«·"¼³´² L2ω (R) ¥ L2ρ (R). AH! ¬ ± #¥ S1,ρ Vp = Vp S 1 ,ω " p A!1±: "L ! « Vp∗ S1,ρ = S 1 ,ω Vp∗ . p ω( xp ) yp × p1 ρ(x) y1 = sup x+y , ∀y ∈ R. sup x≥0 ω( p ) x≥0 ρ(x + y) 1 − − p Rρ,1 = (Rω,1 ) . NS³§!y ³Í»_ ³ α ∈ B − »Q! ± α ∈ B − . Y³´" (f ) ⊂ α,k k∈N ³§·"¼ (2.20)ω,1" Ílµ l¨ p ¥ αρ,1·"L¦§ ³§y ρ µNS¦´ L2ρ (R) ¥ ! : ¥y³´"¼Æ² $ ³ ± !"¦§ p α lim kVp∗ S1,ρ fα,k − e−i p Vp∗ fα,k kω = 0 " k→+∞ α A! Ð lim kS 1 ,ω Vp∗ fα,k − e−i p Vp∗ fα,k kω = 0. hα,k,p = k→+∞ Vp∗ fα,k ·"^ !Tl"¼³§ !" p khα,k,p kω = 1, ∀k ∈ N α lim kS 1 ,ω hα,k,p − e−i p hα,k,p kω = 0. k→+∞ p " ! <- lµ Ë© <lµÀ < (uα,k )k∈N ω 2 ⊂ Lω (R) ! : < ! , # - > = (<( R% * $# 5 − α ∈ Bω,1 . 2 134!5 6 ( i) k uα,k kω = 1, ∀k ∈ N. lµ ¯ ii) lim kSt uα,k − e−itα uα,k kω = 0, ∀t ∈ R. c jY³´" α ± ! B "} ³¿" p ∈ N yÈlĵYl³¿" (h ² $ ³ ± !"D¦§ ³´ " $ lµ L·" Ílµ ¥y p! µqb¥ H"¼ ¥y" k→+∞ − ω,1 ∗ ¥ ! ^ ¥ ³¿"¼ " $¦ # ¥ q ≤ p ! q ∈ N∗ ⊂ L2ω (R) α,k,p! )k∈N ±: p! α α p! k S 1 hα,k,p! − e−i q hα,k,p! kω = k (S 1 ) q hα,k,p! − (e−i p! ) q hα,k,p! kω q p! Y α α −i p! ≤ k k S 1 hα,k,p! − e−i p! hα,k,p! kω . k S 1 − ue p! C{ ¥ ³¿" u∈C, p! u q =1, p! u6=1 Y α k S 1 − ue−i p! k p! "^¬ ± ± B¥y! D«$! W"¾± u∈C, ! " #¥ ³{! Du6=1 ! ± O p! u q =1, k "L ! «Ð α lim k S 1 hα,k,p! − e−i q hα,k,p! kω = 0. q_± O·È"¼ ± « " ³§!1 ³Á±ÄÊ!± ¦´ »! ¥y"L± ¦§ B«$ ! " ¥ ³´ D¥ ! D ¥y³´"¼ k→+∞ q (uα,k )k∈N " $¦´¦§#¥ Ð α lim k S 1 uα,k − e−i p! uα,k kω = 0, ∀p ∈ N∗ ·" k→+∞ p! k uα,k kω = 1, ∀k ∈ N ·"^ !~± α qb¥ O"¼¥y" lim k S 1 uα,k − e−i p uα,k kω = 0, ∀p ∈ N∗ . "O"¼¥l" k→+∞ p∈N ∗ p q∈N »!1± αq α S pq uα,k − e−i p uα,k = Cα,q,p (S 1 − e−i p I) uα,k , $W"S¥y! D«$¬®y³§!± ³´ !)¦´³§! $± ³§ !y³§ "¼ ± ! ¦Á±Ä" ³§! µFS!y« p Cα,q,p αq ·" α kS pq uα,k − e−i p uα,k kω ≤ kCα,q,pk kS 1 uα,k − e−i p uα,k kω , ∀k ∈ N. p αq lim kS pq uα,k − e−i p uα,k kω = 0. F&± ¥y" ± W"ª» !T±) k→+∞ αq αq αq kS− pq uα,k − ei p uα,k kω ≤ |ei p | kS− pq k ke−i p uα,k − S pq uα,k kω , ∀k ∈ N. " αq ?K¬Å¬ Q lim kS− pq uα,k − ei p uα,k kω = 0. "^ $!y I± ! k→+∞ » !1$!1 $ ¥y³´" #¥ R lim kSt uα,k − e−iαt uα,k kω = 0, ∀t ∈ R. k→+∞ xlµ < + Bω,1 ! : < ! , # ω n := z ∈ C | Im z ≤ ln + Rω,1 R ! < et lim n→+∞ n X e2k Im z ω(k)2 o = +∞ . ! - 2134!- 6>! <5 (v ) ⊂ L2 (R) # B/ !+, 1 + , ( α,k k∈N &": ( # <; ,! # ! / α5 ∈!B ω,1 . ( ω Ílµ À i) kv k = 1, ∀k ∈ N. k=0 ∗ α,k ω ∗ Ílµ c |AH!yÈl ¥ ³¿"¼ lim kSt,ω∗ vα,k − e−itα vα,k kω∗ = 0, ∀t ∈ R. ii) (fα,k )k∈N k→+∞ lµ N^¦§ " $¦´¦§#¥ ⊂ L2 (R) + α ∈ Bω,1 "K± ° K¦´^C{ ¬Å¬Hµ Ày»#³´¦·Èl³§W"¼S¥y! ω 1 ω kfα,k k 1 = 1, ∀k ∈ N " ω lim kSt, 1 fα,k − e−itα fα,k k 1 = 0, ∀t ∈ R. A!1±: k→+∞ = A! α ∈ B −1 ,1 Z = ω ω kSt, 1 fα,k − e−itα fα,k k21 ω +∞ −∞ +∞ Z −∞ ω 1 dx ω(x)2 2 1 dx. fα,k (x − t) − eitα fα,k (x) ω(x)2 fα,k (x − t) − e−itα fα,k (x) 2 ·"^ !T± µ|AH!Tl"¼³§ !" vα,k (x) = fα,k (−x), ∀x ∈ R kvα,k kω∗ = 1, ∀k ∈ N Z +∞ 2 kSt, 1 fα,k − e−itα fα,k k21 = vα,k (x + t) − eitα vα,k (x) ω ∗ (x)2 dx ω ω −∞ = kS−t,ω∗ vα,k − eitα vα,k k2ω∗ . A!1±Ð ! « lim kSt,ω∗ vα,k − e−itα vα,k kω∗ = 0, ∀t ∈ R. k→+∞ ! AH!1 $¬Å± ¥y #¥ D ³ ω "L¦§ ³§yB !y³ ± B¦Á±®¶+ ¬®¥ ¦§ ω(x) = ω(−x), ∀x ∈ R. N^¦§ + Rω,2 = lµ < + Bω,2 1 1 − − , Rω,2 = + . Rω,1 Rω,1 ! : < ! $ # ω n = z ∈ C | Im z ≤ + ln Rω,2 et ! 5 2134!5 6+! <- + , ( &": ( # < ,! # ! / α- ∈! Bω,2 lµ i) ky k = 1, α,k ω lµ Ä c EYl³¿" ii) R ! < n X k=0 o e2k Im z ω(−k)2 = +∞ . (yα,k )k∈N ⊂ L2ω (R) . # ∀k ∈ N. lim kSt yα,k − e−itα yα,k kω = 0, ∀t ∈ R. k→+∞ + α ∈ Bω,2 . AH!1± Im(−α) ≥ ln 1 − + = ln Rω,1 Rω,2 µSF& ± ° º¦´5C{$¬¬Å5lµ À<± ¦§³ #¥ 5± ¥ ³§ ² ³ ± !"L¦§$ ³§·"¼$D ·" − −α ∈ Bω,1 (u−α,k )k∈N ⊂ L2ω (R) ω »H³§¦I·Èl³´ "¼5¥y! 6 ¥ ³¿"¼ k u−α,k kω = 1, ∀k ∈ N ·" lim kSt,ω u−α,k − eitα u−α,k kω = 0, ∀t ∈ R. AH!T± k→+∞ kSt,ω u−α,k − e B/ !, 1 ( = Z itα u−α,k kω2 = Z 2 R u−α,k (x − t) − eitα u−α,k (x) ω(−x)2 dx 2 R u−α,k (−x − t) − eitα u−α,k (−x) ω(x)2 dx = kS−t,ω yα,k − eitα yα,k k2ω , yα,k (x) = u−α,k (−x), AH!1y"¼³´$!" ¥ " ¥y" x∈R » ¥yO" ¥y" n∈N µAH!1$!1 $ ¥y³´" #¥ lim kS−t,ω yα,k − eitα yα,k kω = 0, ∀t ∈ R. k→+∞ lim kSt,ω yα,k − e−itα yα,k kω = 0, ∀t ∈ R. k→+∞ xlµ < − Bω,2 ! : < ! ! < ω ,# R n o n X e−2k Im z − = +∞ . = z ∈ C | Im z ≥ ln Rω,2 et ω(−k)2 !" 5 2134!- 6+! <- (z ) ⊂ L2 (R) # B/ ! , 1 − , ( α,k k∈N &": ( # <; $! # ! / α5 ∈! Bω,2 ( ω Ílµ i) kz k = 1, ∀k ∈ N. k=0 ∗ α,k ω ∗ Ílµy¨ ii) lim kSt,ω∗ zα,k − e−itα zα,k kω∗ = 0, ∀t ∈ R. c {Y³´" k→+∞ − α ∈ Bω,2 . A!T± 1 + − = ln Rω,1 Rω,2 Im(−α) ≤ ln µF&± °$¦§CE ¬Å¬6lµ G± ¦§³#¥ ± ¥ ³´ ² $ ³ ± !"L¦§ ³§·"¼ " + −α ∈ Bω,1 (v−α,k )k∈N ⊂ L21 (R) ω »S³§¦&ÃÈl³§ " 5¥ ! 5 ¥ ³´" ω k v−α,k k 1 = 1, ∀k ∈ N " ω lim kSt, 1 v−α,k − eitα v−α,k k 1 = 0 ∀t ∈ R. A!1± k→+∞ kSt, 1 v−α,k − e itα ω = Z ω ω 2 v−α,k k 1 = ω Z R v−α,k (x − t) − eitα v−α,k (x) 1 dx ω(x)2 2 R v−α,k (−x − t) − eitα v−α,k (−x) ω ∗ (x)2 dx = kS−t,ω∗ zα,k − eitα zα,k kω∗ , zα,k (x) = v−α,k (−x), ∀x ∈ R, ∀k ∈ N µAH!T !T ¥ ³¿" ¥y lim kS−t,ω∗ zα,k − eitα zα,k kω∗ = 0, ∀t ∈ R. A!:l"¼³§ !" 2 k→+∞ lim kSt,ω∗ zα,k − e−itα zα,k kω∗ = 0, ∀t ∈ R. k→+∞ xlµ´¨ª < ω ! : < ! $ # R # ! %'&"( 2 1 3 4 ! 6 ! < - >= − 2 18 α ∈ B (u ) ) * $ # α,k k∈N ⊂ Lω (R) (<( ω,1 . ( Ílµ kuα,k kω = 1, ∀k ∈ N, lim k Mφ uα,k − φ̂(α)uα,k kω = 0, ∀φ ∈ Cc∞ (R). k→+∞ 2 3 1 4 ! + 2)8* $# α ∈ Bω,1 6 ! <- (vα,k )k∈N ⊂ L2ω (R) 5 (<( = . ( Ílµ kv k = 1, ∀k ∈ N, lim kM ∗ v − φ̂(α)v k = 0, ∀φ ∈ C ∞ (R). ∗ α,k ω ∗ k→+∞ φ α,k α,k ω ∗ c 3)B* lµ Ä© ,# 5 + α ∈ Bω,2 . 2 134!5 6"! <5 ( (yα,k )k∈N ⊂ L2ω (R) - > = (<( kyα,k kω = 1, ∀k ∈ N, lim k Mφ yα,k − φ̂(α)yα,k kω = 0, ∀φ ∈ Cc∞ (R). 4B ) * , # 5 lµ ¯ kz k→+∞ 2 134!5 6"! <5 − α ∈ Bω,2 . ( α,k kω ∗ (zα,k )k∈N ⊂ L2ω∗ (R) = 1, ∀k ∈ N, lim kMφ∗ zα,k − φ̂(α)zα,k kω∗ = 0, ∀φ ∈ Cc∞ (R). k→+∞ c {AH!yÈl α ∈ B µYl ³§$!" φ ∈ D (R) " ² $ ³ ± !"L¦§ ³§·"¼$ lµ Ë© } " lµ ¯·µAH!:y" ³§ !#"I − ω,1 k Mφ uα,k − ≤ Z φ̂(α)uα,k k2ω +∞ kφk2∞ Z Z = kMφ uα,k − φ̂(α)uα,k k2ω ?K¬¬ ¥y Z +∞ −∞ a −a Sy uα,k (x) − e−iyα uα,k (x) dy −a k∈N 2 ω(x)2 dx, ∀k ∈ N, [−a,a] (x) ≤ kφk2∞ Z Z a Z −a ¥y! ¥ ³¿"¼ 2 φ(y) Sy uα,k (x) − e−iyα uα,k (x) dy ω(x)2 dx a ≤ kφk2∞ " (uα,k )k∈N ⊂ L2ω (R) [−a,a] ± °$¦ ³´! $ ʱ ¦§³¿"¼ SÑ $!y $!:± §¦ ³ #¥ $^% §¦ ±I¬$ ¥ χ µ>Q!T± ¦´³ #¥± !"O¦§H" ½ $ °$¬ I]¥ ³´! ³Íl» !1" ¥l²Ð x2 −∞ 5 = (<( +∞ −∞ 2a dx ·"K¦§±D¶+ ! « " ³§!«$!² ·Èl 2 Sy uα,k (x) − e−iyα uα,k (x) ω(x)2 dx dy a −a kSy uα,k − e−iyα uα,k k2ω dy, ∀k ∈ N. y ∈ [−a, a] » kSy uα,k − e−iyα uα,k kω ≤ kSy − e−iyα Ik ≤ sup kSs k + |e−isα | < +∞, Ê «$I± ¥:"¼½y$ °$¬ D«$!² Ê ! «$ ¬³§! IC{$ |$Ê ¥ »!:«$ ! «$¦´¥y" ¥y s∈[−a,a] lim kMφ uα,k − φ̂(α)uα,k kω = 0. FS¬ $¬ »}$!¢± ¦§³#¥±Ä!#" ¦§ )C{$¬¬Å :lµ l»Blµ ·"Tlµ !<¬Å!" lµ ·»Ílµ © " lµ ¯·µ lµ¿¨¨ < ω ! : ;< ! ,# R ! < φ ∈ C ∞(R) # ! %'&"( c [ lµ À |φ̂(α)| ≤ k Mφ k, ∀α ∈ Aω,1 Aω,2 . c LA! ¬ ± #¥ #¥ T± ° Ŧ ³´! $ʱ ¦§³¿"¼Æ ~?± ¥ «¾½¸ÂWY«¾½K± W"¼Ì » ¥y" ¥y" ± ¥x¬Å ³§! D¥ ! Å $ ³§ Pn e−2k Im z ω(k)2 ·" Pn e ³¿² ÊÅ " ! « z ∈ C k=0 k=0 ω(k) Y³´" φ ∈ C ∞(R) yÈl$ĵQAH! ¥ #¥ α ∈ A T B − µYl³¿" − S + Bω,1 . Aω,1 ⊂ Bω,1 ω,1 ω,1 c C k→+∞ 2k Im z 2 (uα,k )k∈N ⊂ L2ω (R) "¼¥y" k ∈ N » !T± ¥ ! ¥y³´"¼H² $ ³ ± !"L¦Á± ³§·"¼ l µ ·µ ?K¬Å¬ kuα,k kω = 1 »¥ φ̂(α) = < φ̂(α)uα,k − Mφ uα,k , uα,k > + < Mφ uα,k , uα,k >, ∀k ∈ N "L!:y" ³§$!" |φ̂(α)| ≤ | < φ̂(α)uα,k − Mφ uα,k , uα,k > | + kMφ k, ∀k ∈ N. A!1± lim | < φ̂(α)uα,k − Mφ uα,k , uα,k > | ≤ lim kφ̂(α)uα,k − Mφ uα,k kω = 0 "L!)"¼ ¥y² k→+∞ Yl³ α ∈ Aω,1 ?K¬Å¬ q_¥ "_ k→+∞ T |φ̂(α)| ≤ kMφ k. µ ·»!ŬŠ!#" #¥ + »#Ê « S±Ä¥ ¬ $¬S± Ê¥ ¬$!"@ "% Á¦ ± ³´ " l Bω,1 |φ̂(α)| ≤ kMφ∗ k. kMφ k = kMφ∗ k » !Tl"¼³§ !"D |φ̂(α)| ≤ kMφ k, ∀α ∈ Aω,1 . _"¼¥y" z ∈ C ± ¥¬³´! _¥y! B $ $ ³§ Pn e2k Im z ω(−k)2 " Pn e−2k Im z ! « A ⊂ B − S B + µ u « K± ¥È k=0 ³§·"¼ Í µ © {·" lµ ¯k=0 !&¬ω(−k) !"¼ }2 ω,2 ω,2 ω,2 Ílµ xlµ< |φ̂(α)| ≤ kMφ k, ∀α ∈ Aω,2 . ω ! : ;< ! $ # R ! < φ ∈ Cc∞ (R) %6&"( ³´² $ Ê }¬ ¬Å #! |φ̂(α)| ≤ k Mφ k, ∀α ∈ Aω . c BuÄ«$1%¦Á±MO$¬Å± #¥ Tlµ¿¨» !8± y ¦§³ ¥ ± !"± ¥ ³´ ³´ «$ "y" $!#¥8 ! $W"¼ $³§Ê !± !" ω % Z ¥ ! ³§ " "¼± ! ± : $O ½y³´¶ "¼K ³§¦§±Ä"¼ ¼± ¥È)± «$³´$O% $ ³§y ¥ Z +« ¶\µ y» "¼½y$ °$¬ ®·" Í» " ½ $ °$¬D ·» !:" ¥l² + + − − Rω+ = max(Rω,1 , Rω,2 ), Rω− = min(Rω,1 , Rω,2 ). ?K$¦Á±³´¬ ¦´³#¥ #¥ 1¦Á±~¶+ !"¼³´°$ : :¦§±2 ± ! A $ "Å«$ !#" $!#¥ 1± ! q_¥ B" ¥y"¼H¶+ ! « " ³§! φ ∈ C ∞(R) » φ̂ "L$!"¼³´°$ I·"L³§¦jω "^«$¦§± ³§ ¥y c |φ̂(z)| ≤ Ckφk∞ ek Im z ≤ Kkφk∞ , ∀z ∈ Aω , Aω,1 S Aω,2 C > 0 » k > 0 " K > 0 µjAH! $¥y"S ! « Ê « ®± ¥ ³´! $« ³ I ®qQ½ ± ʬ$!lÂzC{³§! ¦ ¶ « ¶\µ Í» µ ¯·» y$ ¥ ³´ D ͵ ÀK¦Ídz§! ʱ ¦§³¿"¼I |φ̂(α)| ≤ kMφ k, ∀φ ∈ Cc∞ (R), ∀α ∈ Aω . µ ` a Ec bc HFH± ! H« " "¼& $«·"¼³§ !»!y¥ H±Ä¦¿Â ¦´! O $¬!"¼ $B¦´JB½ $ °$¬ l µ´¨ » $! ! «$I± ! O¦ ³´!#" ¥ «·"¼³§ !µ L2ω (R) c c lYl³¿" ω ¥ ! ³§ b ¥y R µY³´" M ∈ M µ AH!&²Ä±S¥y" ³§¦§³´ ¦´ ³§ ω » # ¥ ³g±Æ " ®³§!"¼ ly¥ ³´"H± ¥ $y¥y"D «$«¾½ ± ³´" ĵEAH!¼± $¦´¦§ ¥y¦§$ ³´ " ω !"I# ¥ ³´²Ä± ¦´$!"¼D·" L (R) = L (R) µ{F&± ° ¦§±)qQ ³¿"¼³´!ºlµ :± ¦§³ #¥ $ ω ± ¥ ³´ ω ·"D± ¥Z¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ M »j³§¦b·Èl³´ "¼&¥ ! ® ¥ ³´" (φ ) ⊂ C (R) " $¦§¦´ # ¥ ω 0 2 ω 0 2 ω0 ∞ c $W"ЦÁ±~¦´³§¬³´"¼Ð (M ) ± ¥5 ! Ð )¦Á±1"¼ ¦§ ʳ§Ð¶+ " Æ ¼±Ä" $¥ "" $¦§¦´ #¥ ¥ &" ¥y" n ∈ N »φ !6n∈N ± kM k ≤ k kM k , k = sup ω˜ (y) µQYl ³´" φ B n B n 0 |y|≤ µ & F ± ° ^ ´ ¦ J B ½ $ $ ° ¬ l µ l » ± ´ ¦ ³ # ¥ D ± ¥ § ³ B « ! ¼ " ´ ³ ! ¥ » T ! ± a ∈ I ω = Iω ω0 0 n n∈N M n n ω0 1 n ω0 0 2 [ c |(φ n )a (x)| = |φn (x + ia)| ≤ kMφn kBω0 ≤ kn kM kBω0 ≤ kn βω kM kBω , µb?K¬¬Å&¦Á±Æ ¥y³´"¼ (k ) $ "I| !y$ »{ ! $¥y"$» #¥ ³´" "¼%) ¬ ¦§± «$ n n∈N ± ¥ ! 1 ¥ W  ¥ ¿ ³ ¼ " )« $ # ! ² $! ±Ä ¦§Ä»@ ¥ #¥ ((φ «$!² Ê ¥ [ [ ((φn )a )n∈N n )a )n∈N ¦§± "¼ ¦§ ʳ§D¶ ± ³§y¦§ σ(L∞(R), L1 (R)) ² D¥ ! ¶+!y« "¼³´! ν ± ! L∞(R) µjFS ¦§¥y$»! a l"¼³§ !" ¥ "¼ ¥y" x∈R Cω = β ω kνa k∞ ≤ Cω kM kBω , +« ¶\µ lµ¿¨Ï õ^ ¥ L±¹² ! limn→+∞ kn = 1 Z 1 [ (φ lim n )a (x) − νa (x) g(x) dx = 0, ∀g ∈ L (R) 2 »«ª± ·"^ !T $¬Å± #¥ #¥ n→+∞ lim n→+∞ Z R R d d [ (φn )a (x)(f )a (x) − νa (x)(f )a (x) g(x) dx = 0, ∀g ∈ L2 (R), ∀f ∈ Cc∞ (R). AH!º«$! « ¦§¥y" # ¥ (φ « !² Êж ± ³§y¦§$¬$!" ± ! d [ n )a (f )a n∈N µ f ∈ Cc∞ (R) AH ! yÈl f ∈ C ∞(R). ?K¬Å¬ (M f ) ∈ C ∞(R) »!1± φn c a L2 (R) ²$ » d νa (f )a c \ \ (M φn f )a (x) = Mφn f (x + ia) d cn (x + ia)fˆ(x + ia) = φ cn (x + ia)(f =φ )a (x), ∀x ∈ R ·" ± O«$! #¥ !#"$» 2 d \ k(Mφn f )a kL2 = k(M φn f )a kL2 ≤ k1 βω kM kBω k(f )a kL2 , ∀n ∈ N. ¥ V ¥y³´" " %® ¬ ¦Á± « $ (M f ) ± B¥ !y ¥ \Âz ¥ ³´" «$!² !± ¦´ »! $¥y"O ¥ φ a ¥y (Mφ f )a «$ !#² $ ʶ ±Än∈N ³§ ¦´$¬$!"± !y L2 (R) ²$ ¥y! ¶+! « " ³§! h ∈ L2 (R) µ a n∈N A!1± n n Z Ca,g ± ! Mf R (Mφn f )a (x) − (M f )a (x) |g(x)| dx ≤ Ca,g kMφn f − M f kω , ∀g ∈ Cc∞ (R), ∀n ∈ N, "@¥ ! L«$ ! "¼± !"¼ » #¥ ³! O ! #¥ L L2 (R) »!1y" ³§ !#"D ω lim A!:« ! «$¦´¥y" #¥ n→+∞ Z (Mφn f )a (x)g(x) dx = R (M f )a = ha " Z R g µ?K¬Å¬ (Mφn f )n∈N « !² ÊL²$ (M f )a (x)g(x) dx, ∀g ∈ Cc∞ (R). (M f )a ∈ L2 (R) µ?K ¬Å¬ ¥ "¼¥l" g ∈ L2 (R) » d \ [ d lim h(M φn f )a , ĝiL2 = lim h(φn )a (f )a , ĝiL2 = hνa (f )a , ĝiL2 " n→+∞ n→+∞ \ \ lim h(M φn f )a , ĝiL2 = h(M f )a , ĝiL2 , !:y"¼³´$!" n→+∞ d \ (M f )a = νa (f )a . A!:« ! «$¦´¥y" #¥ ¥l"B"¼¥y" ¶+! « " ³§! f ∈ Cc∞ (R) "O" ¥y" a ∈ Iω !T± d \ (M f )a (x) = νa (x)(f )a (x) p.p. ¥y R− < R+ ³ µÄµ A◦ 6= ∅) µ|?K¬Å¬ ω ω ω Yl¥ ! O¬ ± ³´!"¼$! ± !" ± ° L¦´JB½ $ (M f )a ∈ L2 (R) ⊂ S(R)0 , ∀a ∈ Iω , ∀f ∈ Cc∞ (R), °$¬ lµÇ© µ ® E !T± ◦ \ d M f (x + ia) = (M f )a (x), ∀x ∈ R, ∀a ∈ Iω , ∀f ∈ Cc∞ (R) " M $W"I½ ¦§ ¬Å ½ ® ¥ A◦ µEYl ³´" f ∈ C ∞(R), f 6= 0. CE± ¶+!y« "¼³´! ν := Mdf !± d f ω c fˆ ¥y $ $¥ Ð ³§! Ê¥y¦Á± ³´"¼ K±Ä! A◦ "B ·"¼$!yT !1¥y! ¶+! « " ³§! ½ ¦§ ¬Å ½ ^ ¥ A◦ µ ω ω ^¥ L±ª² ! " FS ¦§¥ » ◦ ν(x + ia) = νa (x), p.p. pour a ∈ Iω d M f = ν fˆ, pourf ∈ Cc∞ (R). |ν(α)| ≤ Cω kM kBω , ¥ B"¼¥l" ◦ α ∈ Aω "L!T± ◦ ν ∈ H∞ (Aω ). Uz¦{ "^«$¦Á±Ä³§ #¥ o 1 ≤ |z| ≤ ρ(S) . spec(S) ⊂ z ∈ C | ρ(S −1 ) n AH!T²Ä±Ð $¬!"¼ $O¦Ídz§!y«$¦§¥y ³§ !) $« ³ #¥ ĵ|Yl³¿" a ∈ R "¼$¦ #¥ ¥ B"¼¥y" M ∈ M »¦´$ ω ³¿"¼³§ ! ¨ j·"_ gJB½ ° ¬Å@lµ¿¨g ³´$!"{² ³ $ $µËAH!D ¥ #¥ e−ia ∈/ spec(S) µ N^¦§ (S − e−ia I)−1 "L¥ !1¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ K·"O³§¦ÃÈy³´ " ν ∈ L∞(R) "¼ ¦§¦´ ¥y a >Q! $! ± !" da , ∀g ∈ C ∞ (R). F ((S − e−ia I)−1 g)a = νa (g) c g = (S − e−ia I)−1 f ¥ f ∈ Cc∞ (R) »!:l"¼³§ !" d d (f )a (t) = νa (t)(e−iat − e−ia )(f )a (t), p.p , ∀f ∈ Cc∞ (R). ?K ¦Á±T !"¼¼±ËÎÁ! #¥ ν (t)(e−iat − e−ia ) = 1 µ µ¿»E«$ #¥ g³ $ "®± ¥ ĵ_A!x«$!y«$¦§¥l" #¥ a µ K ? Å ¬ ¬S" ¥ O¦§ B$¦§ ¬Å !"¼B D¦Ídz§!" $ ²Ä± ¦´§¦ h ln 1 , ln ρ(S)i ² $ ³ $!" −ia e ∈ spec(S) ρ(S ) ¦´$ ³´" ³§! S¨ K "S K ¥1JB½ ° ¬Å lµ¿¨» !T± −1 n AH!1$!1 $ ¥y³´" #¥ z ∈C| o 1 ≤ |z| ≤ ρ(S) ⊂ spec(S). ρ(S −1 ) n spec(S) = z ∈ C | o 1 ≤ |z| ≤ ρ(S) . ρ(S −1 ) ` #cÏa Ec c ` FH±Ä! Q «$·" "¼ ± " ³§ »Ä! ¥ ± ¦§¦´! b! ¥ Q ³´!#" $ $ $g± ¥È $ ±Ä"¼ ¥ yÉ-³§$!y$WÂzR^ ¶ ¥ µFH± ! O¦§±® ¥ ³¿"¼ S ²Ä±® ³´Ê! $ ¥ K" ¥y" a ∈ R ¦ÍÇ $ ±Ä"¼ ¥ ! ³{ ¥y L1 (R) L2δ (R+ ) a loc ±ÄL¦§±&¶+ ¬&¥ ¦§ Sa f (x) = f (x − a), p.p. $¥y" ·"¼ Æ«$!y ³§y$ )« ¬¬ÅÅ¥ !6 ¥ \ Âz ± « : » ¥y"¼¥y" x ∈ R− » ¥ B" ¥y" f ∈ L2(R+) µ f (x) = 0 δ CQÇ$ ± «$ L2δ (R+ ) L1loc (R) $! ¼± !" c a ` # cÏa Ec c ` ^¥yB± ¦§¦´! ¥l²$# ¥ }" ¥y" ¼ ±Ä" $¥ b }É-³´$! WÂzR^ ¶ $ "± l« ³§}%^¥ !yK ³´ "¼ ³§ ¥l"¼³§ !µÏ^¥y_! ¥ ³§! ³§ ! Q @¬ " ½ $g$¬ ¦´Ï¸ $ @ ± ! g¦Á± ±Ä "¼³´ «$ $!"¼Ä»l¬Å± ³´³§« ³ ¦Á±H ! ³¿"¼³§ !y ¦Á±® ³´ "¼ ³§ ¥l"¼³§ !)± l« ³§ %®¥y!T $ ±Ä"¼ ¥ B É-³§ ! $\Â\RL ¶_$ "L¬³§!y!±Ä" ¥ ¦§¦§Äµ L "¼!y ¦ÍÇ$ ± «$: ж+ ! « " ³§! y C ∞(R) % ¥ "± !y ]0, +∞[ µJb¥y"Ðj ± | » C0∞ (R+ ) $¬Å± ¥y! #¥ j ± °$O¥ !1¼±Ä³§ ! ! $¬$!"O"¼ °$O ³´¬Å³´¦Á± ³´ ^%« $¦´¥ ³{·È I± ! O¦§±Ð ¥ \ $«·"¼³´!Tlµ »¦§ ³§ δ $ "L ¥y³´²Ä± ¦§ !"L± ¥ ³´ B«$!" ³§!#¥ δ y ! ³ ± O¦Á±&¶+ ¬®¥ ¦§ FS §¦ ¥ » δ1 δ1 (x) = exp $W"L" $¦ #¥ ln δ1 lim K⊂R q± O«$ ! ¥y$!"ª» ¥ ln(δ(x + t))dt . 1 + sup δf 1 (t) = 1 1 0≤t≤ n »! ¥ L±ª² ! » K ⊂ R+ + (t) < +∞. sup δf t∈K 0 < inf δ(x) ≤ sup δ(x) < +∞. x∈K ± 1 2 "^¥ ! ¶+! «·"¼³´!)¦§³ "¼«¾½ ³¿"¼Ì ³§$!y! µ?K ¦Á±$!" ¼±ÄΧ! n→+∞ " ¥ "¼¥l"O«$¬ ± «·" Z x∈K xlµ´¨¹ T ∈ W f ∈ C ∞(R+) / # ! (T f )0 = T (f 0). δ . ( 0 c EYl ³§$!" f ∈ C ∞(R+) " (hn)n≥0 ⊂ R+ ¥ ! I ¥y³´"¼D« !²$ Ê$!" ² 0 µ|AH! 0 (S−hn f )(x) − f (x) − f 0 (x) ≤ 2kf 0 k∞ , ∀x ∈ R+ hn "L$!1¥y"¼³´¦§³´¼± !"¦´H"¼½ ° ¬ÅD D« !²$ Ê$!y«$&y¬Å³´! $Ä» !:y" ³§ !#" lim NS³§!y ³! ¥ B"¼ ¥y²!y ?K¬Å¬ n→+∞ lim A!)² ³´" #¥ δ = 0. T P + S−hn f − T f − T (f 0 ) hn » !1$!1 $ ¥y³´" #¥ n→+∞ T ∈ Wδ P + S−hn f − f − f0 hn δ = 0. T P + S−hn f = T S−hn f = P + S−hn T Shn S−hn f = P + S−hn T f. lim n→+∞ Z +∞ 0 2 (T f )(x + hn ) − (T f )(x) − T (f 0 )(x) δ(x)2 dx = 0. hn Uz¦ !Å ¥ ³´" ¥y (T f )0 . P + S−hn T f −T f hn « !² ÊB² ± ¥Ð ! g $g ³´ " ³§y¥y"¼³´! ·" T (f 0 ) A< δ !8! ,# R+ T ! # / 5 $ # , ! , # L2(R+) 2134!5 6 4! # µ # % # 5 = . ( <( ( T δ 6 < lµ T (f 0 ) = 6 # > 7 T f = P + (µT ∗ f ), ∀f ∈ Cc∞ (R+ ). c Eq_¥ f ∈ C (R) » ! f˜(x) = f (−x), ¥ x ∈ R µ|Y³´" f ∈ C (R) ·"S ³´" z " $¦ # ¥ supp f˜ ⊂] − z , +∞[ ·" S f˜ ∈ C (R ) » ¥ z ≥ z . L¥ ^±¹² ! " (T S f˜) ∈ L (R). ?K$¦§±)$!"¼ ±ÄÎÁ! # ¥ T S f˜ $ " $ʱ ¦ $ #¥ (T S f˜) = T (S f˜) ∞ c 0 f 0 ∞ c 0 f ∞ 0 z 2 loc z z z ±Ä "¼ ¥y"Ð%Z¥ ! ƶ+ ! « " ³§!5« !"¼³§!#¥ ¥ »y! ¥ L±ª²!y z≥z R + «·¶\µ + f » ¿µ ¨ À õFS z§¦ ¥ » ¥ a > 0 f AH!1«$! « ¦§¥y" #¥ (T Sz+a f˜)(z + a) = (P + S−a T Sa (Sz f˜))(z) = (T Sz f˜)(z). n o $W"S« ! "¼± !" ¥y z ≥ z ·"L! (T Sz f˜)(z) f z∈R+ < µT , f >= lim (T Sz f˜)(z). Yl ³´" K ¥ !« ¬ ± « "y R ·" ³¿" z "¼ ¦ #¥ z ≥ 1 " K ⊂] − ∞, z [ µ{?K½y³§ ³§ ! ¥ !y^¶+! «·"¼³´! g ∈ C ∞(R) #¥ ³|$W" K ³¿"¼³¿² »#%I ¥ yK W"±Ä! [z − 1, z +K 1] ·"K"¼ ¦§¦´ #¥ qb¥ f ∈c C ∞(R) »!y¥ ±ª² ! gT (S f˜) ∈ H 1(R)K "K¦§O¦§K¬Å¬B SYl|¦§·² g(zK ) = 1. z +« ¶\µ K³´¬ ¦´³ #¥ #¥ K z→+∞ K ≤C Z |(T SzK f˜)(zK )| = |g(zK )(T SzK f˜)(zK )| 12 Z 21 0 2 2 2 ˜ ˜ g(y) |(T SzK f )(y)| dy + , |(g(T SzK f )) (y)| dy $W":¥ ! Z« ! W"¾± !"¼ÄµO?K$¦§±$!"¼ ±ÄÎÁ!y #¥³´¦^ÃÈl³§ " ¥ ! Z« ! W"¾± !"¼ C > 0 ! ± !" #¥ y K » " $¦´¦§#¥ |y−zK |≤1 |y−zK |≤1 |(T SzK f˜)(zK )| ≤ C(K) ?K¬¬ + Z Z |y−zK |≤1 |(T SzK f˜)(y)|2 2 δ(y) |(T (SzK f˜)0 )(y)|2 dy δ(y)2 |y−zK |≤1 1 < +∞ et t∈[zK −1,zK +1] δ(t) sup sup t∈[zK −1,zK +1] δ(y)2 21 dy δ(y)2 12 δ(t) < +∞, C(K) »K!y " ³§¦j !Ty$«$ ¥ ¦§ #¥ ¥y |(T SzK f˜)(zK )| ≤ C(K)kT k ≤ C(K)kT k !~± ∞ f ∈ CK (R) Z Z 21 Z |(SzK f˜)(y)| dy + 2 |y−zK |≤1 1 |f˜(x)| dx 2 −1 12 + Z 1 −1 |y−zK |≤1 |(f˜)0 (x)|2 dx |(SzK f˜)0 (y)|2 dy 12 21 ≤ C(K)kT k(kf˜k∞ + kf˜0 k∞ ) = C(K)kT k(kf k∞ + kf 0 k∞ ), $ "&¥ ! Å«$! W"¾± !" » ¥y³}! Å $!y ¥y ! ¥ L±ª² ! C(K) " f ∈ C ∞(R) z ≥ zK K K µ>"¾±Ä!#" !y! #¥ ¥ D"¼ ¥ (T Sz f˜)(z) = (T SzK f˜)(zK ), ! ¥ _ $ ¥y³§ ! # ¥ µ $W"¥ ! } ³§W"¼ ³§ ¥y" ³§!jµÏF& ±Ä¥y"¼ ± W "ª» ¥ ! ¥ L ±ª²!y ¥ z > Ty y≥0 (T f )(y) = (S−y T f )(0) = (S−y S−z T Sz f )(0) = (S−z (S−y T Sy )S−y Sz f )(0) = (S−z T S−y Sz f )(0) = (T Sz S−y f )(z). q± O«$ ! ¥y$!"ª»|!1± >g!y ¥ ³¿"¼ » !:² ³´" #¥ Ä» ¥y lim (T Sz S−y f )(z) = (T f )(y). z→+∞ y≥0 " f ∈ Cc∞ (R+ ) » lim (T Sz S−y f )(z) =< µT , S] −y f > z→+∞ "L! ¥ O« ! «$¦´¥ ! #¥ =< µT,x , f (y − x) >= (µT ∗ f )(y) (T f )(y) = (µT ∗ f )(y), y ≥ 0, f ∈ Cc∞ (R+ ). ! ·" f ∈ Cc∞ (R+ ) `_`bc a |` #cÏa Ec c ` zU « ³Í»|!T²Ä± ¥y" ³§¦´³§ B¦Á±¬ ¬ÅD¬ "¼½yl #¥ &« $¦§¦´ #¥Ç!± %Ŭų´ I$!Z$¥y² ¥ l"¼$!y³§b¦Í± ¹Èl³´¬ ±Ä" ³§!y$b¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ µ¹>"¾± !"_ ! ! #¥ #¥ ¦#¥ $_¬Å ³ «ª±Ä" ³§!y " $«¾½ ! ³ #¥ $O$dz§¬ !#"$»y! ¥ B ! !y! B¦§ ·"¾± ³´¦§$µyAH!)! " T ¦ÍÇ $¼±Ë"¼$¥y SÉ-³´$! W RL ¶b ! ³ ± K¦Á±&« !²¦´¥y"¼³´!)±ª² « µ ¥ f ∈ C ∞(R+) µYl³ µµ $W"B¥ ! ³´ " ³§y¥y"¼³´!Æ% ¥ "«$¬ ± «·"ª»Ï!&y³´" #¥ T "_¥ !® $ ±Ä"¼ ¥ b c}É-³´$! WÂ\RL ¶%L ¥ y W"« ¬ ± « "$µ µ 6 < lµÇ© δ 0 ! ! $# R+ T ∈ W # ! 2134!- 6 ! <- (Y ) # / - ,# !>, 6 # 7 ! # : " /,:8δ % 5 &"( = . ( n n∈N (<( lim kYn f − T f kδ = 0, pour f ∈ L2δ (R+ ) n→+∞ kYn k ≤ kT k , ∀n ∈ N. c AH! (U f )(x) = f (x)e » ¥ f ∈ L (R ) » t ∈ R " x ∈ R » !8± y ¦§³ ¥ ± !"¦§Æ" ½ $ °$¬) )«$ !#² $ Ê$! « ~ ¬Å´³ ! $Ä»! ¥y y"¼ ! ! #¥ :¦§)Ê ¥ −itx t (Ut )t∈R AH!T± 2 δ + "^«$!" ³§!#¥ ¥yL¦§±&"¼ ¦´Ê³§L¶+ " $L $ ±Ä"¼ ¥ µ|Yl³´" T (0) = T + T ∈ Wδ " ! T (t) = U−t ◦ T ◦ Ut , ∀t ∈ R. µ q_¥ » " a>0 x>0 f ∈ L2ω (R) ! ¥y^±ª² ! (S−a T (t)Sa f )(x) = (T (t)Sa f )(x + a) = eit(x+a) (T (f (s − a)e−its ))(x + a) = eitx (S−a T (f (s − a)e−it(s−a) ))(x) = eitx (S−a T Sa (Ut f ))(x) = (T (t)f )(x). K? ¦Á±T¬!"¼ #¥ T (t) ∈ W µbFS ¦´¥ $»E³´¦Q$ "&«$¦§± ³§ #¥ kT (t)k = kT k » ¥ t ∈ R µ CQ± ¦´³§«ª±Ë"¼³§ ! T "@«$!" ³§!#¥ Lδ R ± ! W µ#q_ ! Y := (T ∗ γ )(0), (γ ) " ¦§± ¥ ³´" $Ê¥ ¦§± ³´¼± !" Iy ! ³´ ± ! S¦§± ¥ \Âzδ $« " ³§!lµ lnµNS¦§ ¥yn f ∈ L2(Rn+)n∈N »! ¥ δ l"¼$!y! N^³§! ³Í» ¥ n∈N kYn f k2δ ·" lim kYn f − T f kδ = 0. f ∈ L2δ (R ) = k(T ∗ ≤ » ! ¥yO" ¥l²! n→+∞ + Z 0 γn )(0)f k2δ +∞ Z = Z Z +∞ 0 +∞ 2 (T (y)f )(x)γn (−y)dy δ(x)2 dx −∞ +∞ −∞ |(T (y)f )(x)|γn(−y)dy 2 δ(x)2 dx. >g!T± ¦§³#¥±Ä!#"¦Ídz§! ʱ ¦§³¿"¼^ Ñ $! $!~·"L¦§H"¼½y$ °$¬ I]¥y ³§! ³ »y!1y" ³§ !#" kYn f k2δ ≤ Z +∞ −∞ Z ≤ ≤ 2 = kT k Z +∞ |(T (y)f )(x)|2 γn (−y)δ(x)2 dxdy 0 +∞ kT (y)k2 kf k2δ γn (y)dy −∞ Z +∞ −∞ kf k2δ , kT k2 kf k2δ γn (y)dy ∀n ∈ N, ∀f ∈ L2δ (R+ ). ^¥ « ! « ¦§¥ !y #¥ kY k ≤ kT k » ∀n ∈ N µE?K ! ³´ $ ! D¬ ±Ä³§!"¼ !± !"¦Á±Æy³§ " ³´ ¥y"¼³´! ± l« ³§$Ð% Y µEYl³¿" K ¥ n!2«$¬ ± «·" R "D ³´" z ≥ 1 " $¦ #¥ K ⊂] − ∞, z [ µ{>Q! ! ¥ D ²Ä± !"In ¥º¦§$¬¬® ÅYl|¦§·²Z·"D $ ± Ê ¥ ¬ÅK!"¼·È &±Ä! D¦Á± $¥y² ÅK Ц§± qg ³¿"¼³§ !:lµ¿¨»!:y" ³§$!" ¥ f ∈ C ∞(R) » K ≤ C(K)kT k Z |y−zK |≤1 ≤ C(K)kT k C(K) " |(T SzK (f˜gn ))(zK )| 12 Z 2 ˜ |SzK (f gn )(y)| dy + Z 1 −1 |(f˜gn )(x)|2 dx 21 + Z 0 |y−zK |≤1 1 −1 0 |SzK (f˜gn ) (y)|2 dy |(f˜gn )0 (x)|2 dx 12 ≤ C̃(K)(kf k∞ + kf k∞ ), ! D $!y $!" ¥y C̃(K) K µq_± B«$!y #¥ !"ª» ∞ |(T Sz (f˜gn ))(z)| ≤ C̃(K)(kf k∞ + kf 0 k∞ ), ∀z ≥ zK , ∀f ∈ CK (R) "L!:«$!y«$¦§¥l" ¥y µT g n ! ³ ±Ä < µT gn , f >= lim (T Sz (f˜gn ))(z) $ "L¥ !yIy³§ " ³´ ¥y"¼³´!)Ç y ШµF&± ¥y" ± W"ª»! ¥ L±¹² ! z→+∞ (Yn f )(y) = = Z (T (−s)f )(y)γn(s)ds R Z (T (M−s f ))(y)γn(s)ds = < µT,x , f (y − x)e−isx > γn (s)ds R R Z =< µT,x , f (y − x) γn (s)e−isx ds >=< µT,x , f (y − x)gn (x) > e −isy Z R = (µT gn ∗ f )(y), ∀y ≥ 0, ∀f ∈ Cc∞ (R+ ). ]³§!±Ä¦§$¬$!"$»l! ¥ ^ y"¼ ! ! Yn f = P + (µT gn ∗ f ), ∀f ∈ Cc∞ (R+ ), ∀n ∈ N. 21 ?K¬¬ supp µ g ⊂ [−n, n] »«$ ¦Á±«$ ¬ ¦´° " ¦Á± ¥y²Äµ T n 6 < ¢µ ¯ < δ !! ,# R+ T ∈ W / # > ! 2134!5 6! <- % ( ( δ. 5 = ∞ (φn )n∈N ⊂ Cc (R) (<( lim kTφn f − T f kδ = 0, ∀f ∈ L2δ (R+ ) n→+∞ c Y³§ !#" T ∈ Wδ " %Æ ¥ y W"D«$¬ ± «·"ªµYl³´" » θ ≥0 ·" lim ·" n→+∞ 1 0≤t≤ n ¦§±Å ³´ " ³§y¥y"¼³´!1± l« ³§$&% T µA!~ ¥ #¥ µ ¥y! ¥ ³¿"¼&" $¦´¦§ #¥ supp θ ⊂ [0, 1 T] » Z x≥a n n θn (x)dx = 0, ∀a > 0 kθn kL1 = 1, ∀n ∈ N. » !T± f ∈ L2 (R+ ) δ lim kθn ∗ f − f kδ = 0. qb !y n→+∞ AH!)²³¿" #¥ φ T , φn + (t) kT k , ∀n ∈ N. sup δf µT (θn )n∈N ⊂ Cc∞ (R) n qb¥ kTφn k ≤ n Tn f = T (θn ∗ f ), ∀f ∈ L2δ (R+ ). «$ !#² $ ʲ$ T ¥ ¦§±&"¼ ¦§ ʳ§O¶+ " H L ¼±Ä" $¥ B " q_¥y f ∈ L2 (R+) »y!~± (Tn )n∈N = µT ∗ θn ∈ Cc∞ (R). δ kTn f k2δ = kP + (µT ∗ θn ∗ f )k2δ = ≤ Z Z +∞ 0 Z +∞ Z = kP + (θn ∗ µT ∗ f )k2δ R 2 θn (y)(Sy (µT ∗ f ))(x)dy δ(x)2 dx θn (y)|(Sy (µT ∗ f ))(x)|2 δ(x)2 dydx. u « I± ¥)"¼½ ° ¬ÅD I]0 ¥ ³´! ͳ R»l! ¥ L±ª²!y kTn f k2δ ≤ Z 1 n θn (y) 0 ≤ ≤ Z Z 1 n 0 Z 1 n 0 +∞ 2 0 2 |(µT ∗ Sy f )(x)| δ(x) dx dy θn (y)kT (Sy f )k2δ dy θn (y)kT k2 δe+ (y)2 kf k2δ dy C Tn = ≤ kT k 2 + 2 e sup δ (y) kf k2δ . 1 0≤y≤ n ^¥ $ ¥y³§ ! #¥ kT k ≤ sup ·")¦Á±xqQ ³¿"¼³´!<lµ ¯ «$¥ ¦´ + e δ (y) kT k n 0≤y≤ ¥y! I± ´¦ ³#¥±Ä" ³§!Ƴ´¬Å¬$y³Á±Ä" S D¦Á±ÐqQ ³´" ³§!:µ© µ Ec c ` FH±Ä! «$·" "¼) ¥ \ ± " ³§Ä»¦Á± ³´! «$³ ± ¦§By³ «$¥ ¦¿"¼O²Ä± " ^ ± ¦§³ # ¥ $g¦§ }¬ " ½ [email protected] % ·È $ ± !yK¦§ «ª± O L¬&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y K !T« !"¼¥y !± !"L¦§H¶ ±Ä³´" # ¥ `_c a 1 n |` # cÏa U1,δ V−1,δ 6= I. A! δ ∗ (x) = δ(−x)−1 , ∀x ∈ R− . ^¥ O³´!#" ¥ ³´ ! B¦ ±Ä«$ A!: ! ³¿" q_¥ q_¥ L2δ∗ (R− ) n − := f mesurable sur R | [f, g] := [f, g]δ = a∈R + a ∈ R− »y! "¼ ! »y! "¼ ! Ua,δ∗ Z Z R− o |f (x)|2 δ ∗ (x)2 dx < +∞ . + f (x)g(−x)dx, ∀f ∈ L2δ (R+ ), ∀g ∈ L2δ∗ (R− ). ͦ ÇR ¼±Ä" $¥ ByD" ¼± !y ¦Á±Ë"¼³§ !Æ ! ³ ¥y f ∈ L2δ∗ (R− ) (Ua,δ∗ f )(x) = f (x − a), p.p. x ≤ 0. Va,δ∗ ¦ÍÇ ¼±Ä" $¥ K I" ¼± !y ¦Á±Ë"¼³§ !Æ ! ³ ¥y f ∈ L2δ∗ (R− ) ± ± (Va,δ∗ f )(x) = f (x − a), p.p. x ≤ a, (Va,δ∗ f )(x) = 0, x > a. xlµ´¨ª < δ ! : ;< ! , # n R+ % o P 1) * ,# α ∈ Bδ− := z ∈ C | ln rδ− ≤ Im z et limn→+∞ nk=0 e−2k Im z δ(k)2 = +∞ . 2134!- 6 ! <- (u ) ⊂ L2 (R+) - = α,k k∈N ( (<( δ Ílµ i) ku k = 1, ∀k ∈ N. α,k δ Ílµ ii) lim kUt,δ uα,k − e−itα uα,k kδ = 0, ∀t ∈ R. k→+∞ n P := z ∈ C | Im z ≤ ln rδ+ et limn→+∞ nk=0 # α ∈ B+ 2) * , 2134!- 6 ! <- δ(v ) ⊂ L2 (R−) - = α,k k∈N (<( δ ÍlµÇ© i) k v k = 1, ∗ α,k δ∗ ∀k ∈ N. e2k Im z δ(k)2 = +∞ o . ( lµ©y¨ kVt,δ∗ vα,k − e−itα vα,k kδ∗ = 0, ∀t ∈ R. ii) lim c yCE± $¥l²B¥y" ³§¦§³´ g¦§ ± Ê¥ ¬$!"¼ ² $¦§ ± !y¦§@«ª± y ¬®¥ ¿¦ "¼³ ¦§³´«ª±Ë k→+∞ " $¥ +« ¶\µC{$¬¬Å LlµÇ© »lµ ¯l»|lµÀy»lµ õ>Q! ¼±Ä!#" f = χ " g = Pn ei(p+1)α S f p p=0 ³´¦ ¥ Ð" ¥ " D D " $O¦Á±®¬ $¬«$ ! " ¥ «·"¼³§ !µ [0,] n qb¥ T ∈ B(L2 (R+)) !T! Ä"¼ T ∗ ¦ $ ±Ä"¼ ¥ B B(L2 (R−)) " $¦ #¥ δ δ∗ [T f, g] = [f, T ∗ g], ∀f ∈ L2δ (R+ ), ∀g ∈ L2δ∗ (R− ). lµ¿¨ª© < δ ! : < ! ,# R+ % 6 ! < − 2134!- # α ∈ Bδ . ( (uα,k )k∈N ⊂ L2δ (R+) 5 (<( = 1) * , lµ© kuα,k kδ = 1, ∀k ∈ N, lim k Tφ uα,k − φ̂(α)uα,k kδ = 0, ∀φ ∈ Cc∞ (R). k→+∞ 6 + 2134!- 2) )B* ,# α ∈ Bδ . ( ! <- (vα,k )k∈N ⊂ L2δ (R−) - (<( = lµ© kv k = 1, ∀k ∈ N, lim kT ∗ v − φ̂(α)v k = 0, ∀φ ∈ C ∞ (R). ∗ α,k δ ∗ c QYl³¿" (uα,k )k∈N α,k δ ∗ c α ∈ Bδ− ² $ ³ ± !"L¦§ ⊂ L2δ (R+ ) Z = ≤ α,k φ ·"D ³¿" φ ∈ C ∞ (R) µEAH! ¥ #¥Ç³§¦ÃÈy³´ " ¥ ! Ð ¥ ³¿"¼ ³§·"¼[−a,a] $ lµ K " lµ ·µAH!:y" ³§ !#" k→+∞ Z Z +∞ 0 +∞ kφk2∞ a −a a Z k Tφ uα,k − φ̂(α)uα,k k2δ 2 −iyα φ(y) Sy uα,k (x) − e uα,k (x) dy δ(x)2 dx Sy uα,k (x) − e−iyα uα,k (x) dy 2 δ(x)2 dx, ∀k ∈ N. >Q!Z± ¦§³ #¥± 0!"B¦Ídz§!y$ʱ ¦´³´" ^−a Ñ ! !Z "O¦´H"¼½ ° ¬ÅD I]¥ ³´! ³!)"¼ ¥y² ≤ ?K¬¬ ¥y kφk2∞ Z a Z −a ≤ kφk2∞ k∈N " kTφ uα,k − φ̂(α)uα,k k2δ +∞ Z0 a −a 2 Sy uα,k (x) − e−iyα uα,k (x) δ(x)2 dx dy kUy uα,k − e−iyα uα,k k2δ dy, ∀k ∈ N. y ∈ [−a, a] kUy uα,k − e −iyα » uα,k kδ ≤ sup −isα + f δ (s) + |e | < +∞. F&± °$L¦§H" ½ $ °$¬D D«$!² Ê ! «$ ¬³§! » !y¥ L±ª²! s∈[−a,a] lim kTφ uα,k − φ̂(α)uα,k kδ = 0. k→+∞ SF }¬ $¬g$dz§¦ÃÈy³´ " }¥ !y} ¥ ³´" (v ) ⊂ L2 (R−) ² ³ ± !" lµÇ© " ͵© ¨ { ! l"¼³§ !" α,k k∈N δ ¦Í± $ " ³§!1 ·µ xlµ´¨¹¯ < δ ! : ;< ! ,# R+ ! < φ ∈ C ∞(R) ! / ! % c ÍlµÇ©© |φ̂(α)| ≤ k Tφ k, ∀α ∈ Ωδ . c KA!5 $¬Å± #¥ ¥y) ± ° ¦Ídz§! ʱÄ"¼¦´³´" y:?± ¥ «¾½¸ÂWYl½ K± " Ì »g!5l"¼³§ !" ¥y ¥ z ∈ C ± ¥Ð¬Å³´! ¥ !yLy$g ³§ P∞ e−2k Im z δ(k)2 " P∞ e y³´² ÊO·" δ(k) !б Ω ⊂ B − S B + . Y³´" φ ∈ C ∞(R) µY¥ k=0 !y ¥y α ∈ Ω T Bk=0 − µYl ³´" (uα,k )k∈N ⊂ δ δ δ δ δ y ¥ ! D ¥ ¿ ³ ¼ " ² ³ ± ! " lµ }c ·" lµ ·µ?K¬Å¬ » ¥ » !T± 2 L (R) ku k = 1 k∈N ∗ 2k Im z 2 α,k δ ω φ̂(α) = < φ̂(α)uα,k − Tφ uα,k , uα,k > + < Tφ uα,k , uα,k >, ∀k ∈ N "L!)"¼ ¥y² |φ̂(α)| ≤ | < φ̂(α)uα,k − Tφ uα,k , uα,k > | + kTφ k, ∀k ∈ N. A!1± lim | < φ̂(α)uα,k − Tφ uα,k , uα,k > | ≤ lim kφ̂(α)uα,k − Tφ uα,k kδ = 0 " ± B« ! #¥ $!" k→+∞ Yl³ α ∈ Ωδ ?K¬Å¬ T Bδ+ k→+∞ |φ̂(α)| ≤ kTφ k. »y$!1¥y"¼³´¦§³§ ± !"B¦§ B¬ ¬Å O± Ê ¥ ¬Å !"¼O "L¦Á± ³§·"¼ Í µ ·» !T± kTφ k = kTφ∗ k »!:y"¼³´$!" |φ̂(α)| ≤ kTφ∗ k. |φ̂(α)| ≤ kTφ k, ∀α ∈ Ωδ L" ¦Á± $¥y² "L«$ ¬ ¦´° " µ P~± ³§!" $!± !"$»! ¥ L± ¦´¦§! K $¬!"¼ $B¦§ $ ¥ ¦´"¼±Ä" ³´! «$³ ± ¦yI« " "¼ ± W"¼³§Äµ c c _Yl³§ !" δ ¥ ! ³§ L ¥y R+ " T ∈ Wδ µjYl³¿" (φn)n∈N ⊂ ¥ ! ¥ ³´" ~"¼ ¦§¦§ #¥ (T ) «$!² ʲ $ T ¥ ƦÁ±º"¼ ¦§ ʳ§)¶+W"¼Z Cc∞ (R) φ n∈N ¼±Ä" $¥ O " #¥ ³² ³ n ]³¿Èl! + (y) kTφn k ≤ kn kT k, avec kn = sup δf a ∈ Jδ µ ^¥ L±ª² ! 1 0≤y≤ n [ c |(φ n )a (x)| = |φn (x + ia)| ≤ kTφn k ≤ kn kT k , ∀x ∈ R. L¥ ¥l²! L·È" ¼± ³´ ((φ ¥ !yI ¥ \Âz ¥y³´"¼Ä» ¥y³{«$!² Ê ¥ O¦Á±&"¼ ¦´Ê³´ [ n )a )n∈N ¶ ± ³´ ¦§ ∗ σ(L∞(R), L1(R)) ² g¥ ! K¶+ ! « " ³§! ν ∈ L∞(R) µ q_¥ ³§¬ ¦´³ _¦´$! Ä"¾±Ä" ³§! » a « " " ¥ \Âz ¥y³´"¼D $¼±± ¥ ³{! Ä"¼$ ((φ µ^¥ L±ª² ! [ ) ) n a n∈N kνa k∞ ≤ lim ( sup δe+ (t)) kT k n→+∞ ·" lim Z [ (φn )a (x) − νa (x) g(x) dx = 0, ∀g ∈ L1 (R). MO$¬Å± #¥ ! ¥y ¥ Rg ∈ L2 (R) » n→+∞ lim n→+∞ Z R 1 0≤t≤ n f ∈ Cc∞ (R) » d d [ (φ ) (x) (f ) (x) − ν (x) (f ) (x) g(x) dx = 0. n a a a a AH!)«$ ! «$¦´¥y" #¥ » ¥y f ∈ C ∞(R) » (φ « !² Ê ¥ ¦§±I" ¦§Ê³´L¶ ±Ä³§ ¦´ d [ ) (f ) n a a c L2 (R) ² ν (fd) µ?K¬¬Å}!б (T f ) = Pn∈N + d [ )a ) = P + F −1 ((φ n )a (f )a ), ¦§±^ ¥y³´"¼ ((T f )a) a «$ !#² $ Ê ¥ b¦Á±Lφ" a ¦§Ê ³§¶ ± ((φ ³´ ¦§nQ)a ∗(f ² $ µ d L2 (R) P + F −1 (νa (f )a ) a n∈N φ FS ¦´¥ $» !y¥ L±ª²! n n Z R+ |(Tφn f )a (x) − (T f )a (x)||g(x)|dx ≤ Ca,g kTφn f − T f kδ , ∀g ∈ Cc∞ (R), C > 0 ! y $! #¥ Å g ·"& « !²$a,gÊ®±Ä¥1 $! L ^ ³´ " ³§y¥y"¼³´! K² $ ·" (T f )a ∈ L2 (R+ ) µ µ_N^¦§ $»E! ¥y&y" $! ! #¥ (T f ) φ a n∈N µ q ± O « ! # ¥ $! ª " | » 1 ! $ « y ! $ « § ¦ l ¥ " y ¥ (T f ) a n a d (T f )a = P + F −1 (νa (f )a ) HF ±Ä! ®¦Á±1 ¥ ³¿"¼Ä»_! ¥ ¥y J◦ 6= ∅ µ_>"¾± !"y! ! #¥ (φc) $W"¥ ! ¥ ³¿"¼ δ n n∈N H¶+ ! « " ³§! K½ ¦§ ¬Å ½ K¥ ! ³¿¶+ ¬$¬$!"K ! $ B ¥ Ω◦ »y! $¥l"O $¬ ¦§± «$ (φc) δ n n∈N ±ÄI¥y! Å ¥ \Âz ¥y³´"¼ #¥ ³g«$ !#² $ Ê ¥ !y³´¶+ ¬Å ¬Å !#"H ¥ "¼ ¥y"I«$¬ ± «·"I² $ ¥ !y¶+!y« "¼³´! ◦ µNS³§!y ³Í» ¥ g"¼¥l" a ∈ Jδ »¦Á±D ¥ ³¿"¼ (φbn(. + ia))n∈N « !² ÊS² $ ν(. + ia) ν ∈ H ∞ (Ω δ ) ± ¥) $! By$ ³´ " ³§y¥y"¼³´! $µF&Ç¥ !:± ¥l"¼ ^« " »l¦Á± ¥ ³´" ((φ «$ !#² $ ÊH² ν ¥ [ n )a )n∈N a ¦§±" ¦§Ê ³§ σ(L1 (R), L∞(R)) "L! ¥ O $ ¥y³§ ! #¥ ¥ a ∈ J◦ , δ Uz¦{ "^«$¦Á±Ä³§ #¥ ν(x + ia) = νa (x) p.p. + (t)) kT k. kνk∞ ≤ lim ( sup δf n→+∞ 1 0≤t≤ n lY ³ δ $ "I"¼ ¦ ¥y lim »{!x y"¼³´$!" kνk ≤ kT k µbYl³§!y!2! ¥ e+ n→+∞ sup0≤t≤ δ (t) = 1 ±¹² ! kνk ≤ c kT k » c $ "D¦§± $« ! "¼± !"¼Ð ! ³§® ± ! ¦Ídz§!"∞ ¥ « " ³§!jµj?K ¦Á±Æ« ¬Ð ∞ δ δ ¦§°·"¼¦Á± $¥l² µ 1 n C ?RLNHqQUzJOM^> Sfild Hf\d jkilh hji6e ykiyh Sh eBh Hf\di 8f\h6h Hk j h ^h 3 k Sk Sh S dilh c C{$: $ ¥y¦´"¾±Ë"¼)·È :± ! :«$«¾½ ± ³´" !": ² $¦§ :± ! µOYl³¿" E ¥ ! ± « H Ss± !± «¾½ S ¥ ³´" $K« ¬ ¦§ÃÈy K ¥ Z µl^¥ K± ¦§¦´! g! " $ S ¦ÍÇ $¼ ±Ë"¼$¥yK± y §¦ ¦´ ½ ³¿¶ "O !y³ ± Yl !T $ ±Ä"¼ ¥ O³§!² S : CZ 3 x −→ (x(n − 1))n∈Z ∈ CZ . S −1 $W"S ! ³ ±Ä S −1 : CZ 3 x −→ (x(n + 1))n∈Z ∈ CZ . lY ´³ " F (Z) ¦ÍÇ$! $¬®y¦§Ð $I ¥y³´"¼ y !"I" ¥ D¦§ D«$ «$³´$!"¼I¼± ¥l¶¥ !x! ¬® ! ³Q !#" !#¥ ´¦ $µ^ ¥ O ¥ ! ¥y F (Z) "L $!y I± ! E µ < 6yµ´¨ / : / 5 $# ! ,# E - # / - ,# # ! , # E (<( ( ( 5 = ( M Sa = SM a, ∀a ∈ F (Z). ! ! 6 # ! / # M(E) ! , ! ( ( qb¥ z ∈ T = {z ∈ C | |z| = 1} » !y AH!~ $¬Å± #¥ #¥ & ³ ¥ ¦ ± y ¦§³´«ª±Ä" ³§! ( ( : / - , # ! ! $# E% ψz : E 3 x −→ (x(n)z n )n∈Z . » z ∈ T ψz (E) ⊂ E "^ ³E!~ ¥ #¥ ¥ B"¼ ¥y" pn : E 3 x −→ x(n) ∈ C n∈Z "« !"¼³´!¥y »l± ¦§ g¦§O"¼½ ° ¬ÅL ¥ Ê ¼± y½ L¶+ ¬^! ¥ K ! ! ¥y ψ "}¥ !Æ $ ±Ä"¼ ¥ | !y ¥ E µFH± ! D«$«¾½ ± ³´" !2« ! ´³ °$ $D ± « $I Ðs± ! ± z«¾½º² $ ³ ± !"D $¥ ¦´·Â ¬$!"L¦§ B½#¸ "¼½y°$ S ¥ ³´²Ä± !"¼ C » R ¨ F (Z) RH BA!T± RSBA!T± $ "^ ! I± ! S(E) ⊂ E ¥ E ·" ¥ " ¥y" S −1 E ⊂ E ψz (E) ⊂ E, ∀z ∈ T " µ » n ∈ Z pn $W"L«$!"¼³´!#¥T supz∈T kψz k < +∞ E ± !y C µ µ HF ± ! }¦Á±D ¥ ³¿"¼ »# ³ S(E) ⊂ E »l! ¥ }y$ ³´Ê! !y ± spec(S) ¦§L $«·"¼ ^ S¦ ¼±Ä" $¥ %ºy¬ ±Ä³§! E µ}Yl³ S !Ç$ " ± ! spec(S) ³§Ê ! T¦´: $« " T S Q» ¦§±2¶+ ¬·Â S "¼¥ S| µjML± $¦´! ¥y S $W"H¦§± ¦´¥ ·"¼³´" ÃÈ"¼$!y ³§ !~¶+$ ¬Å & S| µ|qQ¦§¥y F (Z) F (Z) «$³§ $¬$!"ª» !1 ¬Å± ³§! S$ " D(S) = {x ∈ E, ∃(xn )n∈Z ⊂ F (Z) t.q. xn −→ x et Sxn −→ y ∈ E} µÏL¥ E! ¥ ! E @¬!"¼ #¥ ! $¥y"_± «$³§ b%O" ¥y"E¬®¥ ¦¿"¼³ ¦§³´«ª±Ä" $¥ ◦ ¥ E ¥ ! 2¶+! « " ³§! L∞ ¥ spec(S) ¥y³$ "T$! ¦§¥y1½y¦§¬ ½ ¥y spec(S) »B ³ ◦ µOq_ ¥ )"¼¥y" k ∈ Z »B! ¥yT± ¦§!y e ¦§± ¥ ³¿"¼2 ¥y Z »B !#":" ¥ :¦§ spec(S) 6= ∅ k «$ «$³´$!"¼Ð !#"!#¥ ¦´Ð%2¦ ÃÈl«$ "¼³§ ! y e (k) #¥ ³B$W"Šʱ ¦B%5¨µUz¦$W"Å« ¦Á± ³´ #¥ )"¼ ¥y" k ¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ $ " ³§!"% F (Z) $ "}¥ !Å ¼±Ä" $¥ @ L«$!² ¦§¥y" ³§!jµ#>g! "ª» ¥ M ∈ " » !T± ¥y N ± $ÌIʼ± !y M(E) a ∈ F (Z) " Sx = y Ma = M N X NS³§!y ³Í»l!:²³¿" ¥y ¥ k=−N N N X X a(k)ek = M a(k)(S k e0 ) = a(k)S k (M e0 ). » n∈Z (M a)(n) = k=−N N X k=−N a(k)(M (e0 ))(n − k) k=−N " ± B« ! #¥ $!" yµ¿¨ M a = a ∗ M (e0 ). FH± ! B¦§± ¥y³´"¼S! ¥ O± ´¦ ¦§!y}! Ä"¼$ M c ¦§±& ¥y³´"¼ M (e0 ) µAH! ¥y"L± «$³§ %®«¾½±¥yI¬&¥ ¦´" ³¿Â ¦§³´«ª±Ä" $¥ ¥y! D $ ³§D ICE± ¥ $!"O¶+ ¬$¦´¦§ M f ! ³§ ±ÄL¦§±&¶+ ¬&¥ ¦§ f(z) = M ^¥ ^± ¦´¦§! O± §¦ M f ¦§D ¸¬®|¦§I $ ³§H¶+ ¬Å ¦§¦´H$! ¼± !" ã(z) = X n∈Z c(n)z n , ∀z ∈ C. M M X n∈Z µ|Nr"¼¥y" a∈E a(n)z n , ∀z ∈ C. C! »! ¥y"H±Ä¥ ³± «$³´$O¥ ! CE± ³§ " yµ¿¨ L !#" ¼±ÄΧ! I¦Á±Å $ $!"¾±Ä" ³§!~ ¥ ³¿² ±Ä!#" &±Ä! H¦ÍÇ$!y $¬& ¦§I $S $ ³§ S EC ± ¥ $!"L¶+ ¬$¦´¦§$} !"L¦§ B«$ «$³´$!"¼L !"L ^ ¥ ³´" $O± ! E µAH!~± ga(z) = M f(z)ã(z), ∀z ∈ C. M L¥ Æ«¾½ «¾½ ! )¦§ Ųı ¦§ ¥ Å r ¥ ¦´$¥y$¦§¦´$Å! $¥y")y! ! ¥ !< $! ± " ¥y" ¥ #¥ ± W"¼¥l"Æ% M f(reiθ ) µ@C{$Æ $ ¥ ¦´"¼±Ä"¼Å % y"¼ !¥yƱ ! Å«$Zy¬ ±Ä³§! ¥ O$ ± «$ ^ DsK± !± «¾½ ± W"¼³§« ¥ ¦§³´$ $»! ¥ O± ¬°$!y$!"L%«$! « " ¥ #¥ D¬ $¬H ¥ B! ½¸ Ä"¼½ ° $K¦§B ¸¬®|¦´B B"¼¥y"@¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ b$W"@¥ ! O¶+! « " ³§! L∞ ¥ @¦Á±S¶+ !"¼³§° B ¥ « "¼ S ·"½ ¦´¬Å ½ D| !y$% ¦ ³´!"¼$ ³§ ¥ ^y¥Z $«·"¼ y S ³_«$® ! ³´$^! " ±Ä ²³§ ĵb^¥ &± ¦§¦´! ¬Å± ³§!" $!± !"I !y! $ ¥ !± $$¥y$& $ ¥ ¦´"¼±Ä"¼I«$!y!¥y$µ_FH± ! ¦§ «$± ± " ³§«$¥y¦§³§ E = lp(Z) »! ¥y^±ª² ! L¦§ ¥ ¦¿"¾±Ä"O« ¦Á± ³ ¥yI ¥ ³´²Ä± !"$µ 6 < yµ´¨< E = lp(Z) 1 ≤ p < +∞ # ! $# 5 M ∈ . % &"( . f ∈ L∞ (T) / M(E) . M fkL∞ (T) ≤ kM k. kM HA !Z ± $¦§¦´ ¥y®± ! ^«$ «ª± ^³§¦E$W"H ³´$!~« ! !#¥ #¥ spec(S) = T. q_ ¥ p = 2 »|! ±~·²³§ ¬Å !#" kM fkL (T) = kM k µC{ ®¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ !#"®± ¥ ³K·"¼Æ·"¼¥ y³§$ ± ! &¦§$ ± « $L% ³§ µ AH!:± y ¦§¦§ ³§ } ¥ Z "¼ ¥y"¼^ ¥ ³¿"¼HW"¼ ³´« " $¬$!" ³¿"¼³´² µyAH!:± l« ³§ %Ð¥ ! ³´ ω ¦§ L ±Ä«$$ ∞ lωp (Z) := {(a(n))n∈Z ∈ CZ | ¬®¥y! ³§B ^!y ¬$ kakω,p = ( X X n∈Z |a(n)|p ω(n)p < +∞} 1 |a(n)|p ω(n)p ) p , n∈Z ¥ 1 ≤ p < +∞ µYl½y³§$¦´ O± $¬!"¼ D± ! E¦´ $ ¥ ¦´"¼±Ä"O ¥y³´²Ä± !"D 6 < yµ >< ω ! ! ,# Z 5 = S ( ◦ ! ! ! 2 2 # ,# lω (Z) spec(S) 6= ∅ %'&"( # ,# M ∈ M(lω (Z)) . 0/ ◦ f(z)| = |M f ∈ H∞ (spec(S)) % M X n∈Z S −1 ! ◦ c(n)z n ≤ kM k, ∀z ∈ spec(S) M ◦ CQǽ#¸ "¼½y°$ spec(S) ·Èl«$¦´¥y"º¥ ! 6¦Á± Ê« ¦Á± 6Ç$ ± «$ x s±Ä!± «¾½µDq_± 6= ∅ ÃÈy ¬ ¦§O± ! Q¦´B«ª± ±Ä "¼³´«$¥ ¦´³§ lp(Z) ¦ ³´!#" $ ³§$¥y ¥Ð $«·"¼ O S $ "g²³§y µ#?K ! ± !"ª» « Æ!Ç$ " #¥ $« $¬¬Å !" #¥ )>Q " $ ¦´Æ±Zy $W²)±Ä! #¥ ƦÍÇ!5±T¦Á± ³¿"¼³´! ¥ ³´²Ä± !"¼Äµ C < yµ ) < l2 (Z) !5 /,: ! 5 = S ( ω ! / ! 1 2 = ρ(S) = r, ( # # ,# M ∈ M(lω (Z)) . ρ(S ) S −1 ! −1 f(rz)| ≤ kM k, p.p. sur T. |M uD ¦Á± « ! ³§ ° D± ! E¥ !y ¦§¥ O¦Á±Ä ÊH« ¦Á± DÇ$ ± «$ S Is±Ä!± «¾½Z% ³§y ¥y³ ° $!":¥ ! ~± ~ 2Yl«¾½±Ä¥ $ µO?SÇ$ "Æ¥ ! Z½¸ Ä"¼½ ° ¦§¥yŶ+W"¼ #¥ R ¨ ·» RH " RS ûg«$¬¬Å:¦´T¬!"¼ ~¦ÍÇ·Èl ¬ ¦´T¯l»}ÃÈ Z¥ ¦¿"¼$ ³§ ¥ ¬Å !#"$µ@CET ³¿"¼¥±Ä" ³§!6±Ä! ¦Á±#¥ ¦§¦§Ä»! ¥ @! ¥y ¦§± $!y} "B|ª± ¥y«$¥ ¦§¥ @Ê ! $ ± ¦§ÄµlCQǽ¸ " ½ °$ RI¨ g "}"¼ ³¿²³Á±Ë ¦§$¬$!"H² $ ³ űÄ! ¦§ $ ± «$ & Ðs±Ä!± «¾½º !"I¦Á±Æ! ¬Å&$ "I ! ³´ ± ¥ ! Ð ³´ «$!² Ê !"¼ µ|VH¥±Ä!#"B%®¦Íǽ#¸ "¼½y°$ RS ûl³´¦ ¥ Ð"O±¹² ³§ k(x(n)) k = k(|x(n)|) k n∈Z n∈Z ¥y #¥Ç$¦§¦´H ³´"K² $ ³ µ Lª± !y¬Å³´! $»³§¦|·Èl³´ "¼D±Ä¥ ³j B$ ± «$ Lys± !±Ä«¾½)± ! B¦´$W ¥y$¦§ k(x(n))n∈Z k 6= k(|x(n)|)n∈Zk " #¥ ³² ³ $!" ¥ "¼± !" R^ ·µ{L¥ H² $ ! I¥ ! ·Èl$¬ ¦§K¥ ¦´" $ ³´$¥ $¬$!"ªµ ^ " ! C = {z ∈ C | |z| = r} » ¥y r > 0 µFH± [email protected]« B«¾½± ³¿"¼ ! ¥ L± ¦´¦§! K $¬!"¼ $B¦§H" ½ $ °$¬Dr ¥ ³¿²Ä± !"ªµ 2lµ´¨ < E 0 !5 /,: , / /,: ,"! <5 ! ! $# Z # B/ ! ( ! ! ! 0 / ) !. /!) ) / %4&"! ( # . ! A! S ! # / # ! = # . # . ρ(S) = +∞ 1) S S −1 . ( !' /! # ρ(S −1) = +∞. S −1 . n o ! / ! 1 2) spec(S) = ρ(S ) ≤ |z| ≤ ρ(S) % < M ∈ M(E) * $# r > 0 5 = C ⊂ spec(S) M f ∈ L∞ (Cr ) / 3) % ( . r f(z)| ≤ kM k . % % ! $# Cr % |M ◦ ρ(S) > 1 M ! # "! $# spec(S). f 4) ( ρ(S ) . FH± ! «$ «¾½ ± ³´" Ä»b!y¥ ®±Ä¦§¦§ ! I± ¥ ³}!y¥ &³´!"¼$ $ $&± ¥lÈx $ ±Ä"¼ ¥ ÅJb ¦´³´"¼Ì ¥ y$B$ ± « $B sK± !± «¾½) S ¥ ³¿"¼$B ¥ Z+ = N µAH!)³§y$!"¼³ x ∈ CZ %&¥y!)$¦§ ¬Å !" CZ ! ¼± !" x(n) = 0 » ¥ n < 0 "S!Z±Ä " &¥ ! «$!² $!"¼³´!2± !± ¦´Ê¥ ¥ ¦§$B ¦§ ¬Å !#" O CZ µ Yl³´" F (Z+) + $ µ F (Z−)I¦ ± « Æ ¥ ³¿"¼$& ¥ Z+ + µ Z− û !#" "¼¥y®¦§ «$ «$³´$!"¼O¼±Ä¥y¶¥ !1! ¬& !y³{ !"^!¥y¦§$µ < <yµ ! 84! ! !! ! # / 5 $# ! S S / # [email protected] # CZ ( , # −1 ( ! ! / 5 ! ( % −1 −1 + − + + P our u ∈ CZ , (S(u))(n) = 0, si n = 0 et (S(u))(n) = un−1 , si n ≥ 1 (S−1 (u))(n) = un+1 , pour n ≥ 0. C L¥ ± ¦´¦§!y^« ! ³´ $ $H¦´$^$ ± «$ s± ! ± «¾½ E + y® ¥ ³¿"¼ H ¥ Z+ »|² $ ³ ± !"¦´$ ³§ " $O ¥y³´²Ä± !"¼ I ¨ CQ ±Ä«$ ")y$! ± ! E + " ¥ "¼¥l" n ∈ Z+ »@¦Í± ¦´³§«ª±Ë"¼³§ ! H F (Z+ ) "L«$!" ³§!#¥ I E + ± ! C µ p : x −→ x(n) n H OA!1± S(E + ) ⊂ E + ¥ S−1 (E + ) ⊂ E + µ qb¥ x = (x(n)) ∈ E + »b! ¥ &±¹² ! " ¥y" " sup kγ k <n∈Z+∞. H z∈T γz (x) = (z n x(n))n∈Z ∈ E + » ¥ z z∈T L¥ Q ¬ ±Ä #¥ ! ¥yB ³ γ (x) ∈ E + » ¥y"¼¥l" x ∈ E + » ± ¦§ γ : E + −→ E + " | !y µËUz¦l "¶ ±Ä«$³§¦´g }² ³§ #¥ Kz ³ S(E +) ⊂ E + »Ä± ¦§ ± b¦§g"¼½ ° ¬Å}z ¥&Ê ¼± ½y}¶+$ ¬ÅÄ» ! y"¼³´$!" #¥ S| »Ä¦Á±^ $W"¼ ³´« " ³§!®y S % E + $W"Q¥ ! $ ±Ä"¼ ¥ | ! K E + ± ! E + µ L¥ @ ³§ ! #¥ SE + $ µ S "} ! ¥ ± ! S(E +) ⊂ E + $ µ S (E +) ⊂ E + õ −1 FH±Ä! ®¦Á±1 ¥ ³¿"¼Ä»_ ³ S| + $ −1 µ S | ) $W"®| !y » spec(S) + $ µ spec(S Dy$ ³´Ê! −1 E −1 ) E ¦´ « " S| $ µ S | ) µbYl³ S + µ S H! " ± D| ! Ä» spec(S) µ −1 E −1 E K ´ ³ Ê ! D¦ § « ¼ " y I¦ § & ± + ¶ $ Å ¬ ·" ¼ ¥ I y + $ µ õ spec(S−1 ) S|F (Z ) S−1 |F (Z ) < xlµ # / 5 $# # ! ,# E + !/ # / 5 $# , 5 < ! ( ( ( = / ; 796: 6 # ( (<( + + + + + + qb¥ (S−1 T S)u = T u, ∀u ∈ F (Z+ ). u ∈ l2 (Z− ) ⊕ E + ! ¥ O³§!" ¥ ³§ ! (P + (u))(n) = un , ∀n ≥ 0, (P + (u))(n) = 0, ∀n < 0. qb¥ L¥ : ! ¼±Ä" $¥ O DJbl ¦§³¿"¼ÌHJ» ! ¥ Uz¦{ "L¶ ± «$³´¦§SyI² ³§ #¥Ç!T± n≥0 » Tb(n) =< T e0 , e−n > Tb(−n) =< T en , e0 > . T u = P + (Tb ∗ u), ∀u ∈ F (Z+ ). ?K¬¬± ! O¦§«$± O $L¬®¥ ¿¦ "¼³ ¦§³´«ª±Ä" $¥ $»! ¥ O ! ³§ !yO¶+ ¬$¦§¦´$¬$!" Te(z) = X n∈Z Tb(n)z n , ∀z ∈ C. C2C + A! ± $¦´¦§ Te ¦§ ¸¬®|¦§Å T µ_>g!x"¼$! ± !"®«$ ¬ " & ³§¬³§¦§³¿"¼¥ y$$!" Ʀ´$&¬&¥ ¦´" ³¿Â ¦§³´«ª±Ä" $¥ S "D¦´$H ¼±Ä" $¥ J_ ¦§³´" Ì®³§¦b$W"D¦§Ê ³#¥ &y$±Ä" " $! %Æ«$ #¥ Te ³´" ¥ ! O¶+! « " ³§! L∞ ¥ @¦Á±H¶+ !" ³§°$ B spec(S) ∩ (spec(S ))−1 µUz¦$W"K ʱ ¦´$¬$!"@!±Ä"¼¥y $¦ Ы$! « " ¥ #¥ Te " ½ ¦´¬Å ½ & ¥ D¦Ídz§!" $ ³´$¥ S−1 spec(S) ∩ (spec(S ))−1 » ³ «$ ! ³´$H!Ç$W" ± S²³´ µUz¦_$W"D«$¦Á±Ä³§ ¥y ³ M $W"D¥ !2¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ B ¥y E−1− ⊕ E + » E − " E + !"®y$& ±Ä«$$& Æs±Ä!± «¾½ ¥y³´"¼ & $ $«·"¼³¿²$¬$!" ¥ Z− " Z+ » ± ¦§ P +M "¥y!Z $¼±Ë"¼$¥yS J_ ¦´³´" Ì ¥ E + µ?K ! ± !"ª»¬ ±Ä¦§Ê I¦§±Å ³§«¾½ y ¦Á±T¦´³´" "¼$ ±Ä"¼¥y ¥ ®¦´$ ¼±Ä" $¥ ® ÆJb ¦´³´"¼ÌÄ»E³§¦@ $¬& ¦§¥j³´¦K!± ± & "¼ "¾±Ä ¦§³ #¥ "¼¥y"@ $ ±Ä"¼ ¥ g OJ_ ¦§³´" ÌB$W"K³´! ¥ ³¿" ± g¥ !¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ ¥ }¥y!Å ± « S Ls±Ä!± «¾½ Ð ¥ ³¿"¼$I ¥y Z ¥y ³§« ³§$¥y $¬$!"I«¾½ ³´ ³ µEAH!x! $¥y" !y« ± &± ¦§³ #¥ $ ³´ « "¼ ¬Å !" ± ¥lÈ ¼±Ä" $¥ ÐJ_ ¦´³´" Ì&¦§ ¼± ³´ !y! $¬$!"¼ #¥ Ð! ¥ I± ¦§¦´! Sy ¦´Ï¸ $D± !y¦§®«$± $¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ $µ|^$± ! ¬³§!y$»j!y¥ D± ¦´¦§! ^± ¦§³ #¥ H¦§&¬ ¬Å® «¾½ $¬Å± $¥l² $!1¦§¬l ³ ± !"ªµyCE± ³§!y«$³ ± ¦§Hy³ «$¥ ¦¿"¼ " ¥y S !j " ¦§¥ O¥ !1 $ ±Ä"¼ ¥ B³§!² ³§ ¦´ ³´!²$ S µ?K ! ± !"ª»y! ¥ B±Ä¦§¦§ ! }!y¥ B ²³§K ¥Æ¶ ± ³´" #¥ S S = I µAH!Æ y"¼³´$!" −1 ¦§H"¼½y$ °$¬−1 ¥ ³¿² ±Ä!#"$µ yµ < E + !5 /,: , / /,: , ! <- ! ! ,# Z+ # B/ ! ( ! ! < / 5 , 5 < ! + H T )% # $# h ( h,# !E % h ) . H ) i H ! 1 1 1) * ,# r ∈ ρ(S ) , ρ(S) . ρ(S) < +∞ ,# r ∈ ρ(S ) , +∞ . ρ(S) = +∞ . / Te ∈ L∞(C ) −1 −1 r 2) 3) S S−1 |Te(z)| ≤ kT k, p.p sur Cr . ! # ! ! 1 ρ(S−1 ) < ρ(S) . o 1 ≤ |z| ≤ ρ(S) . Ω := z ∈ C | ρ(S−1 ) S !' /! # / 4! . n S−1 ! # . S ! # / 4! . ◦ Te ∈ H∞ (U ) . o 1 ≤ |z| . U := z ∈ C | ρ(S−1 ) n 4) / # ! / 796: ( ( S−1 ! /! # . ◦ Te ∈ H∞ (V ), n o V := z ∈ C | |z| ≤ ρ(S) . C ◦ Te ∈ H∞ (Ω) . ` L¥ ± ¦´¦§!yL« ! ± «$ $H« " " ± W"¼³´®%Ŧ§± $ $!"¾±Ä" ³§!y& ³ j $!" $S¶ ± ¬³§¦´¦§ B W ±Ä«$$L Is± ! ± «¾½ #¥ ³² $ ³ !#"^¦§ B½¸ " ½ °$ $ R ¨ û RSK " R^ ·µ ` ¥ ! ³´ ¥y Z µ{Uz¦ "D"¼ °$¶ ± « ³§¦§& ²³´ ¥y¦´$ ±Ä«$$ lp (Z) ¥ 1 ≤ ³ $!"H! S½¸ " ½ °$ $ µE^¥ S± ¦§¦´! O !y! $ ¥y$¦#¥ «ª± ω± W"¼³§« ¥ ¦§³´$ #¥ ³ ³´¦§¦§¥y "¼ $!"H¦´$H ³´" ¥±Ä"¼³´! L ¥ ³¿²Ä± !"¼$ S ·" S −1 !"D| !y$H·"¦§® « " S $W"D ³¿" ¥ !y«$¥ ! ! j³´!#" $ ³§$¥yH! !Z²³§y® ³¿"¥ !« $ « ¦§Ä»¥ ! I ¼±Ä" $¥ S ·" S −1 ! " ±Ä ! ^¥Æ"¼¥yK¦§ K ¥lÈ)!yS !#" ± | !y$$µ VH¥ $¦ #¥ H ³´"¦´ ³´ ω ¦§± ! ¬ÅL $W"^" ¥ ¥y B !y! $ ± Sk Yl ³´" ω ² p < +∞ kS k k = sup FH±Ä! ^¦´$BÃÈy ¬ §¦ $O ¥ ³´²Ä± !"¼O!1«$ ! ³´ °$ n∈Z ω(n + k) . ω(n) E = lω2 (Z) » ω $W"L !y³b« ¬¬ÅH ¥ ³´"$µ jYl³´" µNS¦´ S " S −1 !"| ! " kSk = e » kS −1k = e−1 » ³@¦§ $«·"¼ ) S $W"¦§«$ « ¦§Æ ż±ª¸ ! e "³´¦g$ " dz§!" $ ³´$¥ H²³´ µEFH±Ä! D«$Ыª± ¦´ ¸¬®|¦§y«¾½±#¥ ¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ ^ "I¥ ! ¶+!y« "¼³´! ¥ O«$D«$ « ¦§ µ L∞ ω(n) = en , ∀n ∈ Z " ρ(S −1) = e−1 µbNS³´! ρ(S) = e Yl ³´" » ³ n ≤ 0 µ NS¦§ Q!± kSk = e » kS −1k = 1 » « «$± I¦§Ð $«·"¼ ¥x ½ ³¿¶ "D$ " ¦Á±)« ¥ ! ! $¦´³§¬³´"¼ ±ÄL¦´$B« $ « ¦§ ^ D ±¹¸ !1 $ « " ³´¶ 1 ·" e µ gYl ³´" ω(n) = n!, ³ n > 1 ·" ω(n) = 1 » ³´! !µN^¦§ !²³¿" #¥ kSk = +∞ » " µyC{H « " H ¥ ½ ³¿¶ "$W"O± ¦´ @¦§L«$¬ ¦´$¬$!"¾± ³´ L± ! C ¥ kS −1 k = 1 ρ(S −1 ) = 1 ³´ #¥ ^ S« $!"¼ 0 "}¼±ª¸! 1 µ#C{S ¸¬®|¦´Ly^"¼ ¥y"@¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ $W"½ ¦§¬ ½ B ¥ « ¬ ± ³´! µ » ³ ω(n) = en n>0 ·" µbFH± !y −1 ρ(S) = e ρ(S ) = 1 " ω(n) = 1 Hqb ! » ³ ·" sup e |S(z)| = ³§! !µKN^¦§ Æ!¢± ¥ k > 0 » ·" ρ(S −1 ) = 1 µ#FH± ! @«$·" "¼ ω(n) = n n ≤ 1 ω(n) = 1 ·" µ ^ N § ³ ! ³ ! y ¼ " ´ ³ $!" #¥ ρ(S) = e k −k kS k = 1 + k kS k = 1 ³´"¼¥ ±Ä"¼³´!)!1 $¬Å± ¥y #¥ z∈spec(S) sup C z∈spec(S) |z| = e < kSk. ?K "L·Èl$¬ ¦´I!y¥ O¬Å !#" #¥ D¦ÍÇ$ʱ ¦§³¿"¼ sup z∈spec(S) f(z)| = kM k |M ¥y³ "@²$ ³ ±Ä@" ¥y"}¬®¥ ´¦ " ³ ¦´³§«$±Ä"¼ ¥ ¥y ± ! O¦§«$± ± W"¼³§« ¥ ¦§³´$} $L$ ± «$ lω2 (Z) µ l2 (Z) »l!y $¥y" ± " SÊ $! ¼± ¦´³§ »#¬ $¬ » ¥ _"¼¥y" n ∈ Z µ NS¦§ !Ð $¬Å± #¥ #¥ !1| ! $ µ^¥ L± ¦´¦§!y·Èy± ¬³§! «$D«$± ¦§$B ¼±Ä" ± " ³§« ¥ ¦§³´$ ¥ ² ³§ ³Q¦§JB½ ° ¬Åyµ¿¨Ð$ " #¥± !"I¬ ¬Å®² $ ³ ĵAH!2 ¬ ±Ä #¥ #¥ ·" S 2 "| ! µq± H± ³§¦´¦§$¥y O!± spec(S) = C µj>Q!j·"ª»j ¥ ! #¥Ç³§¦ kS 2 k = 2 ·Èl³§W"¼ λ ∈/ spec(S) µËqb !y A = (S −λI)−1 µAH!&±H± ¦§ AS −λA = I ·" !®¬®¥ ¦¿"¼³ ¦§³Á±Ä!#" «$ " "¼ ʱ ¦§³¿"¼ ± S !Z"¼ ¥y² AS 2 − λAS = S "I± ³´! ³ S = AS 2 − λI − λ2A µE?K$« ³ $ "L±Ä ¥ D«ª± B¦ ¼±Ä" $¥ AS 2 − λI − λ2A $ "L| !y »± ¦´ #¥ S ! ¦ " ± $µq_± «$! #¥ !#"$»l³´¦y "}«$¦Á±Ä³§ #¥ spec(S) = C µ#Y³¦§BJB½ ° ¬ÅLyµ¿¨O "¾±Ä³´"Q²Ä± ¦Á±Ä ¦§Ä» ¦§K ¸¬®|¦§ S 2 #¥ ³Q$W" z2 ± ¥y¼± ³¿"D "¼Ð| !y ¥ C µjUz¦Q$ " ²³´ $!" #¥ Å«Ä " ³§¬ ³´ ¦§ÄµN^³§! ³Í» ! ¥ « ! "¼±Ä"¼ ! ¥y!y "¼ $ ¥ ¦´"¼±Ä"! $¥y" ± " ÐÊ$!y$¼±Ä¦§³§ ¥ ¦§ $ ± «$ y s± !± «¾½1 ¥yL¦´$¥y$¦§O! ³ S ! ³ S −1 ! D !"^ ! $ µ Yl³¿" " !"O"¼+¥y1)B=¦§$O|n| +¥lÈ:1 ! $¥ω(2n) S =" S1 −1 ω(2n ` V¥y$¦§ # ¥ I ³§$!"O¦´$ ³§ +∞ » ͦ Ç$ ± «$ ¼±Ä" ³§ ¶ ±Ä³´"B! B«$! y³´"¼³´! R ¨ ` » ¥ " ¥ p " q "¼ ¦§ #¥ D¦Á±®! ¬Å " ω1 ω2 ® ¬ ¥ !y³{ q p lω1 (Z) ∩ lω2 (Z) » 1 ≤ p < +∞ 1 ≤ q < kxk = max{kxkω1 ,p , kxkω2 ,q } K·" RS õ ¥y! ƶ+! «·"¼³§ !«$!² ·ÈlT« !"¼³´!¥y) "« ³´ ¼±Ä!#" ) ¥ R+ "¼ ¦§¦§#¥ K(0) = 0 " ¥ x > 0. q± BÃÈy ¬ ¦§ » K ¥y" " D¦Á±&¶+!y« "¼³´! xp » ¥ 1 ≤ p < +∞ µ FS$K¶+ ! « " ³§! }$± ¥ « ¥ ) ¦´¥ K« ¬ ¦§³#¥ $}«$¬¬Å ± K·Èl$¬ ¦§ xp+sin(log(− log(x)) , p > √ ² $ ³ $!"S± ¥ ³{¦§ B«$!y ³´" ³§! B #¥ ³´ $ µYl³´" ¥ ! ³§ B ¥ Z µq_ ! 1+ 2 ω Yl³´" K » K(x) > 0 " n lK,ω (Z) = (x(n))n∈Z o X |x(n)| ∈C | K ω(n) < +∞, pour un t > 0 t n∈Z Z n o X |x(n)| kxk = inf t > 0 | K ω(n) ≤ 1 . t n∈Z C QC Ç$ ± «$ l (Z) »K± y §¦ :$ ± $« ~ AH ¦§³´" Ì1% ³§y Í» $ " ¥ !G ± « ~ s±Ä!± «¾½ ² $ ³ ± K,ω !" ! I½#¸ ¼" ½y°$ R ¨ " R^ ·µEqb¥ ¥y RSH ³¿"D²$ ³ ų´¦Q ¥ Ð" #¥ I¦ÍÇ!1± ³´" sup ¥ n∈Z sup ω(n + 1) < +∞ ω(n) ω(n − 1) < +∞. ω(n) qb¥ ¦§¥ ^³´!y¶+ ¬ ±Ä" ³§!yB ¥ ^«$ "z¸ &Ç$ ± « $¦§I¦§ « " $¥ ¥y"H & " $H% _ " µ{^ ¥ ¥y²!y ± ¦§³ #¥ $H¦´®JB½ ° ¬ÅÐyµ¿¨% ·"I± ³§!y ³ !"¼ ± ¥l"¼ ! ¥ lK,ω (Z) l"¼$!y! L¦Á±&¶+ ¬ÅÃÈ ±Ä« "¼D ¥1 $«·"¼ ¥1 ½y³´¶ "ªµ n∈Z Yl ³´" ` ¥y! ¥ ¿³ "¼ $ ¦§¦´ "¼ ¦§¦´ #¥ ! (q(n))n∈Z » a = (a(n))n∈Z ∈ CZ ?K!y ³§y$ ! ^¦ ±Ä«$ kak{q} q(n) ≥ 1 n » ¥ S"¼ ¥y" X a(n) = inf t > 0 | t n∈Z q(n) n∈Z µjq_¥ S¥ ! ® ¥ ³´" o ≤1 . l{q} (Z) = {a ∈ CZ | kak{q} < +∞}. ?SÇ$W"g¥ !Ð$ ± «$B s± !± «¾½j ± °$ Í» µ Uz¦y "g·²l³´ $!" #¥Ç³§¦² ³ ! ½¸ "¼½ ° R ¨ H " RS õEYl³ ¥ S −1 $W" | ! Ð ¥ l{q} (Z) ¦´JB½ $ °$¬yµ´¨ µ$W"I²Ä± ¦§± ¦´®± ! S µ l{q} (Z) ` FS ³´Ê! ! ± C ¦ÍÇ$ ± «$) ®¶+! «·"¼³§ ! &« !"¼³§!#¥ $» 2π  $ ³§ ³#¥ &·"Ð%~²Ä± ¦§ ¥ « ¬ ¦§·Èl H ¥ R µ [0,2π] q_¥y f ∈ C »|!Z!y "¼ fˆ ¦Á±Å ¥ ³´" ® H «$ «$³´$!"¼S ]¥ ³§ ^ [0,2π] µf qb ! ·" C = {fˆ | f ∈ C[0,2π] } µCQǽ#¸ "¼½y°$ RI¨ ^ "²¼± ³´®«ª±ÄS"¼¥l"¼®¶+! «·"¼³´!Z ¥y f ∈ C[0,2π] $W"D¦Á±Å¦´³§¬³´" I¥y! ´³ ¶+ ¬ÅI ¬Å¹¸ ! ! Ð?K$ ± )y ®¼±Æ $ ³´® ]¥y ³§ $µ|NS³´! ³ » kfˆk = kf k∞ , C[0,2π] !T± lim N →+∞ N k 1 X X ˆ f (n)eint − f (t) = 0. N + 1 k=0 n=−k C H$ ³ ! ^R ·µqb¥ A!:« ! "¼±Ä"¼ #¥ ¥y³$W"± ! α∈R ·" »!1± f ∈ C[0,2π] Z 2π 1 ˆ f (t)einα e−itn dt ψeiα (f )(n) = 2π 0 1 = 2π ψeiα (fˆ) Z 2π f (t + α)e−itn dt. "L¦Á±® ¥ ³´" Iy$O«$ «$³´$!"¼L I]¥ ³§$K D¦Á±®¶+! «·"¼³§ ! 0 t −→ f (t + α), µ Uz¦}$W"Ð ²³§ !" ¥y)¦Á±~! ¬ )¦ $ ±Ä"¼ ¥ ψ $W"Ð$ʱ ¦´Æ%¨µ Cgǽ¸ " ½ °$ RS $ "1 ! «Z² ³ $ ± C µUz¦H "T± ¥ ³H"¼ °$ƶ ± « ³§¦´~e ² ³§#¥ 2¦§ ¼±Ä" $¥ S " S −1 !"@"¼¥ g¦´$g $¥ÈÐ ! $µ^ " B"¼½y$ °$¬L ± ¦§³ #¥ B ! «O±Ä! ¦ÍÇ$ ± «$ C » ¥y³ ¥ "¼± !"®!± ±Ä® Æ±Ä y:Yl«¾½± ¥y $·"®!Ç$!"¼ ) ! « ± ® ± ! ®¦´ «ª± y$L$ ± «$ H s± !± «¾½T " ¥ ³§ ± Hu$¦´¦Á± µ L¥ O¶ ± ³§ ! L $¬Å± ¥y$L± ¥~¦´$«·"¼$¥y ¥yI ± ! C ³§¦ÃÈl³§ " D $O ¥y³´"¼ a "¼ ¦§¦§ ¥y C[0,2π] iα lim k k→+∞ k X a(n)en − ak 6= 0. >g!Ðj·"ª»³§¦y$ "gy³§$!Ы ! !#¥#¥Ç$!ÅÊ$!y$¼±Ä¦ ¥ ! B¶+! «·"¼³´!Ы$!"¼³´!#¥ B " 2π $ ³´ly³#¥ K!j " ± S¦§±Å¦´³§¬³´"¼¥y! ³´¶+ ¬ÅI ® ± ³´® ®]¥y ³§ $µ?K $!y± !"ª»¦´$^ ¥ ¿³ "¼$ ! ³§ S !" ³§ ! $! $L± ! C µ n=−k +` a bc Yl³´" E ∗ ¦ ± « ^y¥± ¦ E µ^ ¥ K±Ä¦§¦§ ! ! " $ k.k ¦§±D!y ¬O E ·" k.k ¦Á±D! ¬Å ∗ E ∗ µq_ ¥ y ∈ E ∗ " x ∈ E »j!Z¥l"¼³§¦´³§ ¦§±Å! "¼±Ä"¼³´!~¥y ¥ ¦§¦§ < x, y >:= y(x) µjq_¥ » ! < x, e >= x(−k) µgNS³§!y ³Í»_! ¥ ¥y² ! Å« ! ³§ $ #¥ e "Å¥ ! k ∈ Z $¦§ ¬Å !"O E ∗ µ|AH!: ¬ k ± #¥ ¥yI ³{! |||x||| = sup kψ (x)k » ! ¥ Okl"¼$!y! ¥ ! ®! ¬® ¥ E #¥ ³¿² ±Ä¦§$!" %ƦÁ± !y ¬ k.k µCg±ª²Ä± !"¾± Ê&z∈T¦Á± z !y ¬ |||.||| " #¥ ! ¥ L±ª²!y sup |||ψz (x)||| = 1, ∀z ∈ T. q± O«$ ! ¥y$!"ª»¼± ! $ " y Ê ! $ ± ¦§³¿"¼Ä» ! ¥y"Sy !±ª² ±Ä!#"^«$ ! ³´ $ $ ¥y ψ $W" ¥ ! ~³´ ¬ " ³´) E ± ! E » #¥ ¦ #¥ T ³¿" z ∈ T µ}^¥yƱ ¦´¦§! ¥y² $Ŧ§1¦§$¬z ¬ ¥ ³¿² ±Ä!#"$µ |||x|||=1 xyµ¿¨ * $# 1 X x(n) en − x = 0. k + 1 n=−p p k X lim k→+∞ c {]³¿Èl!y 0/ x ∈ E. p=0 µCE±¶+ ! « " ³§! x∈E Ψx : T 3 z −→ ψz (x) ∈ E "Æ« !"¼³´!¥y µ}>g!8 "ª» ¥ x ∈ F (Z) Á¦ ±2«$!"¼³´!#¥ ³´" T "Æ·²³§ !#" ~·" ¥ x ∈ E #¥ ¦§«$ ! #¥ :$¦§¦´Æ «$¥ ¦´Æ³§¬¬$ ³§±Ä"¼ ¬Å !#" ½¸ "¼½ ° R ¨ û RS ·" SR ·µL¥ ³´!#" ¥ ³´ ! B¦´$O! ¹¸±Ä¥lÈ1 I] $ (g ) ⊂ L1 (T) y ! ³´ ± O¦Á±&¶+ ¬®¥ ¦´ k k∈N it gk (e ) := k X p=0 1 X imt e k+1 ML± $¦´! #¥ I! ¥ FS ´¦ ¥ $ » ¥ ·" ¥ (k+1)t 1 sin( 2 ) 2 , ∀ t ∈ R. = k+1 sin 2t L±ª²!y kg k 1 = 1 » ¥y k ∈ N " k L (T) Z gk (eit )dt = 0, ∀δ > 0. lim » |n| ≤ k k→+∞ δ≤|t|≤π 1 gˆk (n) = 2π |n| > k, u « I%Ц§±«$!"¼³´!#¥ ³´" L¥ H±Ä¦§¦§ ! B $« µq_ ¥ m(T) = 1 |m|≤p Z π −π gk (eit )e−int dt = 1 − |n| k+1 gˆk (n) = 0. Ψx »! ¥ L±ª² ! lim k(gk ∗ Ψx )(1) − Ψx (1)k = 0. ³§ ds ± ¥~¦´³§ ¥1y dm(s) » m $W"S¦§±¬$ ¥ I RS±± B"¼¥y" n ∈ Z » !~± Z (gk ∗ Ψ)(1) (n) = gk (s)ψs−1 (x)ds (n) T Z = gk (s)s−n x(n)ds = gˆk (n)x(n). k→+∞ N^³§! ³Í» ! ¥ Oy"¼ ! ! (gk ∗ Ψx )(1) = O ¥ T " $¦´¦§ #¥ T k X n=−k X 1 X |n| x(n) en . 1− x(n)en = k+1 k + 1 n=−p p=0 k p >Q"¼± !"L ! ! #¥ Ψ (1) = x »« $«$³E« ¬ §¦ ° " H¦Á± $¥y² y¥T¦´$¬¬Åĵ ^¥ L± ¥y ! O|$ ³§!1x ± ¥ ³{ ¥1¦§ ¬Å¬H ¥ ³´Ä² ± !"ªµ !5 /,: , / /,: # B/ ! : < ! ( 796:; lµ < ) % * ,# x ∈ E E / M ∈ M(E) .8( ) . ) Mx : T 3 z −→ (ψz ◦ M ◦ ψz −1 )(x) ∈ E ! : < % c ]_³¿Èl! x ∈ F (Z) " M ∈ M(E) µUz¦ "L¶ ± «$³´¦§HyD² ³§ #¥ yµ c) ∗ x, ∀z ∈ T. Mx (z) = (ψz ◦ M ◦ ψz )(x) = ψz (M >g!1 "$» ¥ B¥ !1«$$W"¾± ³´! k ∈ N ± $ÌIʼ±Ä! »!1± −1 X c(n − p)z −p x(p), ∀n ∈ Z. M (ψz ◦ M ◦ ψz −1 )(x) (n) = z n ?K$«$³{³´¬ ¦§³#¥ ¥y ¥ " ¥y" |p|≤k x ∈ F (Z) »¦Á±&¶+ ! « " ³§! T 3 z −→ (ψz ◦ M ◦ ψz −1 )(x) ∈ E $ "@« !"¼³§!#¥ ĵqQ ! ± !"K !Ы$¬ "¼B¦§±H $!y ³´" O $ "L« !"¼³§!#¥ D T ± ! E » ¥ " ¥y" x ∈ E µ F (Z) ± ! E »! ¥ Q ¥ ³´ ! #¥ FH± ! ¦§± ¥ ³´" » ! ¥ K± ¦§¦´! @! ¼" M »l¦ÍÇ ¼±Ä" $¥ @ H« !²¦´¥y"¼³´! ± µ^ ¥ L¹± ² ! L ³´!: T ¥ ¦´$¬φ¬ÅS ¥ ³¿² ±Ä!#"$µ φ∗E ⊂E 1) ) 0 % / , # lµ < !5 /,: , / /,: # B/ ! : < ! ( E M ∈ M(E) . x ∈ E % Mx φ ∈ F (Z) »l ³ ) . ) lim kMk x − M xk = 0, ! 8 / # / 97 # ( ( k→+∞ k ∈ N . Mk Mk = k X p=0 p k X |n| c 1 X c n M (n) S = 1− M (n)S n . k + 1 n=−p k+1 n=−k ? ( ! . kM k k ≤ kM k, ∀k ∈ N % ! /! # / 4! S ! # M −1 c(n) = 0, $# S . . ! # / 4! S !' /! # M . . c(n) = 0, ,# n > 0 % 2) 3) ! n < 0. / # ! = ! ( S −1 c EAH!T± " ± B « ! #¥ $!" E µ}?K ! ± !"ª» ¦ ± "¼³´!¢¨ ® $« ¥ ¦§1³§¬¬$ ³§±Ä"¼ ¬Å !#"® T¦§± $! ³´" T ³´¦Q$ "&¬Å ³§! D ³´¬ ¦§Çy" $! ³´I¦´Å« !"¼¦§Ð ¦Á±:! ¬ÅÐ $¥y² Æ¥y" ³§¦§³´ ¦´$ ± Ê¥ ¬$!"¼D Q± ± " $I%)! " Ð ³¿"¼¥±Ä" ³§! #¥ ³Q "I|ª± ¥ « ¥ º ¦§¥yÊ$! ¼± ¦´ µ{L¥ ± ¦´¦§!yB! ¥ O ²³§L D! ¥l²ª±Ä¥~ ^! ¹¸± ¥lÈ1 ] $O ! ³´L ± ! L¦Á± $¥y² & ¥1¦§ ¬Å¬ $« $ !#"$µ]³ Èl! M ∈ M(E) µÏ?K ¬Å¬ ¥ {"¼¥y" x ∈ E »¹¦§±K¶+! «·"¼³§ ! Mx "b«$ !#" ³§!#¥ T ± ! E " M (1) = M x »|!y¥ L±ª²! |n| k+1 limk→+∞ 1 − c(n) = M c(n) » M ¥ n∈Z ± ! F (Z) µbCE± Mk x lim k(gk ∗ Mx )(1) − M xk = 0, ∀x ∈ E. µ>g!T"¼$! ± !"H«$ ¬ " y ©yµ´¨¨ ·» ¥ x ∈ F (Z) P~±Ä³§!"¼ !± !"ª» yÈl! !! k→+∞ Z k∈N gk (z)Mx (z −1 )dz T Z Z c) ∗ x)dz. = gk (z)ψz −1 (M ψz (x))dz = gk (z)(ψz −1 (M (gk ∗ Mx )(1) = ?K ¦Á±Ð$!"¼ ±ÄÎÁ! T T (gk ∗ Mx )(1) = q_± L± ³§¦´¦§$¥y $»l!: ¬ ± #¥ ¥y ¥ Z T c gk (z)ψz −1 (M )dz ∗ x. » !T± |n| ≤ k Z Z c c(n)dz gk (z)ψz −1 (M )dz (n) = gk (z)z −n M T ± ¦´ #¥ ¥ |n| > k T c(n), c(n) = 1 − |n| M = gbk (n)M k + 1 »y!~± Z CQÇ$ʱ ¦´³´" ³´¬ ´¦ ³#¥ #¥ T c)dz (n) = 0. gk (z)ψz −1 (M dk = M k X n=−k dk = M Z 1− |n| c M (n)en , k+1 c)dz . gk (z)ψz −1 (M P~±Ä³§!"¼ !± !"ª»³´¦ ²³´$!"O¶ ± «$³´¦§H D¬Å± T $ µ|A!1± kMk k Z d c) ∗ a)dz kMk ak = kMk ∗ ak = gk (z)(ψz −1 (M = Z T T gk (z)(ψz −1 ◦ M ◦ ψz )(a)dz »!y¥ S y"¼Ã ≤ "L¦Á±® ! ³¿"¼D Yl¥ ! #¥ ¦ÍÇ ¼±Ä" $¥ F (Z) ± !y S −1 Z T |gk (z)| kψz −1 kkM kkψz k kakdz ≤ kM kkak, ∀a ∈ F (Z) E ! ¥ O ! ! !Ç$ " ± b ! g"¾± !y ³§ #¥ Mk = $ " | ! Ä»{³§¦Q " «$¦Á±Ä³§ #¥ ! ¥ Lª± ²!y S k−1 Mk = 1 − ¥y kMk k ≤ kM k, ∀k ∈ N. k X S Mk " ! µË]³ Èl! k∈Z µ?K¬¬ |n| c M (n)S n k+1 "&± ¥ ³g¥ ! ¼±Ä" $¥ D| ! ĵb>Q"¼± !"&y! ! #¥ n=−k k−1 1− S k X k c |n| c M (−k)S −1 + 1− M (n)S n+k−1 , k+1 k + 1 n=−k+1 k X 1− |n| c M (n)S n+k−1 k+1 $ "K ! ^ " ¥y S −1 !Ç$ " ± } ! »! ¥yK«$ ! «$¦´¥ ! ¥y M c(−k) = 0. FSH¬ $¬ »#$! «$¬ ¼± !" M " S p » ¥ p = k − 2, k − 3, ...., 1 »l ! ¬ !"¼ #¥ M c(−n) = 0 » ¥ k µYl³ S −1 $ "Ð| ! Ʊ ¦´ #¥ S ! " ± ®| ! Ä» ±Ä&¦§Å¬ ¬Åż± ³´ !y! $¬$!"ª» n > 0 ! ¥ ¥y² ! O«$¬ ¦§·"¼ B¦Á± $¥y² & ¥1¦§ ¬Å¬ µ µ n=−k+1 lµ© < !5 /,: , / /,: # B/ ! : < ! ( < = ) % ! φ ∈/ -F (Z) ( φ! ∗E8⊂- E.! ! , 1 ! / !+ / ! 1) # ,# S S −1 ( # . ( # ( E e |φ(z)| ≤ kMφ k, ∀z ∈ Ω := 2) S ! /! # / 4 ! = . 3) S ! # / 4 ! = . n o 1 ≤ |z| ≤ ρ(S) . ρ(S −1 ) ! # ! φ ∈ F (Z− ) . n 1 o e |φ(z)| ≤ kMφ k, ∀z ∈ O := z ∈ C | |z| ≥ . ρ(S −1 ) S −1 ! /! # ! φ ∈ F (Z+ ) . n o e |φ(z)| ≤ kMφ k, ∀z ∈ U := z ∈ C | |z| ≤ ρ(S) . S −1 ) . ) / # ! / ( 6 > ! / ! c gYl¥ ! I ±Ä #¥ ·" S S −1 µFS$¥È1«ª± O $ $!"¼ !#"& spec(S) ¨µUz¦{ÃÈl³§ " D¥ ! D ¥ ³¿"¼ (h ) ⊂ E #¥ ³{² $ ³ !#"I"¼ ¥ D¦§$D ¥lȺ| ! $µbYl ³´" z ∈ p p∈N lim p→+∞ (S − zI)hp = 0 khp k = 1, ∀p ∈ N. lµjCQ $ ±Ä"¼ ¥ H±Ä ³§!"S ¥ ½ ³¿¶ " S ∗ ± ¬ " z ¥ ^²Ä± ´¦ $¥ ®·"H !y«®³§¦b·Èl³§W"¼ " $¦ #¥ S ∗y = zy µ y ∈ E ∗ \{0} Yl¥ ! #¥ I! ¥ O ¬Å¬$B±Ä! O¦§ $¬³§$K«ª± µjAH!: ¬ ±Ä #¥ #¥ !~± N^³§! ³ ¥y"¼¥y" lim kS k hp − z k hp k = 0, ∀k ∈ Z. p→+∞ φ ∈ F (Z) »¥ N >0 e kφ ∗ hp − φ(z)h pk ≤ L¥ L$!1 $ ¥y³§ ! N X ( sup |φ(k)|)kS k hp − z k hp k. k=−N |k|≤N e lim kφ ∗ hp − φ(z)h p k = 0. ?K¬¬ ³´¦$!1 $« ¥ ¦´ #¥ ± $ÌIÊ ± ! » !~± p→+∞ e e e |φ(z)| = kφ(z)h p k ≤ kφ(z)hp − φ ∗ hp k + kφ ∗ hp k, e |φ(z)| ≤ lim kφ̃(z)hp k ≤ kMφ k. P~±Ä³§!"¼ !± !"L ¥ ! ¥j!1$p→+∞ W"S ± ! O¦§ ¥lÈl³§° ¬Å«$± $µAH!1± Mφ∗ (y) = X φ(n)(S ∗ )n (y) n∈Z = ·"^³´¦$W"L ²³§ !" #¥ X φ(n)(z n y) = φ̃(z)y n∈Z L¥ L«$! « ¦§¥ ! #¥ ¥ " ¥y" |φ̃(z)| ≤ kMφ∗ k = kMφ k. φ ∈ F (Z) »!1± e |φ(z)| ≤ kMφ k, ∀z ∈ spec(S). lY ³O¥ ! Å $ ±Ä"¼ ¥ S " S −1 !Ç$W" ± Å| ! »Q¦§± ¥y²T$ "Ð" °$ ³´¬Å³´¦Á± ³´ ĵY³ S !Ç$W" ± } ! Ä»y± ¦´ # ¥ S −1 "K ! »! ¥yK¥y" ³§¦§³´ [email protected]¦´^ « " Sy S "K¦§[email protected]¬ $¬$ ± Ê¥ ¬$!"¼ µNS³´! ³{ ³ 1 = ρ(S) » ¦Á± $¥y² "^± «¾½ ·² µ ρ(S ) C −1 ML·²$!y! b± ¥D«ª± ED¦´${ $ ±Ä"¼ ¥ S " S −1 !"E"¼¥y{¦§${y$¥lÈD| ! ${·"_ ¥ ! ¥y 1 < ρ(S). ]_³¿Èl! φ ∈ F (Z) µYl³´$!" 0 < R1 < R2 y$¥lÈ $¦§I"¼ ¦§ ¥yƦ§ «$$ «$¦´$ρ(S C ) " C ±ª¸!T $ « " ³´¶ R " R ± ± W"¼³´$! ! !"H% spec(S) µ?K¬¬¦Á± 2 ¶+! « " ³§! φRe $W"½ R¦§¬ ½ B ¥ C\{0} " 1|φ(z)| » ¥ z ∈ C ∪ C »± ° e ≤ kMφ k R R ¦§ ³§! « ³ H ¥1¬ ±ËÈl³´¬®¥ ¬1»l!1y" ³§ !#" −1 1 2 1 2 n o e |φ(z)| ≤ kMφ k, ∀z ∈ ΩR1 ,R2 := z ∈ C | R1 ≤ |z| ≤ R2 . >Q"¼± !"L ! ! #¥ D¦§ O«$$ «$¦´$ Cρ(S) " C 1 ρ(S −1 ) !"L³§! « ¦§¥ B±Ä! spec(S) »!1± e |φ(z)| ≤ kMφ k, pour z ∈ Ω. >g!®" $!±Ä!#"«$¬ "¼K ¥®¶ ± ³´"#¥ B ³ φ ∈ F (Z−) »Ä¦Á±^¶+ ! « " ³§! z −→ φ(z e −1 ) $W"g½ ¦§ ¬Å ½ ¥ C " #¥ L ³ φ ∈ F (Z+) » φe $W"K½y¦§¬ ½ B ¥y C »#¦Á± $¥y² S ¥¦§ ¬Å¬B L«$ ¬ ¦´° " ¬ ¬Å O± Ê¥y¬Å !#" $µ ± B¦§ O c |Yl¥ ! #¥ S ·" S −1 !"@"¼¥y}¦´[email protected] $¥lÈ| ! $ µ Yl³´" M ∈ M(E) µl^¥yB± ¦§¦´! @¥y"¼³´¦§³´ [email protected]¦Á± ¥ ³¿"¼ (M ) « ! W"¼ ¥y³´"¼S± ! ¦§± $¥y² D ¥ k k∈N CE ¬Å¬yµ lµML± $¦´! #¥ c yµ lim kMk x − M xk = 0, ∀x ∈ E k→+∞ " k, ∀k ∈ N. DkM W" kk#¥ ≤ kM q_ M k = M φk . A! yÈl r ∈ [ 1 , ρ(S)] µF ρ(S −1 ) q_¥yH± ¦´¦§$Ê $B¦§ O! "¾±Ë"¼³§ ! ! φ = M dk , ¥ k ∈ N » k ¥ r > 0 " a = (a(n)) ∈ E »! " ! (a) (n) = a(n)rn µ r ± °$L¦§DC{ ¬Å¬yµÇ© » n∈Z ! ¥ S±ª² ! ] |(φ k )r (z)| ≤ kMφk k ≤ kM k, ∀z ∈ T, ∀k ∈ N. #¥ #¥ ³´" "¼º%5 $¬ ¦§± «$ (φ ± :¥ !yº ] k )r »Ä!®l"¼³§ !" #¥ k∈N «$!² Ê ] ] (φk )r (φk )r ^¥ : ! ¥ ´³ ! ¥j!!y "¼ ¼±± ¥ ³ ² $ ^¥ !yI¶k∈N +!y« "¼³´! σ(L∞ (T), L1 (T)) lim " k→+∞ Z T νr ¥ ´³ " 2«$!² !± ´¦ ¥ _¦Á±^" ¦§Ê §³ qQ¦§¥ k∈N $«$³´ ¬Å !#"$»!1± ∈ L∞ (T). ] (φ ) (z)g(z) − ν (z)g(z) dz = 0, ∀g ∈ L1 (T) k r r Uz¦{$W"^« ¦Á± ³´ #¥ ≤ kM k. kνr k∞ Z g g ] lim (φ ) (z) (a) (z)g(z) − ν (z) (a) (z)g(z) dz = 0, ∀g ∈ L2 (T), ∀a ∈ F (Z) k r r r r k→+∞ T ·" ± L«$ ! ¥y$!" ¥ a ∈ F (Z) » (φ «$ !#² $ Ê ¥ O¦Á±"¼ ¦´Ê³§H¶ ±Ä³§ ¦´ g ] k )r (a)r k∈N L2(T) ² $ ν (a) g . Cb±®" ¼± ! W¶+ ¬Å±Ä" ³§!Æy ]¥y ³§ B l2 (Z) ± !y L2 (T) ! ³´ ± r r F : l2 (Z) 3 (a(n))n∈Z −→ ã|T ∈ L2 (T) " ! «®¦Á± ¥ ³¿"¼ (M a) = (φ ) ∗ (a) φk r k r r « % yµ ·» !)" ¥y² k∈N ¦´Ê³§L¶ ± ³§y¦§H l2(Z) µ|u k∈N "D¥ ! &³§ ¬Å·"¼ ³´D ¥yL¦§±&"¼ νbr ∗ (a)r « !² Ê®² $ lim | < (Mφk a)r − (M a)r , b > | ≤ lim kMφk a − M ak k(b)r−1 k∗ = 0, ∀b ∈ F (Z). ?K ¦Á±Ð$!"¼ ±ÄÎÁ! k→+∞ k→+∞ (M a)r (n) = (νbr ∗ (a)r )(n), ∀n ∈ Z, ∀a ∈ F (Z). N^³§! ³Í» !:¬!"¼ #¥ c)r ∗ (a)r = νbr ∗ (a)r , ∀a ∈ F (Z) (M ·" c)r = νbr . (M q_± O«$! #¥ !#"$» !~± f(rz) = M X c(n)r n z n = M X νbr (n)z n = νr (z), ∀z ∈ T. ?K¬¬ kν k ≤ kM k »³§¦ !~ «$¥ ¦´ ¥y M f $W"S $!"¼³´$¦§¦´$¬$!"L ! ± r ∞ " ¥y"L«$ «$¦´D Ω µ|Yl³ ρ(S) = 1 »³´¦j$W"L ²³§ !" #¥ spec(S) = C = Ω µ n∈Z n∈Z kM k ¥y ρ(S) ρ(S −1 ) N ± W "¼³´® :¬ ±Ä³§!"¼ !± !"ª»_! ¥ Ð ¥ ! #¥ ρ(S) > 1 µ>Q"¼± !" ! ! #¥ ρ(S ) $ W @ " ¥ ! B y ¥ ´ ³ ¼ " B y ¶ + ! « " § ³ y ! ½ ´ ¦ Å ¬ ½ ¥ ! ¿ ³ + ¶ ¬ $ ¬ $! "Q| ! $ Q ¥ Ω◦ »± °$ f (φk )k∈N ¦´Ð" ½ $ °$¬Ð PT!"¼ ¦Í»{! ¥y ¥y² ! I$! ÃÈ"¼¼±Ä³§ Ð¥ !yÅ ¥ W ¥ ³¿"¼ #¥ ³g«$!² $ Ê ¥ ! ³  ¶+ ¬Å ¬Å !" ¥ D"¼¥y"I« ¬ ± « "D Ω◦ ² $ I¥ ! «$ "¼± ³§! ж+! «·"¼³´!2½ ¦§ ¬Å ½ ν µb^¥ ! Ä"¼! _«$·" "¼} ¥yWÂz ¥ ³´" K±Ä¥ ³ (f µËNS¦´ » ¥ r ∈] 1 , ρ(S)[ »Ï¦§±L ¥ ³´" ((φ ] φk )k∈N ρ(S ) « !²$ Ê&¥y! ³´¶+ ¬Å ¬Å !"B ¥ T ² $ ^¦§±&¶+! « " ³§! z −→ ν(rz) "O!y¥ Ly"¼ ! ! k )r )k∈N −1 −1 L¥ L«$! « ¦§¥ ! #¥ q_± O«$! #¥ !#"$» ν(rz) = νr (z). f(rz) » ν(rz) = M f(z) = ν(z) = M f $W"^½y¦§¬ M ¥ X n∈Z z∈T "L!1± ◦ c(n)z n , pour z ∈ Ω. M ½ H ¥ Ω◦ µ P~± ³§!" $!± !"$»¬Å !#" !y #¥ spec(S) = Ω µLYl¥ ! ¥j³´¦H·Èl³§W"¼ C ⊂ Ω "¼ ¦ r ¥y Cr !Ç$W" ± 1³§! « ¦§¥ )±Ä! spec(S) µOY³´" α ∈ Cr "¼$¦ #¥ α 6∈ spec(S) µBNS¦§ ·" ¥ r > 0 »{ ³ C ⊂ Ω »{³§¦ÃÈy³´ " ¥y! ¶+ ! « " ³§! ν ∈ L∞(T) (S − αI)−1 ∈ M(E) r r "¼$¦´¦§#¥ gr (z), ∀z ∈ T, ∀a ∈ F (Z). F ((S − αI)−1 a)r (z) = νr (z)(a) >g!1 ¬ ¦Á± $± !" a ±Ä (S − αI)a » !1" ¥y² gr (z) = νr (z)F ((S − αI)a)r (z) (a) gr (z), ∀z ∈ T, ∀a ∈ F (Z), = νr (z)(rz − α)(a) H" !± (rz − α)ν (z) = 1 µjA!Z± α = rz , z ∈ T µjqb¥ S" ¥y" > 0 »|³§¦E·Èl³§W"¼ z ∈ T "¼$¦ ¥y |rz − rz |r ≤ " |ν (z )| ≤ kν k0 µb0?K ¦Á±Æ³´¬ ¦´³#¥ ¥y 1 ≤ kν k "D ! ¥ ± ¥y"¼³´ ! K% ¥ ! 0S« !"¼ ± ³§«·"¼r³´! µ^¥yK$!r ∞ $ ¥y³§ ! #¥ C ⊂ spec(S), Ωr ⊂∞ spec(S) r "L ! « !~± spec(S) = Ω µ Yl³! ¥ @ ¥ ! #¥Ç¥ !Æ } $ ±Ä"¼ ¥ S ·" S −1 !Ç$ " ± g| ! Ä»! ¥ @y" $! ! ¦§$O¬ $¬$B $ ¥y¦´"¾±Ë"¼O$!1 ¬ ¦§± ª±Ä!#" Ω ± O ¥ U µqb¥ O¦§ O !y³´"¼³´! B O ·" U ¦´ ¦§$«·"¼ ¥ $¥y"B W"¼ O± ¥:C{$¬¬ÅSyµ©yµAH!) ¬ ± #¥ #¥ #¥± ! spec(S) = O »l! ¥ $ ¥y³§ ! ρ(S) = +∞ » ± ¦§ #¥ ¥ ± ! spec(S) = U »! ¥ L±ª²!y ρ(S −1 ) = +∞ µ ` #cÏa E c ` HF ± ! }«$ " "¼ ± " ³§Ä»!y¥ @ $¬!"¼ ! K¦´^JB½y$ °$¬Hyµ lµq± @¦§L¬ $¬L ± ³§ ! ! ¬Å !" ¥yI ± ! O¦Á± $¥y² ¥TC{$¬¬ÅDlµ´¨» ¥y"¼¥y" x ∈ E + »y!1" ¥l² lim k→+∞ k X n=0 n 1 X x(p)ep − x = 0. k + 1 p=0 Yl³ φ ∈ F (Z) $W""¼ ¦ #¥ P +(φ ∗ E +) ⊂ E + »! ¥ ! Ä"¼! T ¦ÍÇ ¼±Ä" $¥ ^ ¥ E + !y³ ± T x = P +(φ ∗ x), ¥ x ∈ F (Z+). >Q!x± ¦´³#¥± !"φ¦´$D¬ ¬Å ¬Å·"¼½ ¥yI« $¦´¦§$ByI¦φ §± $«·"¼³´! «$$y$!"¼Ä»! ¥ ^ y"¼ ! ! O¦§ $ ¥ ¦´"¼±Ä"O ¥ ³¿²Ä± !"ªµ{µ 8yµ ¯ < !5 /,: , / /,: # B/ ! : < ! (H1) (H2) ( . (H3) % 0 / 5 , 5 < ! < ! /+! <- (φ ) φ !B 8 1) T E + %&"( # ( # $ # , # n n∈N . n ( / # / 79 # ( ( n p E+ φn = # ( ! # # ! p=0 X 1 Tb(k)ek n + 1 k=−p lim kTφn x − T xk, ∀x ∈ E + , 2) 3) X n→+∞ kTφn k ≤ kT k, ∀n ∈ N. S S ! # / # !>= S !' /! # Tb(k) = 0 ,# k < 0 . . % !' /! . # ( / 4! S−1 ! # Tb(k) = 0. ,# k > 0 % −1 . . 8yµÀ < !5 /,: , / /,: # B/ ! : < ! ( (H3) S % S ! # ! ,# - φ ∈ F (Z) 6 !>/ ! 1) −1 . . E+ n e |φ(z)| ≤ kTφ k, ∀z ∈ Ω := z ∈ C | 2) S !' /! # / # > ! = . ( 3) S ! # / 4! . o 1 ≤ |z| ≤ ρ(S) . ρ(S−1 ) ! # , # . 6 ! / ! !' /! # $# 5 .n 6 ! / ! φ ∈ F (Z− ) . o n 1 e ≤ |z| . |φ(z)| ≤ kTφ k, ∀z ∈ V := z ∈ C | ρ(S−1 ) S−1 φ ∈ F (Z+ ) . o e |φ(z)| ≤ kTφ k, ∀z ∈ U := z ∈ C | |z| ≤ ρ(S) . c S−1 (H1) . (H2) ^¥ gÃÈ ! K $¥ ¦´$¬$!"@¦Á± $¥y² S D¨ õ#CE $¥l²$ S·"® S ¬ ¦§³´$!"D¦§ ¬ ¬Å ±Ä Ê¥ ¬$!" $µbYl¥ y ! #¥ S ·" S !"I ! $µ −1 Yl ³´" λ ∈ spec(S) ∩ (spec(S−1 ))−1 . ?K¬¬ λ ∈ spec(S) »³´¦jÃÈy³´ " I¥y! ¥ ³´" (f ) » f ∈ E + " $¦´¦§#¥ n n∈N n yµ© lim kSfn − λfn k = 0 et kfn k = 1, ∀n ∈ N n→+∞ ¥1 ³´$!:³§¦ÃÈl³§ " yµ ¯ ∗ a ∈ E + \{0}, S∗ a = λa. Yl³{!1± y µÇ© ·» ± ¦§ B!:y" ³§$!" yµÀ lim n→+∞ A!1± kSk fn − λk fn k = 0, ∀k ∈ N. lim kSk−1 fn − λ−k fn k n→+∞ ?K¬Å¬ yµ ≤ lim kSk−1 k|λ−k |kλk fn − Sk fn k = 0, ∀k ∈ N. n→+∞ λ −1 ∈ spec(S∗−1 ) » ³§¦ÃÈl³§ " D¥ ! I ¥ ³¿"¼ » (gn )n∈N gn ∈ E + ∗ #¥ ³² ³ lim kS∗−1 gn − λ−1 gn k∗ = 0 et kgn k∗ = 1, ∀n ∈ N n→+∞ ¥: ³§ !1³´¦·Èl³´ "¼ yµ b ∈ E + \{0}, (S∗−1 )∗ b = S−1 b = λ−1 b. Yl³ yµ } "L² ³ ±Ä¦§ $» !)" ¥y² yµ lim k(S∗ )k g − λk g k = 0 et lim k(S∗ )k g − λ−k g n n ∗ −1 n n k∗ = 0, ∀k ∈ N. A !T²¥ y¼± ³¿"D±ª²³´ lµ À O¥ lµ õjqb¥ H ±Ä ¥y $ #¥± ¥¬³§! L¥ ! y®«$ ³§·"¼ $ "^²¼± ³´I³´¦{ ¥ Ð"S &¬!"¼ $ ¥y lµ ¯·" lµ ÃÈy« ¦§¥ !"¬®¥y" ¥ $¦´¦§$¬$!"$µ|Yl¥ ! ¥y&! ¥yH±ª¸!yS ³´¬®¥ ¦¿"¾± ! ¬Å !" yµ ¯O·" yµ ·µYl³¿" a ∈ E +∗ \{0} " $¦ #¥ S∗ (a) = λa "L ³¿" b ∈ E +\{0} " $¦ #¥ S b = λ−1b µq_ ¥ u ∈ E +∗ ·" n ≥ 0 » ! n→+∞ n→+∞ −1 ?K¬Å¬ u(−n) =< en , u >=< Sn e0 , u >=< e0 , S∗n u > . F (Z+ ) $W"L $! ± ! a(−n) = λn a(0), n ≥ 0 " a 6= 0 ± uk ∈ F (Z+ ) »¦ ± ¦§³´«ª±Ä" ³§! u −→ (u(−n))n≥0 $ "L³´! $«·"¼³´² µAH!T± ?K¬Å¬ E+ ·" b(n)λn = b(0), n ≥ 0. b 6= 0 uk = »b! ¥y±ª²! k X n=0 » a(0) 6= 0 b(0) 6= 0 µqb¥ k ∈ N X n 1 X b(p)ep = 1− b(n)en . k + 1 p=0 k+1 n=0 n k » ! ³§ !y L¥ {"¼ ¥y² ! lim " ± {« ! #¥ $!" k→+∞ kuk −bk = 0 q_± L± ³§¦´¦§$¥y $»l!1± lim < uk , a >= lim k→+∞ k→+∞ k X n=0 1− limk→+∞ < uk , a >=< b, a > µ n b(n)a(−n) k+1 n −n λ b(0)λn a(0) k→+∞ k + 1 n=0 k + 1 a(0)b(0) = +∞. = lim "b! ¥ E« ! «$k→+∞ ¦´¥ ! #2¥ @¦§ ³§·"¼ yµ ¯j " lµ @ ¦´³§Ê±Ë"¼³´ $¬$!" yµÀ ¥ yµ õNS³´! ³ »! ¥ k X = lim 1− K? «$³#$W"± y ¥ g j !"b³§! « ¬ ±Ä"¼³´ ¦§ ·"K ! «O! ¥ }±ª²! ¥y² ! K± ¦§³ #¥ $Q¦´$ ¬ $¬$L± Ê¥ ¬$!"¼ #¥ I«$ ¥lÈ1 D¦Á± $¥y² y¥~C{$¬¬Dyµ À·"^!y¥ O$!1 $y¥ ³§ ! #¥ e |φ(λ)| ≤ kTφ k, ∀φ ∈ F (Z), ∀λ ∈ spec(S) ∩ (spec(S−1 ))−1 . >Q!~¥l"¼³§¦´³§ ± !"¦§ ³§!y«$³ ¥:¬Å±ËÈl³§¬®¥ ¬1»y!y¥ B"¼ ¥y² ! yµ´¨ª e |φ(λ)| ≤ kTφ k, ∀φ ∈ F (Z), ∀λ ∈ Ω. Yl³ S "| ! »b¬ ±Ä³§ S !j " ±Ä&| ! Ä» #¥ ¦ #¥ Æ ³´" λ ∈ spec(S) ³§¦g·Èl³§W"¼Æ¥y! ¥ ³´" (h ) » h ∈ E +−1"¼ ¦§¦´ #¥ lim " kh k = 1 ¥Zy³§$!~³§¦ n n∈N n n→+∞ kShn − λhn k = 0 ÃÈy³´ " c ∈ E +∗\{0} " $¦ ¥y S∗ c = λc µFS ¬ ¬Å #¥ ®± !yS¦Á± ¥y²n& ¥C{$¬¬Å yµÇ© » !1¬!"¼ #¥ e |φ(λ)| ≤ kTφ k, ∀φ ∈ F (Z+ ), ∀λ ∈ spec(S). $W"Ð ! Ä»! ¥y¥y"¼³´¦§³§ ! ¦§ « " : S Ð " ¦Á±T¬ ¬ÅƬ " ½ µFH± ! ¦´$ −1 ¥lÈ1«ª±Ä$»! ¥yL« ! « ¦§¥ !yL !T!y¥ O ²Ä± !S" y¥ ³§! « ³ H 1 ¥ ¬ ±ÏÈy³´¬®¥ ¬1µ Yl³ S−1 P~±Ä³§!"¼ !± !":! ¥ :±ª² ! Æ"¼ ¥ )¦§ Æ¥y" ³§¦§Å! $« $ ¼± ³´ $ ¥ ÆÃÈ $)¦§± ¥y²2 ¥ JB½ ° ¬Å yµ lµ c c DCE± $¥l² ¥l"¼³§¦´³§ Z¦§ 1¬ $¬$1± Ê¥ ¬$!"¼ #¥ º¦Á± $¥y² xy¥JB½ ° ¬Åºyµ¿¨µBLª± !y¬Å³´! $» #¥ $¦ ¥y$1¬Å ³ «ª±Ä" ³§! Å$dz§¬ $!":·":! ¥ ± ¦´¦§!y ! !y$L¦§$ ³§!y«$³ ± ¦§ B "¾± $µF&± | » ¥ ! #¥ S ·" S !"O"¼¥ O¦´$ ¥lÈÅ ! $µlYl³´$!" T ¥ ! $ ±Ä"¼ ¥ g LJ_ ¦´³´" ÌL ¥ E + " (φ ) ⊂−1F (Z) ¥ ! L ¥y³´"¼ k k∈N #¥ ³{² ³ ·" lim kTφk a − T ak = 0, ∀a ∈ E + k→+∞ kTφk k ≤ kT k, ∀k ∈ N. _q ¥ r > 0 " a ∈ E + »! Ä"¼! ±¹² ! OÊ «$I± ¥1§¦ ¬Å¬ $« $ !#" (a)r (n) = a(n)r n . ]³¿Èl! r ∈ [ ρ(S1−1 ) , ρ(S)] µQ^ ¥ ] |(φ k )r (z)| ≤ kTφk k ≤ kT k, ∀z ∈ T, ∀k ∈ N. ¦§)Ç·È" ¼± ³´ : (φ ¥ ! : ¥ W ¥ ³¿"¼ #¥ ] ) k r zU ¦B$ " ! « ³´ ³B«$ !#² $ Ê ¥ ¦§± "¼ ¦§ ʳ§ σ(L∞(T), L1 (T)) ² ~¥y! « $ "¼± k∈N ³§!yº¶+!y« "¼³´! ν ∈ L∞(T). qb¥ Z±Ä¦§¦§ Ê$ rµb?K$¦§±: !"¼¼±ËÎÁ! #¥ Ä» ¥y ¦§$!y "¾±Ä" ³§!yH«$·" "¼ ¥ W ¥ ³¿"¼ ¼±1± ¥ ³Q! " $ (φ ] k )r k∈N » « ! ² Ê ¥ ® § ¦ T ± ¼ " ´ ¦ Ê ´ ³ ж ±Ä³§ ¦´ L2 (T) ² ν (a) g g. ] a ∈ F (Z) (φk )r (a)r k∈N FS$ ³§Ê! ! ±Ä νb = (νb (n)) ¦Á± ¥ ³´" S «$ « ³§$!" ]¥ ³§ @y ν µyCE± " ¼± ! r W¶+Wr ¬ ±Ä" ³§!& O]¥ ³´r$_ "¾±Ä!#r"@¥ !yB³n∈Z§ ¬Å·"¼ ³´K l2(Z) ± ! L2 (T) » ¦Á±S ¥ ³´" (φr ) ∗ (a) k r r «$!² ÊH² $ νb ∗ (a) ¥yK¦Á±D"¼ ¦´Ê³§B¶ ± ³´ ¦§L l2 (Z) µyF&¥y!:± ¥y"¼ S« "¼Ä» T ak∈N φ k∈N «$!² ÊD² Tra ¥ rO¦Á±&" ¦§Ê ³§Sy E + µq± O« ! #¥ $!"ª»|! ¥y^±ª² ! k lim |(Tφk a)r (n) − (T a)r (n)| k→+∞ ≤ lim r n |Tφk (a)(n) − T (a)(n)| = 0, ∀n ∈ N, ∀a ∈ F (Z+ ). A!:« ! «$¦´¥y" #¥ ?K¬Å¬ n→+∞ (T a)r = P + (νbr ∗ (a)r ), ∀a ∈ F (Z+ ). (T a)r = P + ((Tb ∗ a)r ), ∀a ∈ F (Z+ ), !6y"¼³´$!" Tb(n)rn = νb (n), ∀n ∈ Z. CgÇ$W"¼³´¬ ±Ä" ³§! kν k ≤ kT k ³§¬ §¦ ³ ¥y#¥ :¦Á± ¶+! « " ³§! Te $ "L !"¼³§ ¦§¦´r$¬$!"O ! ± kT k ¥ O«¾½±#r¥ I«∞$$ «$¦´I³´! «$¦´¥ ı ! Ω µ Yl³j! ¥ ¥ ! # ¥ ρ(S) > 1 »l ¬ $¬ # ¥ ± ! ¦§± ¥y²y¥)JB½ $ °$¬ ρ(S ) y µ´¨ » !: $ ¬!"¼ ¥y ¦§& ± ¶+! « " ³§! Te $ "L½ ¦§¬ ½ H y¥ Ω◦ µ −1 g> !~ $¬ ¦§± ª± !" Ω ³¿" ± U ´³ " ± V "S$!Z & $ ²Ä± !"y$H¬ $¬$^± Ê ¥ ¬Å !"¼$» !:y"¼³´$!"O¦§ O $ ¥ ¦´"¼±Ä"¼B ³ S ¥ S ! Ç$ " ±ÄL| ! µ −1 ! ?RLNHqQUzJOM^> © ^fzild Hf\d jkilh \f h5h Hk jh ^h 3 k ^k S h ¾e |ild\e e h[f\e [email protected]\h h}i e H k |i k { fWd\h a `b` Ec c b` a `_a HA !) ³¿" #¥Ç¥ !:Ê ¥ L"¼ ¦´Ê³#¥ L± |$¦´³§ ! G $W"B¦§«ª± ¦´$¬$!"}«$¬ ± «·" ³ G !Æ"¾± !" #¥Ç$ ± «$^"¼ ¦§ ʳ#¥ "}¦§«ª± ¦´$¬$!"g«$ ¬ ±Ä« "ª»« #¥ ³ ¥y³´²Ä± ¥y"}± ¥¶ ± ³´" #¥ L¦ÍÇ¥ ! ³¿"¼B ° O¥ !²³´ ³§! ± ÊK« ¬ ± « "$µC{BÊ ¥ B ¥± ¦l¥y!Ê ¥ Lu¢¦´l«$± ¦§ Å ¬ !"« ¬ ± « " G ± |$¦´³§ ! Cb?KN ·»³ µÄµ ¦ÍÇ$!y $¬& ¦§ O¬Å ½ ³´ ¬$B«$ !#" ³§!#¥ O G ±Ä! T =: {z ∈ C, | |z| = 1}, ! Ä "¼ Gb $ "I±Ä¥ ³¥ !Ê ¥ Cb?KN ¥ H¦Á±"¼ ¦§ ʳ§I ®¦§± «$!² $ Ê $! « Å¥ !y³´¶+ ¬ÅI ¥ " ¥y"D«$ ¬ ±Ä« "ªµq± ÃÈy ¬ ¦§ »j!º± Rb = R » Zb = T ·"D¦§Ê ¥ ® ¥±Ä¦Q ®"¼¥l"DÊ ¥ ³´ « "@ "}«$¬ ± «·"ªµFS ¦´¥ $» !ű Gbb = G " G\ c c ³ G1 » G2 ·" G !#" 1 × G 2 = G1 × G2 , ^Ê ¥ $LCb?KN&µL¥ L±ª²!y^¦´H"¼½ ° ¬Å DW"¼ ¥y« "¼¥y I ¥ ³¿² ±Ä!#" ²³´ µ¿¨¨¹ õ G© µ¿¨ ) # / : / : " /,: G ! , /+79 # / # = : ;( ( ! ( ! # +: " /,: H 5 = ( G /H Rp × G 1 . G1 ( 1 ! < 4! : # < ) # % / : / : " /,:" : 621$ ! , / 79 # Rp × H : " ( /,:( ( : 621$ ( . ! H # % AH!®! Ä"¼ C(G) µ C (G) E¦ÍÇ$ ± «$} $_¶+! « " ³§!y_« !"¼³§!#¥ ¥y G $ µ ¦ÍÇ$ ± « &¶+! «·"¼³´! &« !"¼³´!¥y$® ¥ 0 G #¥ ³K« !² Ê !#"² $ 0 %~¦ ³´! ! ³·»{±ª² «)¦Á±1«$!² !"¼³§ ! ³ " «$¬ ± «·" ·µbAH!º! ¬ ± ¦´³§ ®¦§ ¬Å ¥ $&yÅR^±± dx ¥ G " C0 (G) = C(G) G ¥ Gb «Ä "S%Ðy³§ ¦´$O¬$ ¥y $O³´!#²Ä± ³Á± !" $ ± " ¼± ! ¦Á±Ä" ³§! } ¶ ± $ !T%Ðy" $! ³´B¦Á± dχ ¶+ ¬®¥ ¦´³´!²$ ³´!Ty ]¥y ³§ ¥ O¦Á±®¶+ ¬ g(x) = Z b G F (g)(χ)χ(x)dχ, p.p. sur G, ¥y b » g ∈ L1 (G), F (g) ∈ L1 (G) b F : L1 (G) −→ C0 (G) $ "L ! ´³ ± B¦Á±®¶+ ¬®¥ ¦§S¥ ¥ ¦§¦´ F (g)(χ) = Z g(x)χ−1 (x)dx, p.p. ∀g ∈ L1 (G). HF ± ! Å «$1«ª± ŦÁ±Z"¼ ± ! ¶+ ¬Å±Ä"¼³´!6 ~]¥y ³§ F : L2(G) −→ L2 (G) b "Æ¥y! ³´"¼± ³§ µQCE± "¼¼±Ä! ¶+ ¬Å I]¥ ³´$j¥ !yI¶+!y« "¼³´! f ¼ ±Ð ¥y² $!"^! "¼ fˆµ G A!:! Ä"¼ K Cc (G) ¦ ! ¬® ¦§y$O ¥yWÂz ! ¬® ¦´$L«$¬ ± «·"¼B A!1± ` µq_¥y K∈K » ! CK (G) = {f ∈ C(G) | supp(f ) ⊂ K}. CK (G) ≈ {f ∈ C(K) | f |F r(K) = 0}, y$ ³´Ê! D¦Á±&¶+ !"¼³´°$ y F r(K) " G K µqb ! Cc (G) = ∪K∈K CK (G) kf k∞ = max |f (x)|, ∀f ∈ Cc (G). NS¦§ (C (G), k.k ) "L¥ ! I± ¦´Ê°$yx∈G Ds±Ä!± «¾½ ¥y"¼¥y" K ∈ K µAH!T¬&¥ ! ³´" C (G) K ∞ ®¼±Æ" ¦§Ê³´D³§! ¥y« "¼³¿²®¦´l«$± ¦§ ¬Å !"S« !²·Èl!±Ä" ¥ ¦§¦§Ä»«Ä "I%Æ ³´ ¦Á±Å" ¦§cÊ ³§ ¦§«ª± ¦´$¬$!"« !²·Èl ¦Á± ¦§¥y ! ¥ O¦Á±#¥ ¦§¦§S¦Ídz§! « "¼³´! $ "&«$!" ³§!#¥ ¥ I"¼¥l" ± ¦´³§«ª±Ë"¼³§ !Ʀ§³´! ª ± ³´ iK : CK (G) −→ Cc (G) K ∈ K µQ?K·" "¼"¼ ¦´Ê³§ "«$± ¼±Ä« "¼ ³´ $ ± I¦§Ð¶ ± ³´" #¥Ç¥ ! φ : Cc (G) −→ F ± ! ¥y!®$ ± «$² $«·"¼ ³§$¦ F ¦´l«$± ¦§ ¬Å !"« !²·ÈlB "Q« !"¼³§!#¥ K ³y·"Q ¥ ¦§ ¬Å !"Q ³ φ| C (G) $ "L« !"¼³§!#¥ ¥ " ¥y" K ∈ K µ M^± y ¦§! #¥Ç¥ ! ± " ³§ B j¥ !x$ ± «$Ų$«·"¼ ³§ ¦Q"¼ ¦§ ʳ#¥ E $W"& ³¿"¼Ð| ! Å ³ ¥yD" ¥y"D² ³§ ³´!± Ê V 0 ³´¦Q·Èl³§W"¼ λ > 0 "¼ ¦ #¥ λB ⊂ V µY³ E ·" F !"&y$¥lÈ $ ± «$ L² $« " ³´$¦´O" ¦§Ê ³#¥ $ »y!: ³¿" #¥¥y! &± ¦§³´«ª±Ä" ³§! ¦§³§!yª± ³´ K φ : E −→ F " ! $Ð ³ φ(B) $W" ¥y!x| !yy F ¥yH"¼¥y"D| ! B y E µJb¥y"¼± y ¦§³´«ª±Ä" ³§! ¦´³§! $± ³§ φ «$!" ³§!#¥ H E ± ! F $W"B ! S¬Å± ³§@¦Á±I $« ³ #¥ ^$W"B$!ÆÊ $! ¼± ¦|¶ ± ¥ µ FSS¬ ¬ÅL ³ φ : E −→ F $W"B«$ !#" ³§!#¥ Ä» ± ¦´ φ "B #¥ !#" ³§ ¦§¦§ ¬Å !"«$ !#" ³§!#¥ Ä»l¬Å± ³§@¦Á± $«$³ #¥ I$ "^¶ ± ¥ I$!~Ê $! ¼± ¦ µ|?K $! ±Ä!#"S«$ S ¥lÈT $« ³ #¥ ^ !"^" ³¿²l³§± ¦§ ¬Å !" ²¼±Ä³§$O ³ E "L! ¬ÅÄ» «$ #¥ ³{ ! ! I¦§ ¥ ¦¿"¾±Ä"Oy³§$!:« ! !#¥T ¥ ³´²Ä± !"$µ 6 < º© µ¿¨ < E !5 /,: : / : 21$ ! < φ : C (G) −→ E 6 / : / ; < / # # ! ! : <;( !"( ! / 5 ! ! . = / 5 ! c %6&"( ( ( % !( # ( φ ) % : < ! 8 ! = ) φ ! : < (<( % ) φ % c jAH!Æ ± ³´" #¥ \¨ g·" Í g !" «$!y #¥ ! «$ ^y ·µlYl¥ ! #¥ \¨ "@²$ ³ ĵlYl³´" K ∈ K µ?K¬¬ i : C (G) −→ C (G) "}«$!" ³§!#¥ »# ¦§¦§B "@| ! µ FS ! « φ ◦ i "| !y$® ! «®«$ !#" K³§!#¥ K ¥ ^"¼¥y" Kc " φ $ "H«$!"¼³´!#¥ µjFS®¬ $¬ »| ³ O "^²$ ³K »± ¦§ φ ◦ i $ "^ #¥ $!"¼³´$¦´¦§$¬$!"S«$!"¼³´!#¥ »| ! «I«$ !#" ³§!#¥ ¥yL" ¥y" K " ^ " $ « ! ¼ " ´ ³ # ! ¥ µ K∈K φ AH!: ¬ ± #¥ #¥ ± ! B¦§± ³´" ³§!Å«$³¿Â $ ¥ »!)! « ! ³´ °$ I% ³´ ³ #¥ D $ | !y$ ± W"¼³§« ¥ ¦§³´$ } C (G) »y«Ä "^%® ³´ ¦§ $!y $¬& ¦§ Ly¦Á± ¶+ ¬Å i (B) »y B $ " ¥ !1| ! y C (G) µ FSD¬c ¬ÅÄ»y!1! «$ ! ³´ °$ % ³§ ³ #¥ D L ¥ ³´" $KO«$ !#² $ Ê$!"¼ K ±Ä "¼³´«$¥ ¦´³§° $K ¦§ ¬Å !#" B Cc(G) » « Ç$W"S%®y³§ ¦´$ ¥ ³´" $O ¦§±&¶+ ¬ (iK (fn))n≥1 » $W"&¥y! ¥ ³´" Å« !²$ Ê$!" ) ¦§ ¬Å !#" & C (G) ±ª² « K ∈ K µ_q_¥ ¦§Ð«$ !y¶+ " (fn ) ¥)¦´$«·"¼$¥yK! ¥yB± ¦§¦´! @¬Å !#" K± !y¦Í± ! ! ÃÈl&¨#K¥ H¦´$| ! $} C (G) "¦´$} ¥ ³¿"¼$ « !²$ Ê$!" $ZÇ$¦´$¬$!"¼: C (G) !"1 2¦Á± ¶+ ¬Å« ³¿Âzy$ ¥ $µ^?K cT $ ¥y¦´"¾±Ë"¼) !#" « $ "¼± ³§!y$¬$!" ³§ ! « ! !#¥ »@¬Å± c³´! ¥ Ð!±ª²! ± ¥6"¼ ¥y² $Ð : ·¶+$ ! «$ $«$³´ ±Ä! ^¦§±®¦§³¿" "¼ ¼±Ä" ¥ ¥yB¦§$OÊ ¥ $B! ! σ «$¬ ± «·"¼ µ Yl ³´" ya `bc a G`_c ¦ÍÇ$ ± « $¶+! «·"¼³´! H¬$ ¥ ¼± y¦§$D% IJ ± ¦´$¥ «$¬ ¦´·Èl$ ¥ G " $¦§¦´$ #¥ f |K ³´"г´!#" $Ê ± ¦´ ¥ ЦÁ±Z¬$ ¥y 1 1RS±± ¥ м" ¥y"Ы ¬ ± « " K ⊂ G µgq_¥ »y ³¿" S ¦ $ ±Ä"¼ ¥ B ! ³E ¥ L1 (G) ± x∈G x loc qb¥ L1loc (G) b »!:! χ∈G " Sx f (y) = f (y − x), p.p. Γχ ¦ÍÇ $ ±Ä"¼ ¥ L1loc (G) 3 f −→ χf. 2C Yl³´" E ⊂ L1loc (G) ¥ !1$ ± « Is±Ä!± «¾½» "O ¥ ! #¥ I¦Í± ¦§³§«$±Ä"¼³´! ³´ $!" ³´"¼ i : E −→ L1loc (G) ³¿"Ы$!"¼³´!#¥ µ@F&± °$ ¦´)"¼½ ° ¬Å: ¥ ʼ± ½ )¶+$ ¬: ³ S (E) ⊂ E » ¥ x ∈ G » ¦ÍÇ ¼±Ä" $¥ S $W"S| !yIy E ± ! E µFS&¬ $¬ » ³ Γ (E)x ⊂ E »|±ª² « χ ∈ Gb »¦ ༱Ä" $¥ Γ $W"x¥ !Z ¼±Ä" $¥ ^ ! I E ± ! E µ^¥ Sχ± ¦§¦´! O«$ ! ³´ $ $Sy$H ± « $ Is± !± «¾χ½ E ¼±Ë"¼³§W¶ ± ³§ ± !"O¦§ B«$! y³´"¼³´! B ¥ ³´²Ä± !"¼ D R ¨ $! ± ⊂ E ⊂ L1loc (G) ! C c(G) µ E RH Hq_¥ " ¥y" µ K⊂G RSq_¥ B" ¥y" »¦´$® ¥lÈ6³´! «$¦´¥ ³´! &·"¾± !"Ы !"¼³§!#¥ $»·" » ·" x ∈ G Sx (E) ⊂ E b » Γχ (E) ⊂ E χ∈G ·" supx∈K kSx k < +∞ Cc (G) " » ¥ H" ¥y"D«$ ¬ ±Ä« " supχ∈Gb kΓχ k < +∞. _q ! |||f ||| = sup kΓ f k, ¥ f ∈ E µCE± ! ¬Å |||.||| $W"H ¥y³´²Ä± ¦§ !"¼®%ŦÁ± ! ¬H E ·"O¼± ! $Wχ∈ "¼Gb DÊχ ! $ ± ¦§³¿"¼ »! $¥y"L«$ ! ³´ $ $B± !yL¦§±® ¥ ³¿"¼#¥ Γ $W" χ ¥ ! D³§ ¬Å·"¼ ³§H ¥ E ¥yO" ¥y" χ ∈ Gb µ ^ " ! #¥ ~ ³S« ! ³¿"¼³´! RHÐ$ " ²$ ³ » ¥y #¥ ¦Ídz§!y«$¦§¥y ³§ ! E ⊂ L1 (G) ³¿" «$!"¼³´!#¥ D³§¦ ¥ Ð" #¥Ç³§¦j·Èl³§W"¼D¥ !:²³´ ³§! ± Ê«$ ¬ ±Ä« " K yI¦ ¥y! ³´" S"¼ ¦ ¥y loc Z © µ¿¨ |f (x)|dx ≤ CK kf kE , ∀f ∈ Cc (G), ±¹² $« C > 0 ¦§±Z«$! y³´"¼³´!5$W" ·²³§ ¬Å¬$!"Ð! «$$ ¼± ³´ ·µ}>g! "¥ ! ± Ê¥ ¬$!" $! ³´" &K¬!"¼ #¥ ® ³b¦Ídz§!y$ʱ ¦´³´" «$³¿Â $ ¥ S$ "S²$ ³ ¥ ^" ¥y" f ∈ C (G) »| ¦§¦´ " ²l ± ³§ ¥ E µ|Yl ³´" V ¥ !:² ³§ ³´!± ÊyI¦ ¥y! ³´" H«$¬ ± «·"B² ³ ± !" +© µ¿¨ ·µ c q_¥ L"¼¥l" K « ¬ ± « "^ G »³§¦EÃÈy³´ " x , ..., x ∈ K "¼ ¦§ ¥y K ⊂ ∪ x + V 1 k 1≤j≤k j "L!1± ¥yO" ¥y" f ∈ E » K Z = X Z 1≤j≤k V K |f (x)|dx ≤ |f (s + xj )|ds = X Z 1≤j≤k < 6 © µ¿¨ / (<( V X Z 1≤j≤k xj +V |f (x)|dx |(S−xj f )(s)|ds ≤ CV ( ( : / 5 $ # ! ,# M : E −→ E E X 1≤j≤k kS−xj k kf kE . 5 # / 5 , # < / # # ( 5 = ( Sx M = M Sx , ∀x ∈ G. / ( # >, ! ( ; ( : / 5 $# ! ! ,# ! # /6 M(E) % Yl ³´" B(E) ¦ ±Ä¦§Ê° ¶+$ ¬$) ! Ê ! ± {S } . AH!6 ³§Ê ! $ ± ± Ab»¦ !l $¬® ¦´ &«$± ¼± «·"¼° $&¥y! Ʊ ¦´Ê°$y Æs±Ä!± «¾½ Ax µ_x∈G AH! ! Ä"¼ σ({S } ) ¦§ $« " x x∈G ³§¬®¥ ¦¿"¾± !y D¦Á±&¶ ± ¬³§¦´¦§ {S } ! ³ ±Ä x x∈G E \ σ({Sx }x∈G ) = {(γ(Sx ))x∈G , γ ∈ B(E)}. FH±Ä! H¦§I«ª±Ä ± W"¼³´«$¥ ¦´³§$ E = Lp(G) » ¥ 1 ≤ p < ∞ »³§¦E$W"H ³´$!T«$! !#¥ #¥ ¥ B"¼ ¥y" M ∈ M(E) » ³´¦j·Èl³´ "¼ hM ∈ L∞(G) b " $¦ #¥ khM k∞ ≤ kM k "O !~± +© µ d M f = hM fb, ∀f ∈ Cc (G). CE±~¶+! «·"¼³´! h $ "± ¦§ Ʀ§ÆW¸¬® ¦§Æ M ·"¦Í± ¦´³§«$±Ä"¼³´! M −→ h $W"Ð¥ ! M M ³´ ¬ " ³§S M(E) ¥ L∞ (G) b ³ E = L2 (G) µ Yl ³´" Ge ¦ÍÇ$!y $¬& ¦§ S¬ ½ ³§ ¬Å B«$!"¼³´!#¥ O G ± ! C∗ "^ ³¿" Gf+ ¦ÍÇ$! $¬®y¦§ Ŭ ½ ³§ ¬Å «$!" ³§!#¥ G ± ! R+ = [0, +∞[. A! ¬®¥ !y³´" Ge 1¦Á±~" ¦§Ê³´ «$± ¼± «·"¼ ³§ $ ± B¦§±®± I ² ³§ ³´!± Ê B D¦ÍÇ¥ ! ³¿"¼® e | θ −1 (K) ⊂ V }. W (V, K) = {θ ∈ G Uz« ³ V " K ± «$¥ $!"^ $ « " ³´² ¬Å !"¦ÍÇ$! $¬®y¦§D $L²³´ ³´!± Ê ^ I¦ÍÇ¥ ! ³¿"¼D C∗ " ¦ ! ¬® ¦§y$B²³´ ³´!± Ê B«$¬ ± «·"¼B ¦ ¥ !y³´"¼S G µyUz¦$W"L¶ ±Ä«$³§¦´S H² ³§ #¥ Ge ¬&¥ ! ³ «$·" "¼ "¼ ¦§ ʳ§ "H¥y!ZÊ ¥ D"¼ ¦´Ê³ ¥y µ|AH!~ $¬Å± #¥ ¥y ¥ f ∈ C (G) " c e »¦Ídz§!" $Ê ± ¦§ θ∈G Z f (x)θ −1 (x)dx ! G f˜(θ) = R f (x)θ−1 (x)dx » G "B ³´$!Å !y³§ µyq_¥y f ∈ C (G) » ¥[email protected]" ¥y" θ ∈ Ge µ c AH!& $¬Å± #¥ #¥ f˜| = fˆ " f˜ " \¦Á±" ¼± ! W¶+ ¬$g }]¥ ³§ {Ê$! ¼± ¦´³§ $ Q f ! ³§ ¥ Ge µAH!~²Ä± $«¾½ Gb«¾½ ¥ !Z ¥ WÂ$ ± «$ Gf Ge " $¦ ¥y ¥ ^" ¥y" M ∈ M(E) " E ¥ L"¼¥y" f ∈ Cc(G) ¦§±¶+!y« "¼³´! (M f )θ−1 ± ±Ä "¼³´$! !y®% L2 (G) ¥y^"¼ ¥y" θ ∈ GfE µ ?K «$³ ¬ " "¼¼±B @ ! ³§{[email protected]¬Å± ! ³´°$ Q!±Ä"¼¥y $¦´¦§¦§± z"¼ ± ! W¶+ ¬$Q @]¥ ³§ Ê$!y$¼±Ä¦§³§ $ M f ¥ Gf µ>Q!"¼ !± !"Q« ¬ "¼ g± Ê ¥ ¬Å !"¼ ·" ͻĥ !«ª±Ä! ³§ ±Ä"!±Ë"¼¥ $¦ "^¦ÍÇ$! $¬® ¦´E Mf ∈ M(E) fE = {θ ∈ G e | |f˜(θ)| ≤ kMf k, ∀f ∈ Cc (G)}, G $W"^¦ ¼±Ä" $¥ B D« !²¦´¥y"¼³´! E 3 g −→ f ∗ g §³ K¦´H ¥y"O DY$« " ³§!1 ·µ L¥ O± ¦´¦§[email protected]² ³§¥y¦´"¼ ³´$¥ $¬$!" #¥ Gf ! " ±Ä²l³´ µ Uz¦B$ "·²l³´ $!" #¥ χGf = Gf ¥ ®" ¥y" χ ∈ Gb µgL "¼ ! ¥y1 ³ GE $W"Å¥y! Ê ¥ «$¬ ± «·"ª»±Ä¦§ ®¦Ídz§¬Å± ÊE |θ|E$W"Ц§Æ ¥ \ÂzÊ ¥ )«$¬ ± «·"®"¼ ³´²³Á±Ä¦} R+ ¥y&"¼¥l" fE "} ! « G fE = G b µyA!Å ¬ ± #¥ #¥ ^ ³ x1 , ..., xn ∈ G !" z³§! ! ± !" ¾»#³ µÄµ θ∈G ³y¦´}W¸ "¼° ¬Å χ(x ) = » 1 ≤ i ≤ n ±S¥y! ¦§¥y" ³§! χ ± ! Gb ¥y_"¼ ¥y" ( , ..., ) ∈ Tn 1 n ± ¦§ ¦ÍÇ$!y $¬& ¦§ i i +² fE } {(θ(x1 ), ..., θ(xn )), θ ∈ G l" ²³´_¦ ± !y! ·ÈlO^yK«$K«¾½± ³¿"¼ $ "¥ ! ¬Å± ³´! g BMO$³§!y½± Uz¦{$ "L« ¦Á± ³´ #¥ I ³ θ ∈ Gf »± ¦§ ·µÄq_ ! E f+ = G fE ∩ G f+ µ G E |θ| : G 3 x −→ |θ(x)| ± ± W"¼³´$!"L% Gf+ µ Uz¦{ "L«$¦§± ³§ ¥y Gf = Gf+ Gb µ E E E < <© µ # / = ! ( : 2 $ 1 " ! / 9 7 6 : ; ∗ ! R ( ( C , 796:; > ! ! $ # G / , # ! / ! ( G 3 x −→ φ(x)λ ψ(x)1−λ ∈ R / / # C ,# - λ ∈ [0, 1] % ^¥ ¥y² $ ! ^¦´H"¼½ ° ¬Å ¥ ³´²Ä± !"$µ )©yµ < E !5 /,: , / /,: # B/ ! ! ! ) ) ( . ) % ! f+ [email protected] , : " /,: : 21$ i) G ( . ( % E ! / ! ii) © µ f = {θ ∈ G e | |θ −1 (x)| ≤ sup η(x)−1 , ∀x ∈ G}. G E A!)²Ä±²³´ ¥y #¥±Ä! g η∈G E $ "L¥ !1Ê ¥ ³§ «$ ·"L!~± G © µÇ© fE = {θ ∈ G e | |θ −1 (x)| ≤ ρ(Sx ), ∀x ∈ G}, G $« #¥ ³B$W" ±Ä¥ ³B ²³§ !" #¥± !y G $ "« ¬ ± « "$µQ^¥y« ! « "¼¥y ! #¥ :¦Á±~¶+ ¬®¥ ¦´ © µÇ© K$W"O²l ± ³§ ¥ " ¥y"LÊ ¥ DC_?KN µ Yl³´" U ¥ !1¥y² "^ $ ±y³´"¼D±Ä!± ¦´¸#" ³#¥ ¥y Cp U µX^! ¶+! «·"¼³´! e Π : U 3 λ −→ Π(λ) ∈ G » ³ ¥y"¼¥y" x∈G » ¦Á±®¶+! «·"¼³´! U 3 λ −→ Π(λ)(x) ∈ C "}± !± ¦¿¸#"¼³#¥ } ¥ U µ#AH!Ð! " d ¦Á±H¬$ ¥ y³§ « °·"¼B ¥y Gf+ µL¥ Ql"¼$!y! Q¦§K $ ¥ ¦´"¼±Ä" E ¥ ³´²Ä± !"ªµ )© µ < E !5 /,: , / /,: # B/ ! ! ! ) ) ( . ) % θ ∈ Gf * ,# 5 f ∈ C (G) (M f )θ−1 ∈ L2 (G). * ! ! i) M ∈ M(E) c . ,# - δ ∈ Gf+ ' $# # ! = E% - χ ∈ Gb . E ! 2134!5 68796: &"( # ( \ g M f (δχ) = (M f )δ −1 (χ). fE , d ⊗ m) hM ∈ L ∞ ( G 5 = (<( ^ (M f ) = hM f˜, ∀f ∈ Cc (G) ! 6+: ! / 56> / /!>, khM k∞ ≤ CkM k, C M% < , 6 9 7 6 : / / p < fE ii) U C % Π : U −→ G # ( 5 = 5 6 796:; H ∈ L∞(G, ∞ b H (U )) (<( ,# λ ∈ U $# M,Π b χ ∈ G. g M f Π(λ)χ = HM,Π (χ)(λ)f˜ Π(λ)χ , ∀f ∈ Cc (G). ! = 2 134!- # % ! = ( 5 L¥ ± ¦§¦´! H¬Å± ³§!" $!± !"I !y! $ #¥ $¦ #¥ $ ³§!"¼ "¾±Ë"¼³§ ! ¥ºJB½ $ °$¬Å©yµ yµEqbÄ ]³ Èl! φ " L = {z ∈ C | Re z ∈ [0, 1]}. f+ µ|Yl¥ y ! #¥ ψ∈G E φ 6= ψ µCE±¶+ ! « " ³§! Π : L 3 λ −→ φλ ψ 1−λ "1± !±Ä¦´¸#"¼³ ¥y~ ¥y L◦ µB?K$«$³S¬Å!" #¥ ¦´ZJB½ ° ¬ÅZ©yµ ³´³ ! ! y$ ³´ " $ ± !±Ä¦´¸#"¼³´«$³´" ® h y°$ #¥ Gf+ ! " ± D ¥ !º ³´! ʦ´ "¼ !¬ ¬Å ³ Gf ±:¥ !2³§!"¼ ³§ ¥ E M E ²³§yI ± ! Ge µq_ ! AH!:²³¿" ¥y ¥ Ωφ,ψ = Π(L). θ = φλ ψ 1−λ ∈ Ωφ,ψ »±ª² « λ∈δ » !T±Ð¦Á±® $ $!"¾±Ä" ³§! φ(x) θ(x) = φ(x)Re λ ψ(x)1−Re λ × ei Im λ ln ψ(x) , ∀x ∈ G. Uz!"¼ ly¥ ³§ ! O¦§¬ ½ ³§ ¬Å φ(x) ·" ! γ : G 3 x −→ ei ln ψ(x) ∈ T b | χ(x) = (γ(x))t , ∀x ∈ G, avec t ∈ R}. γ R = {χ ∈ G A!:! Ä"¼ Sφ,ψ A!в ³´" #¥ ¦ ! ¬® ¦´«$!²ÃÈl& ! ³ ± n o + t 1−t f = η ∈ GE | η(x) = φ(x) ψ(x) , ∀x ∈ G, avec t ∈ [0, 1] . Sφ,ψ Ωφ,ψ "}³§ ¬Å ½ O% ` J_¥l" fE φ∈G z∈C " Sφ,ψ × γ R µ#^ ¥ K±Ä¦§¦§ ! ³´ «$¥l"¼$ #¥ ¦#¥ [email protected]·Èl ¬ ´¦ $ µ $W"^Gy=! ! Z ± φ : Z 3 n −→ z n ∈ C, 1 ρ(S −1 ) ≤ |z| ≤ ρ(S) µYl³ ρ(S) > 1 , ρ(S −1 ) ! ¥y"O«¾½ ³´ ³§ φ ·" ψ "¼$¦´ #¥ φ(n) = ρ(S)n , ∀n ∈ Z, Ay $W²!y #¥ Sφ,ψ NS³§!y ³Í»l!1y"¼³´$!" ψ(n) = ρ(S −1 )−n , ∀n ∈ Z. "L³§ ¬ ½yD± ¥: $Ê ¬Å !#" [ ρ(S1−1 ) , ρ(S)] " b = T. γ1R ≈ Z n Ωφ,ψ ≈ z ∈ C | o 1 ≤ |z| ≤ ρ(S) ρ(S −1 ) "^¦§DJB½ $ °$¬I© µ ! !y ÃÈ ±Ä« "¼ ¬Å !"H¦§D $ ¥y¦´"¾±Ë"S« ! «$ !±Ä!#"S¦§ ^¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ K ¥ ¥ !T ±Ä«$I Is± !± «¾½1 D ¥y³´"¼ $» ¬Å !#" D± !yL¦´I?K½± ³¿"¼ Dyµ ` J_¥l" φ ∈ Gf "b G! =!y R ± φ(x) = eax, ∀x ∈ R » ¥y{¥ !D«$$W"¾± ³´! a ∈ [ln 1 , ln ρ(S )] µ 1 ρ(S ) j µ S F ± ! S « Yl¥ ! E#¥ E = L2 »|±¹² $« ω ¥ ! ³§ L "S ¥ y ! #¥ 1 < ρ(S1 ) ρ(S ) «ª± O³§¦j$ "L!±Ë"¼¥ $¦{ D«¾½ ω³´ ³´ −1 −1 φ(x) = eln ρ(S1 )x , ∀x ∈ R, ?K$¦Á± !#" ¼±ÄΧ! "L³§¦$ "L·²l³´ $!" #¥ ψ(x) = e− ln ρ(S−1 )x , ∀x ∈ R. h Sφ,ψ ≈ ln i 1 , ln ρ(S1 ) , ρ(S−1 ) n h Ωφ,ψ ≈ z ∈ C | Im z ∈ ln io 1 , ln ρ(S1 ) . ρ(S−1 ) q ± Ы$ ! ¥y$!"ª»@¦§)JB½ ° ¬Å:© µ³§¬ ¦§³ #¥ +¬ ± ³´ :¬ ±Ä! ³§° Æ! ! « ! " ¥ «·"¼³¿² ¦§ "¼½ ° ¬Å®y® $!"¼±Ä"¼³´!2 $¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ O ¥ L2 (R) ÃÈ ± ! ¦´?K½± ³´"¼ ω lµ ` =Z HA ! e = (e G, ..., ±ª²$« ej,k ), j j,1 AH!1 $ ¥y³´"O ©yµ© #¥ k ej,i = 0 ¥y i 6= j » " ej,j = 1 µAH!: ! ³´" S j = S ej . fk ≈ σ(S , ...., S ). Z 1 k E ?K½± ¥y φ ∈ Zfk $W"S I¦Á±¶+ ¬Å φ = φ » z = (z , ..., z ) ∈ C∗k " φ $ "S ! ³E ¥ z 1 k z ± g¦Á±H¶+ ¬®¥ ¦§ φ (n , ..., n ) = zn1 ...z » ¥ nk F^! «L ³! ¥ !y n Zk . z = z1n1 ...zknk z 1 k 1 k " ¥y" z = (z , ..., z ) ∈ Ck ·" n = (n , ..., n ) ∈ Zk , «¾½± #¥ φ ∈ Zfk $W" y¦§±Æ¶+ ¬ 1 k 1 k » µEq_ ! n ¥ " ¥y" k » ¥ O¥ !:« $ "¼± ³§! φ = φz φz (n) = z n∈Z z ∈ C∗k fk }. FE = {z ∈ C∗k | φz ∈ Z E AH!1 $¬Å± ¥y #¥ F ² $ ³ D¦§ ³§·"¼ B ¥ ³¿² ±Ä!#" $ µ E \¨ » ¥ K" ¥y" (z , ..., z ) ∈ F " ¥ }"¼¥y" (θ , ..., θ ) ∈ Rk µ iθ iθ (z1 e , ..., zk e ) ∈ FE BCgÇ$!y $¬& ¦§ {(log |z |, ..., log |z |)} 1 k "^«$E!²ÃÈl& "L« ¬ ± 1« "$µ k 1 k (z ,...,z )∈F >Q! ± " ³§« ¥ ¦§³´$ F◦ $W"^¥y!1y¬ ±Ä³§! ^ IML ³§! ½±Ä y"B¦´ÊÄ«$!²ÃÈlI ³ F◦ 6= ∅ µAH!) ¥ E #¥ F◦E 6= ∅ "L!E 1 k 1 k E ◦ fk . Π : FE 3 z −→ φz ∈ Z E Uz¦ "" ³¿²l³§± ¦ #¥ Π $W"L± !± ¦¿¸#"¼³#¥ ĵyL "¼ ! F (Zk ) ¦ÍÇ$! $¬® ¦´H B ¥ ³¿"¼$ ! ³§ ¥y Zk µ AH!Ð $y¥ ³´"g ¥ÐJB½ ° ¬ÅB©yµ #¥ ¥ "¼ ¥y" M ∈ M(E) ³§¦l·Èl³§W"¼B¥ !yO¶+ ! « " ³§! H ∈ M,Π ◦ ² $ ³ ± ! " ¥ ¼ " y ¥ " » k ·" ¥y $#¥ I" ¥y" ∞ c ∞ c k k u ∈ F (Z ) χ∈Z L (Z , H (F )) E ◦ gu(φz χ) = HM,Π (χ)(z)ũ(φz χ), ∀z ∈ FE . M ?K½ ³§ ³´ ! ck ²$ χ = (χ1 , ..., χk ) ∈ Z " zχ = (z χ , ..., z χ ) » −1 (χ−1 1 1 k k 1 , ..., χk ) N^¦§ φzδ ³ ±Ä!#"ƦÁ±º¶+ ¬®¥ ¦´:«$³¿Â $ ¥ µOAH! ¥ z = (z , ..., z ) ∈ Ck µFS ! ³´ ! 1 k θM : FE 3 z −→ HM,Π (χ)(φz χ−1 ) ∈ C. ◦ Yl ³´" " $¦#¥ µBUz¦H$W"1«$¦Á±Ä³§ #¥ 2! ¥ 1±¹² ! ¥ z ∈ FE » θM ∈ H∞ (FE ). δ ∈ Tk χ = φδ » ¥ ¼ " y ¥ " » ∗k µ ^¥ Ol"¼$!y! ¥y = φ z φδ z∈C u ∈ F (Zk ) gu(φz ) = M gu(φzδ−1 φδ ) = θM (z)ũ(φzδ−1 φδ ) = θM (z)ũ(φz ). M N^¥y"¼ $¬$!": ´³ "Ƴ§¦L·Èl³§W"¼ θ M ◦ ¬®¥ ! ³yI¦§±¬Å ¥ g k+ Z × Tk E χ−1 = ¼" ¦#¥ θ (z) = h (φ ) µ µ} ¥ M M z µL¥ L±ª²! L !y«I¦´ ¥ ¦¿"¾±Ä"O ¥ ³´²Ä± !" d⊗m ◦ ∈ H ∞ (F E ) ◦ FE ≈ 6< ¢© µ¿¨ < E !5 /,: , / /,: , ! <- ! ! , # Zk # B/ ! ( ◦ : <; ! ) ) ) ! ! = Zfk 6= ∅ # ! , # M ∈ M(E) . % %&"( . . ( 2134!- θ ∈ H∞(F◦ ) - = ,# 5 f ∈ F (Zk )E M E ( . ` ◦ gf φz = θM (z)f˜ φz , ∀z ∈ FE . M G = Rk ?K½±#¥ I$¦§ ¬Å !"O "L D¦Á±®¶+ ¬Å fk R FS !y³§ ! ψa : x −→ e−i<a,x> , avec a ∈ C∗k . fk E : C∗k 3 a −→ ψa ∈ R E µ f −1 ∗k k E (ψ) ∈ C , ∀ψ ∈ R L" ¬ ± #¥ ! #¥ Yl¥ ! ¥y Rf◦k 6= ∅ µAH! E A!: ± $¦§¦´#¥ I¦ÍÇ$! $¬® ¦´ E UE = E ◦ g k+ R E −1 ◦ ◦ g k f k RE ≈ R + iRk+ E . "L¦§ ÊÄÂz« !²·ÈlĵjY³´" fk . Π : UE 3 a −→ ψa ∈ R E _q ¥ Ð"¼ ¥y" x ∈ Rk »g¦§±¶+ ! « " ³§! a −→ Π(a)(x) = e−i<a,x> $W")± !±Ä¦´¸#"¼³ ¥y1 ¥ ]³¿Èl! M ∈ M(E) µ>Q!~± y ¦§³ ¥ ± !"B¦§JB½ ° ¬ÅD© µy»!:l"¼³§ !" g M f (ψa χ) = HM,Π (χ)(a)f˜(ψa ), ∀a ∈ UE , p.p., ck , H∞ (UE )) µ}]³¿Èl! HM,Π ∈ L∞ (R " $¦ #¥ χ = ψ µAH!T± δ ∈ Rk δ ck χ ∈ R ² $ ³ ± !"ŦÁ±2¶+ ¬®¥y¦§:«$³ Âz ¥y$µKYl³´" g M f (ψaδ−1 ψδ ) = HM,Π (ψδ )(a)f˜(ψaδ−1 ψδ ), ∀a ∈ UE , ∀f ∈ Cc (Rk ). A!:l"¼³§ !" A! FS! «D!1± gf (ψa ) = HM,Π (ψδ )(a)f˜(ψa ), ∀a ∈ UE , ∀f ∈ Cc (Rk ). M JM ∈ H∞ (UE ) JM (a) = HM,Π (ψδ )(a), ∀a ∈ UE . " g M f (ψa ) = JM (a)f˜(ψa ), ∀a ∈ UE , ∀f ∈ Cc (Rk ). ! UE µ ?K ¦Á±Ð $ ¬!"¼ ¦§«$ ¦´¦Á± ³´ S ¥ ³´²Ä± !"$µ <¢© µ < E !5 /,: , / /,: , 796: ! ! $# Rk # B/ ◦ ! ! = f ) . ) ) % RkE 6= ∅ % "& ( # ! . ,# M ∈ M(E) . ( 2 134!5 5 = JM ∈ H ∞ U E ( gf (ψa ) = JM (a)f˜(ψa ), ∀a ∈ UE , ∀f ∈ Cc (Rk ). M L¥ ±Ä¦§¦§ ! Q ! !y$K¬Å± ³´!#" $!±Ä!#" #¥ ¦#¥ $}·Èl$¬ ¦´[email protected] ±Ä«$$K Hs±Ä!± «¾½Å² $ ³ ± !" ¦´$O«$!y ³´" ³§! R ¨ ·» RSK " RS õ ϲ ` EY³´" ω ¥ ! ¶+! «·"¼³´!:¬Å ¥ ± ¦§ ³´"¼³¿² ¥ G µqb¥ 1 ≤ p < +∞ » ! Lpω (G) n := f mesurable sur R | kf kω,p = Z Z G |f (x)|p ω(x)p dx Uz¦b$W"H«$¦§± ³§ #¥ ¦ÍÇ$ ± «$ ®sK± !G± «¾½ "L²$ ³ D ³{ "L $¥ ¦´$¬$!"L ³ +© µ ¯ 0 < sup ess Lpω (G) y∈G o |f (x)|p ω(x)p dx < +∞ , p1 , for f ∈ Lpω (G). ¼±Ä" ³§W¶ ± ³´" R ¨ B " ^R ·µ|CE±Å« ! ³¿"¼³§ ! HR ω(x + y) < +∞, ∀x ∈ G. ω(y) g³´«$³¥y!T«$± ± " ³§« ¥ ¦§³´$« ! «$ "$»:! $¥l"^«$± ¦§« ¥ ¦§ ·" E = lω2 (Z2 ) µ N^¦§ B!T± fE G ÃÈ §¦ ³§« ³´" $¬$!"ªµAH! ω(n, k) = emax(n,k) , ∀(n, k) ∈ Z2 kSn,k k = max(en , ek ) ·"^j ± °$L¦§JB½y$ °$¬I©yµ l» ·" ρ(Sn,k ) = max(en , ek ). Uz«$³ f2 ≈ σ(S1,0 , S0,1 ) Z E f2 ≈ {(z1 , z2 ) ∈ C2 | 1 ≤ |zi | ≤ e, i ∈ {1, 2}, |z1 ||z2 | = e}. Z E AH!: ¬ ± #¥ ¥y σ(S , S ) 6= spec(S ) × spec(S ) "O¦Ídz§!" $ ³´$¥ } σ(S , S ) 1,0 0,1 1,0 0,1 1,0 0,1 ◦ ◦ "B²³§ ^ ³§ ! #¥ spec(S ·" y µ K ? $! y ± ! ª " » ´ ³ | ¦ B " « Á ¦ ± ´ ³ # ¥ 2+ ± spec(S0,1 ) 6= ∅ Zg 1,0 ) 6= ∅ E ± ¥1¬³§! K $¥lÈ1 ¦§ ¬Å !#" $µ ` {Y³´" ω ¥ ! ³§ B« !"¼³§!#¥1 ¥ G µq_ ! C0,ω (G) = {f ∈ C(G) | f ω ∈ C0 (G)}. A !2¬®¥y! ³´" C (G) ЦÁ± !y ¬ kf k = kf ωk µjUz¦Q$ "D« ¦Á± ³´ #¥ «$! y³´"¼³´! R 0,ω¨ K·" RSK "L³§¦ ¥ Ð" ¥y ω ± ³´"∞¦Á± ³´ " ¥y ¥y C0,ω (G) ±Ä"¼³´ ¶ ± ³¿"H¦§ ω(x + y) < +∞, ∀y ∈ G, ω(x) x∈G 0 < sup ² $ ³ RS·µ ` Y³´" A ¥ ! H¶+! «·"¼³´!) $¦§¦´H«$ !#" ³§!#¥ ¥ [0, +∞[ »l"¼ ¦§¦´ #¥ A(0) = 0 " y −→ A(y) $W"®« ³´ ¼±Ä!#" Å ¥ R+ µ_Yl³¿" L (G) ¦ÍÇ$ ± «$Å $D¶+! «·"¼³´! I¬Å ¥ ± ¦§ A ¥ G " $¦´¦§$ y #¥ C0,ω (G) ¥yB¥ ! t>0 "O ³´" Z A G |f (x)| n t kf kA = inf t > 0 | Z dx < +∞, A |f (x)| t o dx ≤ 1 , ¥y f ∈ LA(G) µ#NS¦´ LA(G) $W"¥ !Å$ ± «$S SsK± !± «¾½)± ¦§L$ ± «$^ Hs}³§ !#±Ä¥ ¬Ð A ¦§³´«$Ì +« ¶\µ ·µyUz¦{$ "L« ¦Á± ³´ #¥ L (G) ¼±Ë"¼³§W¶ ± ³´" R ¨ ·» RH " RS õ G A FH± ! )« "¾±Ä ¦§³´& "¼½ ° ¬Å ¶+! « " ³§! fE G " " « "¼³´!<! ¥y:± ¦§¦´! y! ! Æ¥ ! ! ¥y² ¦§¦§~«ª± ± « " $ ³´¼±Ä" ³§!8 Gf ·" E ³§·"¼$y GfE #¥ ³² !#" ¥ $¥ !6¦§Æ³´¬ W"¾± !"® ± ! ЦÁ± $¥l²1 ¥ ³´! «$³ ± ¦@ « )«¾½± ³¿"¼ ĵAH!5 $¬Å± #¥ ¥y ¥y φ ∈ C (G) » g ∈ E »b¦Á± K $ "L¥ !y³´¶+ ¬Å ¬Å !#"«$!"¼³´!#¥ D ¥ Z G 3 x −→ φ(x)Sx g ∈ E G ·" kφ(x)Sx gkdx ≤ kφk∞ kgk sup kSx km(K) < +∞. A !-« ! «$¦´¥y"#¥ R φ(x)S gdx ":¥ !yº³´!"¼$Ê ¼± ¦´Z 2sK«¾½ ! 1« !² Ê !#" ¥ )¦§± "¼ ¦§ ʳ§¶+W"¼^ $K ¼±Ä" $¥x «·¶\µ » ?K½ ± ³´" S ·µl^¥yB±ª²! K¦Á±I¶+ ¬®¥ ¦´^ ¥ ´³ ²Ä± !" Z © µÀ M = φ(x)S dx. K x∈K φ x g> ! "$» ³¿" K ¥ ! ¥ WÂ$ ± «$^«$¬ ± «·"g G µ#AH!ű M (C (G)) ⊂ C ·" φ K K+supp(φ) (G) ¦Á±& $ " ³´« "¼³´!Æ R φ(x)S dx % C (G) ¥y" ·"¼ « ! ³§ $« ¬¬Å^¥ ! ³§!" $Ê ± ¦§^ x K G G sK«¾½ ! ¥y C (G) % ²Ä± ¦§ ¥ @± !y C µy?K¬¬ÅL¦§ K³´!#" $Ê ± ¦§ }ysK«¾½ ! $ K K+supp(φ) (G) « ¬Å¬&¥y"¼ !#"^±ª² « ¦´$¶+ ¬Å B¦§³´! ª± ³´ «$!" ³§!#¥ $ »!:y"¼³´$!"ª» ¥ g ∈ C (G) » Mφ g(x) = (φ ∗ g)(x) = Z Z G φ(y)g(x − y)dy = Z c φ(y)(Sy g)(x)dy supp(φ) φ(y)Sy g (x), ∀x ∈ G supp(φ) $!" ¼±ÄΧ! S¦§± ¶+ ¬&¥ ¦§ © µÀ õyY³´" = ·"B¦Á±I $! ³´" H C (G) ± ! E ¦Í± ½ $!y«$H±Ä! A(E) c &¦ ±Ä¦§Ê° D$!yÊ$! y $ ± {M } µAH!Z! " ρ (A) ¦´ ±ª¸!Z « " ¼± ¦ M(E) φ φ∈C (G) Ç¥ !T$¦§ ¬Å !" A y A(E) µ ^¥ L±ª² ! ^¦§± ³´" ³§!) ¥ ³´²Ä± !A(E) "¼Äµ 6 < © µ f ∈ C (G) f ≥ 0 f 6= 0 / # ! ρ (M ) > 0 c f . . . ( % A(E) c AH! yÈl f ∈ Cc(G) "¼ ¦#¥ f ≥ 0 ·" f 6= 0 µ#Yl³¿" V ¥ !² ³§ ³§!±ÄÊB«$¬ ± «·" 0 "¼$¦ #¥ supp(f ) ⊂ V "^ ³¿" F ¥ !~ ! ¬® ¦§ ! ³E G "¼ ¦ ¥y V + V ⊂ F + V µ AH! c AH!1 $¬Å± ¥y #¥ +© µ L "¼!y kf k = R 1 nV := {s1 + ... + sn , s1 , ..., sn ∈ V }. nV ⊂ (n − 1)F + V, ∀n ≥ 1. G f (x)dx. AH!1± Z f ∗n $W"¦§± ¥ ³§ ¼± ! « K!l³§° ¬ÅnVQ +© µ K " Ílµ Z f ∗n (x)dx = kf kn1 , ∀n ≥ 1, f ± ! b¦Í± ¦§Ê °$ Q }« !²¦´¥y"¼³´! f (x)dx = f ∗n (x)dx (n−1)F +V X Z = S−s f ∗n (x)dx s∈(n−1)F Yl ³§$!" ¦§«$± y³§!± ¦j k ·" ± O«$! #¥ !#" kf kn+1 1 = Z G F " µÄA! ±^j ± °$ Z ∗n nV L1 (G) ≤ CV V X s∈(n−1)F kS−s kkf ∗n kE . µyNS¦§ B!T± D = maxs∈F kS−s k X kS−s k ≤ k n−1 D n−1 s∈(n−1)F f ∗(n+1) (x)dx ≤ CV k n D n kf ∗(n+1) k ≤ CV k n D n k(Mf )n kkf k, ∀n ≥ 1. C A!: !T ¥ ³¿" kf kn+1 1 k(Mf ) k ≥ CV kf k(kD)n n "L ! « !~± ρA(E) (Mf ) ≥ kf k1 > 0. kD A!¢ $W²#¥ º¦Á± ³¿"¼³§ ! «$$y$!"¼x $¬!"¼ º ! ±Ä "¼³´«$¥ ¦´³§ #¥ A(E) ! " ± O¼±Ä ³§«$± ¦§ÄµAH!T $¬Å± #¥ #¥ S ◦ M = M » ¥ B"¼ ¥y" φ ∈ C (G) µF^! « x φ S (φ) c » y ¥ ¼ " y ¥ " ·" ¥ " y ¥ " | µ l Y ´ ³ " µA!T± [ R ◦ T ∈ A(E) R ∈ B(E) T ∈ A(E) γ ∈ A(E) x ¥y γ(R ◦ T1 ) γ(R ◦ T2 ) = , γ(T1 ) γ(T2 ) » PT± ³§!"¼ !± !"® !yÈl R ∈ B(E) T1 , T2 ∈ A(E) \ Ker(γ). "L!1 !y³´" Mφ ∈ / Ker(γ) ∆γ : B(E) −→ C ± B¦Á±&¶+ ¬®¥ ¦´ © µ ∆γ (R) = Uz¦{$ "L« ¦Á± ³´ #¥ ∆γ (R1 R2 ) = ¥y © µ R1 , R2 ∈ B(E) © µ¿¨¹ φ ∈ Cc (G) »E"¼$¦ #¥ γ(R ◦ Mφ ) , ∀R ∈ B(E). γ(Mφ ) γ(R1 ◦ R2 ◦ Mφ2 ) γ(R1 ◦ R2 ◦ Mφ ) = = ∆γ (R1 )∆γ (R2 ), γ(Mφ ) γ(Mφ2 ) µ?K¬¬Å ∆γ (I) = 1 » !1± \ ∆γ ∈ B(E). 5 = ( AH!: ¬ ± #¥ ¥y 1 ≤ |∆γ (Sx )| ≤ ρ(Sx ), ∀x ∈ G. ρ(S−x ) ^¥ L±ª² ! ^ ¦´H¦§ ¬Å¬H ¥ ³´²Ä± !"ªµ 2© µ´¨< E !5 /,: >, / /,: # B/ [ 5 ( = fE . ( 2134!5 γθ ∈ A(E) i)+* $# θ ∈ G $# φ ∈ Cc (G) θ(x) = Mφ ∈ / Ker(γθ ) ) γθ (Mφ ) , γθ (Sx ◦ Mφ ) / 1 ≤ |θ −1 (x)| ≤ ρ(Sx ), ∀x ∈ G. ρ(S−x ) ) % ii) [ A(E) / ( : / [ fE 3 θ −→ γθ ∈ A(E) T : G ! # ,4! , ,# / 5 >, 7 / ( ( ( % c {]³¿Èl!y θ ∈ GfE L" ! ³§ !y Z + © µ´¨ ¨ γ (M ) = φ(x)θ −1 (x)dx, ∀φ ∈ C (G). θ φ fE G ! ,# c Uz¦ ¥ ¦¿"¼L S¦§± !y³´"¼³´! Gf #¥ γ ·"¼$!y ± g«$!" ³§!#¥ ³´" L% A(E) "}³§¦y $ ¥ ¦´" ^y¥ " ½ $ °$¬B B]¥ ³´! ³¥yB«$ " "¼BÃÈEl" $! ³§θ!Ð "g¥ ! ¦§$¬$!"Qy A(E) [ µ ^¥yQ± ¦§¦´! ¥l²$ #¥ G +© µ´¨¹ AH!T± Z Sy φ(x)θ −1 (x)dx G Z Z −1 = φ(x − y)θ (x)dx = φ(x)θ −1 (x + y)dx G G Z = θ −1 (y) φ(x)θ −1 (x)dx = θ −1 (y)γθ (Mφ ). G +© µ¿¨Ï·µA!: $¬Å ± #¥ #¥ © µ K³´¬ ¦´³ #¥ γθ (MSy φ ) = N^³§! ³Í» !:l"¼³§ !" +© µ´¨ª γθ (Mφ ) . γθ (Sx ◦ Mφ ) θ(x) = 1 ≤ θ −1 (x) ≤ ρ(Sx ), ∀x ∈ G, ρ(S−x ) " ¼±Ä" ³§!) D³·µ « # ¥ ³E± «¾½y° ²I¦Á± ¬Å ! ]³ Èl! γ ∈ A(E) [ " ψ ∈ C (G) " $¦§ #¥ Mψ ∈ / Ker(γ) c θγ (x) = µ|AH! γ(Mψ ) , ∀x ∈ G. γ(Sx ◦ Mψ ) Yl ³´" (φ ) ⊂ C (G) ¥ ! I ¥y³´"¼ #¥ ³E«$ !#² $ Ê ¥y! ³´¶+ ¬Å ¬Å !"O ¥ µqb¥ Bn"¼n≥0 ¥l" g ∈ EK»!1y"¼³´$!" ·"g«$$¦§±S$!" ¼±ÄΧ! #¥ kMφn g − Mφ gk ≤ limn→+∞ kMφn Z K ² $ φ ∈ CK (G) kφn − φk∞ sup kSy kkgkdy µq±Ä«$!y #¥ !"ª»¦Í± ¦§³§«$±Ä"¼³´!I¦§³§!yª± ³´ −Mφ k = 0 K y∈K Cc (G) 3 φ −→ Mφ ∈ A(E) "I ¥y$!"¼³´$¦§¦´$¬$!"D«$ !#" ³§!#¥ ·" y! ««$ !#" ³§!#¥ ¦´³§«$±Ä"¼³´! x −→ Sx (φ) Cc (G) ± ! A(E) µE?K¬¬®¦Í±  $ "L« !"¼³§!#¥ D G ±! Cc (G) !)²³¿" #¥ ¦ ± ¦§³´«ª±Ä" ³§! x −→ Sx ◦ Mφ = MSx (φ) $ "H«$!"¼³´!#¥ G ± ! A(E) µAH!$! ¥ ³´"¥y¦Á±¶+ ! « " ³§! A!)²Ä±Ð¬ ± ³´!"¼$! ± !"B¬Å !#" ¥y θ ∈ Gf µq_ ! γ Cg± ¦§³§«$±Ä"¼³´! η $W"¥ ! @¶+ ¬Åg¦§³´! ª±Ä³§ Q« !"¼³´!¥yK ¥y «·¶\µ »|?K½± ³¿"¼ D" $¦´¦§#¥ µ ?K$¦Á±¬!"¼ ¥y ¥ f » η(φ) = G Z G (φ ∗ f )(t)dµ(t) φ(x)f (t − x)dx dµ(t). >g!1¥y" ³§¦§³´¼± !"K¦§H"¼½y$ °$¬Iy ]¥Gy³§! G³ »y!:y" ³§ !#" γ(Mφ ◦ Mf ) = = " © µ¿¨ª© γ(Mφ ) = q± O«$ ! ¥y$!"ª»|!:«$!y«$¦§¥l" ¥y "L ! « Z θγ G Z Z Z G G φ(x) G Z G f (t − x)dµ(t) dx φ(x)γ(Sx ◦ M f )dx φ(x)θγ−1 (x)dx, ∀φ ∈ Cc (G). φ(x)θγ−1 (x)dx = |γ(Mφ )| ≤ kMφ k f . FS ! ³§ ! ∈G E ± B¦Á±&¶+ ¬®¥ ¦´ R(γ)(x) = G "_³§¦ÃÈy³´ " K !y«K¥ ! }¬Å ¥ » !T± γ(Mφ ◦ Mf ) = Z Z Cc (G) φ(x)dµ(x), ∀φ ∈ Cc (G). φ ∈ Cc (G) = $W"«$!" ³§!#¥ ¥y E η : Cc (G) 3 φ −→ γ(Mφ ). Z θγ [ −→ G, e R : A(E) γ(Mφ ) [ ∀x ∈ G, , ∀γ ∈ A(E), γ(Sx ◦ Mφ ) µ ¥ φ ∈ Cc(G) "¼ ¦ #¥ Mφ ∈/ Ker(γ). ^¥yO±¹² ! !yÈl φ »¿µ´µ¿µ¿» φ ∈ C (G) " > 0 »!1± 1 k c sup ¥ B"¼ ¥y" i=1,...,k fE θ∈G sup AH!~« ! « ¦§¥y" #¥ "¼ ¦ #¥ Z G θ0−1 (x)φi (x)dx |θ0−1 (x) − θ −1 (x)| < − Z T R = RT = I µ Yl³´" fE µYl³ θ0 ∈ G θ −1 (x)φi (x)dx < , G . (1 + supi=1,...,k kφi k∞ )m(∪i supp(φi )) "^«$!"¼³´!#¥ µAH!1²Ä±Ð¬ ±Ä³§!"¼ !± !"O¬Å !#" #¥ R "L«$ !#" ³§!#¥µYl³¿" [ ¥ !yÆ ¥ ³¿"¼ÆÊ $! ¼± ¦´³§ #¥ ³« !²$ ÊƲ γ ∈ A(E) [ ¥ ®¦§±1" ¦§Ê³´ (γα ) ⊂ A(E) ~u$¦´¶ ±Ä! µQAH! yÈl f ∈ C (G) "¼$¦ #¥ γ(M ) 6= 0 µQUz¦B·Èl³§W"¼ β ·" δ > 0 "¼ ¦ #¥ c f µ A T ! ± γ (S ◦M ) µAH!T y"¼³´$!" −1 |γ (M )| ≥ δ, ∀α ≥ β R(γ )(x) = x∈∪supp(φi ) β f T α α x f γα (Mf ) |R(γα )(x)−1 − R(γα )(y)−1 | ≤ δ −1 kγα kkSx ◦ Mf − Sy ◦ Mf k ≤ δ −1 kSx ◦ Mf − Sy ◦ Mf k, ∀x, y ∈ G, ∀α ≥ β. AH![ !y$ ¥ ³¿" #¥ º¦Á±¶ ± ¬³§¦´¦§ (R(γ )) "T#¥ ³´«$!"¼³´!#¥ x ¥ G µL?K ¬Å¬ R(γ ) α α≥β « !²$ Ê ³´¬ ¦§$¬$!"g² R(γ) » R(γ ) «$!² Ê¥y! ³´¶+ ¬Å ¬Å !"K ¥ }" ¥y"« ¬ ± « "} α α ² $ µ S F ! « L " $ « ! " § ³ # ! T ¥ ·" L ¦Á± $¥y² & ¥:¦§ ¬Å¬H "L«$¬ ¦´° "¼Äµ G R(γ) R EN ²Ä± !"D ¥y² ¦§&JB½y$ °$¬© µ »j! ¥yD± ¦§¦´! L ! ! #¥ $¦ ¥y$ ¥ ¦¿"¾±Ä" ¦§³  ¬³§!±Ä³§ $µ 6 < © µ < G # +5 = ! < φ : G −→ R # ,4! </ # # ! φ ! : < ! 8! ( . ! φ ! : / # %'&"( ( ( ( % c Uz¦$ "_«$¦§± ³§ ¥[email protected]" ¥y"b¬Å ½ ³´ ¬«$ !#" ³§!#¥& "b¦§«ª±Ä¦§$¬$!"{| ! µÄYl¥ y ! #¥ φ $W"B¦§«ª± ¦´$¬$!"@ ! L "B ³§ !" U ¥ ! ² ³§ ³§!± Ê ^ 0G " M > 0 " $¦ #¥ φ(U ) ⊂ µYl³§ !" > 0 " n ≥ 1 " $¦ #¥ M < µ Uz¦ÃÈy³´ " ¥ !)²³´ ³§! ± Ê V 0 " $¦ #¥ [−M, M ] »|Ç n|φ(x)| ≤ M ¥ L"¼¥l" x n∈ V µjq± ^«$!y #¥ !"ª» |φ(x)| < ¥y^"¼ ¥y" nx ∈ U » "L« $¦Á± !#" ¼±ÄΧ! #¥ φ $W"^« !"¼³´!¥1$! 0 "L !y«I« !"¼³´!¥1 ¥y G µ x∈V G < ©yµ < E ⊂ L1 (G) !5 /,: , / /,: # B/ ) ) % loc < χ ∈ \ ! / : / ; B(E) %'&"( # .8( ( ! : ;< % G 3 x −→ |χ(Sx )| ∈ R+ !1± c EYl ³´" A!1±y» ¥ K = K −1 ¥ !:²³´ ³§! ± Ê«$ ¬ ±Ä« "L 0G µF&± °$L¦Íǽ#¸ "¼½y°$ RS·» 1 ≤ M := sup kSx k < +∞. » x∈K x∈K |χ(x)| ≤ M, |χ(x)|−1 ≤ M. A!:l"¼³§ !"ª» − log M ≤ log |χ(x)| ≤ log M, ∀x ∈ K, " !º$!º $y¥ ³´"D Ц§± ³´"¼³´! $«$ $!" #¥ Цͱ ´¦ ³§«$±Ä"¼³´! x −→ log |χ(x)| $ " D «$!"¼³´!#¥ D ¥ G µ P~± ³§!" $!± !"$»! ¥ ^± ¦§¦´! B¥y" ³§¦´³§ ^R ·µAH!1¼± $¦´¦§ #¥ I«$·" "¼I«$ ! ³´" ³§!:³´¬ ¦´³#¥ ¥y GfE ±¦§± ³§·"¼ ¥ ³´²Ä± !"¼ fE = G fE , ∀χ ∈ G. b χG Uz¦} "®y³§$! «$! !#¥ #¥Ç¥ !5 ¬ ± ³´! ÆMO$³´! ½± y" +« ¶\µb¦Í± ! ! ÃÈl) X «$ !#" $!±Ä!#" 0 " ¬Å!y¬Å³§± ¦§ ¬Å !"g«$!²ÃÈlI ³j "B $¥ ¦´$¬$!"O ³« Ç$ "¥ !) ¬Å± ³§!yS¦§ ÊÄÂz« !²·ÈlH« !"¼ !± !" ¦§ ¦´¸ ³§ #¥ D ±ª¸! Y(z) » ¥ B" ¥y" z ∈ X \ {0} µ|FS ¦§¥ »|¥y!~ ¬Å± ³´! ML ³§!l ½± l" X " $¦ #¥ 0 ∈ X $W"Ƭ! ¬³Á±Ä¦§$¬$!"« !²ÃÈy2 ³^·"Æ ¥ ¦§ ¬Å !"Æ ³ X ")¦´ ¬Å± ³§! & Å« !² Ê ! «$Æj¥ !yÅ $ ³´Ð$!"¼³´°$ «·¶\µ ·µbCE± $¥l² y¥ JB½ ° ¬ÅЩ µ ¥y³ ¥ ³¿" $ $!"¼Ð D± !± ¦´Ê³§ H±ª² «¦Á± $¥l²Ð ® $ ¥y¦´"¾±Ë"¼ ³´¬Å³´¦Á± ³´ «$ "¼± ³§! ¬Å !" ³§ !1« ! !#¥ ¥ B¦´$O ¬Å± ³§!y$ IMO$³´! ½± y"O«$!" $!#¥ S ± ! C∗k µ c |c HF&± °$)¦§±5qg ³¿"¼³§ !8© µ ":¦´C{$¬¬ÅZ©yµ´¨»}³§¦ $ "1« ¦Á± ³´ # ¥ Gf !Ç $W" ± )²l³´ µLAH!-® ¬ ¥ !y³´" Ge 2¦Á± "¼ ¦´Ê³´~ ¦§±«$!² Ê ! «$ ¥ ! ³´¶+ ¬1 ¥ Å" ¥y"Å«$¬ ± «·"ªµ}PT!" ! ±Ä # ¥ Gf " «$¬ ± «·"ªµ}^¥ ± ¦§¦´! ±Ä :¬Å!" $ # ¥ Gf "L# ¥ ³´«$!"¼³´!#¥T ¥ G µYl³´" θ ∈ Gf µ|AH!~± E + E + E + E 1 ≤ θ −1 (x) ≤ ρ(Sx ), ∀x ∈ G. ρ(S−x ) q± O«$ ! ¥y$!"ª» ¥ "¼¥l"B²³´ ³§! ± Ê«$ ¬ ±Ä« " 1 C−V0 V0 0G ≤ θ −1 (x) ≤ CV0 , ∀x ∈ V0 , C = sup ρ(S ) < +∞ " C = sup ·Èl³§W"¼ V n > 0 "¼x∈V ¦#¥ (Cx ) − 1 < δ −V" 1 − 1x∈V 0 0 0 V0 » !T± 1 n 0 1 (C−V0 ) n µE]³¿Èl! δ > 0 µ{Uz¦ ³§! x −→ nx " ρ(S−x ) < +∞ µ|CQ ± ¦§³´«ª±Ä" <δ « !"¼³§!#¥ Å ¥ G ·"& ! «Å³´¦Q·Èl³§W"¼¥ !x² ³§ ³§!±ÄÊ µ NS¦´ » x∈W Wδ )"¼ ¦ ¥y nx ∈ V0 , ¥ " ¥y" δ 1 ·" ≤ θ −1 (nx) ≤ CV0 , ∀x ∈ Wδ C−V0 1 ?K ¦Á±Ð$!"¼ ±ÄÎÁ! #¥ 1 (C−V0 ) 1 n ≤ θ −1 (x) ≤ (CV0 ) n , ∀x ∈ Wδ . 1 − δ ≤ θ −1 (x) ≤ 1 + δ ·" ı ^« ! #¥ $!" θ−1(0) − δ ≤ θ−1(x) ≤ θ−1(0) + δ » ¥ B"¼¥y" x ∈ W µNS³§!y ³{³§¦{ " « ¦Á± ³´ #¥ Gf+ "D#¥ ³´«$!" ³§!#¥2$! 0 "Dy! « Gf+ "D#¥ ³´«$!" ³§!#¥2 ¥ G µEδAH!2±Æ²¥ #¥ E E $ S " ! y ¥ ^ ¼ " y ¥ " ·"H³§¦E $ ¥y¦´"¼ &¦§±Å ! ³¿"¼³§ !T ¦ ! ¬® ¦§ {θ −1 (x)}θ∈G x∈G g f+ $W"B¶+$ ¬Å^± ! G fE ¥y G e ¥ ¦Á±I"¼ ¦§ ʳ§O ¦§±&«$!² $ Ê $! « ¥ !y³´¶+ ¬ÅL ¥ " ¥y" G E « ¬ ± « "$µ_Uz¦@ ¥ ¦¿"¼Æ±Ä¦§ Ij¥ !y ²Ä± ³§± !"¼ "¾±Ä! ± ¥ " ½ $ °$¬ÅjN^ «$ ¦§³ #¥ ± ¥y$ " ³§!º! f+ " «$¬ ± «·"ªµEqQ ¥l²! ¬ ±Ä³§!"¼ !± !" #¥ G f+ " ¦§ÊËÂz«$ !#² ·Èl µbCE G E E #¥ D ³ Gf+E ±± ¥:¬³§! By$¥lÈ: ¦§ ¬Å !#" $µ|Yl ³§$!" η1 " η2 ∈ GfE "¼$¦´ ¥y |η1 | 6= |η2| µ AH! L = {z ∈ C | Re z ∈ [0, 1]}. qb¥ λ ∈ L »!1 ! ³´" + E qb¥ f ∈ Cc (G) θλ (x) = |η1 (x)|λ |η2 (x)|1−λ , x ∈ G. " ¥ » !T± x ∈ supp(f ) sup sup |f (x)θλ−1 (x)| ≤ kf k∞ sup λ∈L l∈[0,1] CE±¶+! « " ³§! x∈supp(f ) |η1 (x)|l sup x∈supp(f ) |η2 (x)|1−l < +∞. G × L : (x, λ) −→ f (x)θλ−1 (x) ∈ C "g ± $¬$!"g«$!" ³§!#¥ ·"Q¥ ! ³¿¶+ ¬$¬$!"_ ! $K "Q ! «¬$ ¥y¼± ¦´ ¥y G×L ·µ AH!1 $ ¥y³´"^± ¦§ L" ½ $ °$¬$O PT $ ± "L I]¥ ³§!y³ #¥ ¥ " ¥y" f ∈ C (G) »¦Á± c ¶+!y« "¼³´! F ! ³§ ±Ä "B±Ä!± ¦´¸#" ³#¥ O ¥ @¦Á±D± !y F : λ −→ ≤ max sup a∈R G f (x)θλ−1 (x)dx µF&± ° K¦´ ³§!y«$³ O ^qg½y¼± ʬ$!Â\C{³§! y$¦§Ä¶\» !Åy"¼³´$!" L ◦ |F (λ)| ≤ Z Z max Z Re λ∈{0, 1} G f (x)θλ−1 (x)dx |η1 (x)|ia f (x)|η2 (x)| dx , sup |η2 (x)|ia a∈R G Z f (x)|η1 (x)| G |η2 (x)|ia dx . |η1 (x)|ia >g! $!± !"$!I«$¬ "¼g¦´g¶ ±Ä³´" #¥ |η | ∈ Gf+ "#¥ 2 E SF D¬ ¬Å |η |1−ia |η |ia ∈ Gf "L!:y" ³§$!" 1 2 ia η1 η2 E b »¹ ∈G !&± fE . |η2 |1−ia |η1 |ia ∈ G |F (λ)| ≤ kMf k, ∀λ ∈ L. ?K$¦Á±³´¬ ¦´³#¥ ¥y fE , ∀λ ∈ L. θλ ∈ G A !)²Ä±Ð¬ ± ³´!"¼$! ± !"B¬Å !#" ¥y ¦ !1± n © µ¿¨Ï¯ f = θ∈G e | |θ −1 (x)| ≤ G E Yl³´" θ ∈ Gf+ "¼ ¦ ¥y © µ¿¨¹À o sup |η −1 (x)|, ∀x ∈ G . g η∈G E f+ µN^¦§ B³§¦ÃÈy³´ " φ ∈ C (G) " > 0 θ∈ /G c E Z φ(x)θ −1 (x)dx > kMφ k + . "¼ ¦§ #¥ Yl³´" K = supp(φ) µ?K¬¬ÅO¦Á±H¶ ± ¬³§¦´¦§ nφη−1, η ∈ Gf+ ∪ {θ}o $ "K#¥ ³´«$!"¼³´!#¥ ^ ¥ ¥y"¼¥y" x ∈ K » ³§¦j·Èl³§W"¼D¥ !:²³´ ³§! ± Ê V y x ± ! E K "¼$¦ #¥ G K » x sup |φ(y)η −1 (y) − φ(x)η −1 (x)| < f+ ∪ {θ}. , ∀η ∈ G E 3m(K) bC ±H¶ ± ¬³§¦´¦§ {V } "@¥ ! $« ¥y² $¬$!"Kj¥l²$W"¼g K » !y«L³´¦yÃÈy³´ " a , ..., a ∈ K x x∈K "¼$¦´ #¥ K ⊂ ∪p V |µ AH! V c = {x ∈ K | x ∈/ V } µF^ ! ³´ ! K 1= V p " y∈Vx i=1 ai ¥y 1<i≤p 1 ai ai Ki = Vai ∩ (∪j6=i Vaj )c , µ|AH!1± Z −1 K φ(x)η (x)dx − XZ p X φ(ai )η −1 (ai )m(Ki ) i=1 p = i=1 XZ p ^¥ L±ª² ! $ » ¥ ≤ f+ » η∈G Z i=1 Ki (φ(x)η −1 (x) − φ(ai )η −1 (ai ))dx p Ki X dx = m(Ki ) = . 3m(K) 3m(K) i=1 3 E K Z −1 φ(x)η (x)dx − −1 K φ(x)θ (x)dx − p X φ(ai )η −1 (ai )m(Ki ) ≤ , 3 p X φ(ai )θ −1 (ai )m(Ki ) ≤ i=1 ! i=1 3 a1 ·" Z −1 φ(x)η (x)dx− Z φ(x)θ (x)dx ≥ ·"^y! «D! ¥ Ol"¼$!y! K p X K φ(ai )η −1 (ai )m(Ki ) − q_± O«$! #¥ !#"$» !~± i=1 K p X i=1 φ(x)η (x)dx − Z φ(x)θ −1 (x)dx > , K φ(ai )θ −1 (ai )m(Ki ) > . 3 f+ G E n o fE . C = (log |η(a1 )|, ..., log |η(ap )|), η ∈ G $W"^« ¬ ± « "$» C $W"S« !²ÃÈy "B¶+ ¬I·" (log θ(a1 ), ..., log θ(ap )) ∈ / C. FS ! « »³§¦j·Èl³§W"¼D¥ ! ¶+ ¬Å¦´³§! $± ³§ AH! −1 f+ . (θ −1 (a1 ), ..., θ −1 (ap )) 6= (η −1 (a1 ), ..., η −1 (ap )), ∀η ∈ G E qb !y ?K¬¬ Z −1 " $¦´¦§ #¥ ¥ L Rp L (log θ(a1 ), ..., log θ(ap )) > sup L (log |η(a1 )|, ..., log |η(ap )|) . g η∈G E n o ∆ = (α1 , ..., αp ) ∈ Rp |α1 log θ(a1 )+...+αp log θ(ap ) > sup (α1 log η(a1 )+...+αp log η(ap )) . g + η∈G E ?K¬¬ ¥ sup | log η(ai )| < +∞, g + η∈G E $W"H¥ !Z ¥y² "H·"H« ¬Å¬ λ∆ ⊂ ∆ » ¥ λ > 0 » ∆ µUz¦{ $ ¥y¦´"¼D D¦§±® !y³´"¼³´!) ∆ #¥ I¦ÍÇ!:± (n , ..., n ) ∈ ∆ ∩ Zp 1 » 1≤i≤p p p θ −1 (a1−n1 ...a−n ) > sup |η −1 (a1−n1 ...ap−np )|. p ?K ¦Á±Ð¬Å!" #¥ g η∈G E « #¥ ³{ ¬Å !#" n o e | |θ −1 (x)| ≤ sup |η −1 (x)| ⊂ G fE , θ∈G ©yµ´¨Ï¯ ·µ g η∈G E ∆ ∩ Zp 6= ∅ µYl ³´" 6< Z© µ© / fE = {θ ∈ G e | |θ −1 (x)| ≤ ρA(E) (Sx ) , ∀x ∈ G}, G ρA(E) (Sx ) = supγ∈A(E) [ |∆γ (Sx )|. % )% 6< Z© µ ¯ fE G # / = (<( ∆γ ! : 621$"! ! ! ( Ea bc !+ 8 / # / 97 # ( ( b G !: 621$ % HF ± ! Q« " " O $«·"¼³´!»! ¥ g± ¦§¦´! _ ! ! #¥ $¦ ¥[email protected] $ ¥ ¦´"¼±Ä"¼«$! « $ !± !"@¦Á±H !l "¾±Ä" ³§!:Ç¥ !1¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ K !T"¾±Ä!#" #¥Ç $¼±Ë"¼$¥y^ I«$ !#² ¦§¥l"¼³§ !µ|Yl³¿" B ¦ ! ¬® ¦´ $ ± W"¼³´$B| ¦§³§ ! ! Ly G µXL! D¬Å ¥ D ¥ G $W"L¥ ! I± ¦´³§«$±Ä"¼³´! G µ : BG,µ −→ C :¦Á±Z¶+ ¬ µ = µ − µ + iµ − iµ »Q µ "Å¥ 1 2 3 4 i " $¦ ´ § ¦ # ¥ » ¥ ¼ " y ¥ L " «$¬ ± i = 1, ..., 4 µi (K) < +∞ ! 1¬Å ¥ ³´" ³´²: ¥ «·" K ⊂ G ·"L G ¥ BG,µ = {U ∈ BG | sup µi (U ) < +∞, i = 1, ..., 4}. ^ " ! M (G) ¦ ! ¬® ¦§º Z¬Å ¥ $T ¥ G µLC{x" ½ $ °$¬ x $!"¼±Ä"¼³´!r ML³´$ ̺¬!"¼ #¥ «¾½ ±#¥ ¶+ ¬Å2¦§³´! ª±Ä³§ « !"¼³´!¥y ¥y C (G) $¥y" "¼ $ !l c "¼$ Ƭű ! ³§° Å¥ ! ³#¥ ¥ ¦§±T¶+ ¬Å L(f ) = R f (x)dµ (x), µ ∈ M (G) µQLÄ L G "¼! M (G) ¦ÍÇ$!y $¬& ¦§1 $ŬŠ¥ $Æ ¥ G %2²Ä± ³§±Ä"¼³´!6 ! $Ä»g¬®¥ L! ³L 1¦Á±x!y ¬ b ¥y³ $¥l" ·"¼ ³´ $!" ³ Ы$ ¬Å¬«$³ Âz ¥y&±ª² «Å¦´ ¥± ¦Qy Cc(G) µ kµkM (G) = |µ|(G), ^¥ g ³´Ê! $ ! ± M (G) ¦ ! ¬® ¦§[email protected]¬Å ¥ $}%I ¥ "@«$ ¬ ±Ä« "@ ¥ G µ#CQ W c ¥j¥ !y ¥ ³¿"¼~Ê $! ¼± ¦´³§ ± «$ M (G) $¼±¬&¥ ! ³L Z¦Á±º"¼ ¦§ ʳ§Æ²Ä± Ê¥y µML± $¦´! «$!² ÊD²Ä± Ê¥ ¬Å !"L² $ µ ∈ M (G) ³{ "L $¥ ¦´$¬$!"L ³ (µ ) ⊂ M (G) b α Z f (x)dµ(x) = lim α Z f (x)dµα (x), ∀f ∈ Cc (G). ^¥ @ ±ª² ! K|$ ³§! [email protected] ! ³´" ³§!yQ ¥ ³¿² ±Ä!#" $ ¥ g³§!"¼ l ¥y³§ B¦´$g ¥ ¦¿"¾±Ä" g HuD± ¥l ¸)ÃÈ $L± ! õ G G < 6 © µ < n K DK (G) = u ∈ Cc (G) | u = : " /,:B, ∞ X i=1 G% ! fi ∗ gi , fi , gi ∈ CK (G) et ∞ X i=1 o kfi k∞ kgi k∞ < +∞ . < kukDK (G) , /6 DK (G) ( # ∞ ∞ ∞ nX o X X = inf kfi k∞ kgi k∞ |u = fi ∗gi , fi , gi ∈ CK (G)et kfi k∞ kgi k∞ < +∞ . < <© µ© < D(G) = limD (G) , / 5 +<5 < : ( : / : 21$ / = /!→ ! K,# ! ! ( <!> ( / D(G) 0 , D(G) % ( ( % (<( ( ( ( AH! ¬ ±Ä #¥ #¥ &¦ÍÇ$ ± «$ D(G) " ! &± ! C (G) «·¶\µ ·µ^ ¥ ±ª²!yH¦§ c " ½ $ °$¬D ¥ ³¿² ±Ä!#"$µ x©yµ© a Ec ) * ,# 5 M # / 5 ,# , C (G) / ! !5 /,: , ! ( c = " : - / : ! / 5 ! , : / : 6 796:; ! ! ( , C,# (G)M (G) 213.4!5 6 = / ! ! ,# ( µ 5 # = $# . ( (<( c i=1 i=1 i=1 M f = µ ∗ f, ∀f ∈ Cc (G). AH! $¬Å± #¥ #¥ ¥ D"¼ ¥y" f ∈ C (G) »{¦Í± ´¦ ³§«ª±Ë"¼³§ ! g −→ f ∗ g " $« !" ³§!#¥ D(G) ± ! D(G) " µ ∗ f » ¥ f ∈c C (G) " ¥ ! #¥±Ä ³§¬$ ¥ Å ! ³§ ± I¦Á± c ¶+ ¬®¥ ¦´ Yl³ < µ ∗ f, g >=< µ, f ∗ g >, ∀g ∈ D(G). »±Ä¦§ ¦Á±®¶+! «·"¼³´! f ∈ D(G) µ ∗ f : G 3 y −→< µx , f (x − y) > "O«$ !#" ³§!#¥ H ¥ G µ F ± °$B¦´SJB½ ° ¬Å©yµ© »¦Á±I "¼ ³§«·"¼³§ !)% C (G) Ç¥ !)¬®¥y¦´"¼³ ¦§³´«ª±Ë " $¥ O ¥y Lp(G) » G $ "L¥ !:Ê ¥ DCb?KNr " p ≥ 1 $ "Oy ! ³´D«$c¬¬ÅS¦Á±&« !²¦´¥y"¼³´! ±ª² «Ð¥ ! #¥± ³´¬Å ¥ «·¶\µ » ·µjF&± ¥y"¼ ± "$»>Q K± yIy! ! ¥ !" ½ $ °$¬® $ !"¾±Ä" ³§!1² ±Ä¦Á± ¦´ ¥ O¦§ B¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ } C (G) ± !y M (G) µqb !y c c ·" b B(G) = {F f, f ∈ L1 (G)} kF f kB(G) = kf k . L¥ Sy$ ³´Ê! !y ± P (G) ¦´D ¥± ¦b B(G) µjC{$^L$(¦´G)$¬$!"¼L P (G) !"S± $¦´$Sy$ $¥ y¬Å ¥ $ µ^L "¼ ! P 1(G) ¦ ! ¬® ¦§~ Z"¼¥l"¼$)¦´$ $¥y ¬$ ¥y $)" $¦§¦´$#¥ » ¥ f ∈ C (G) µL¥ L±ª²!y^¦´H"¼½ ° ¬Å ¥ ³´²Ä± !" «·¶\µ ·µ s ∗ f ∈ Mb (G) c 2© µ ¯ &a c ) ! # / 5 $# ! < / # ! : < ! T , C (G) / ! M (G) = : 5 / : ! 1) b ( # / ! / ; ! ! 2( 1 /,:5 : 1 , / 79 # c ( ( 1 b T f = s ∗ f, ∀f ∈ Cc (G), C 2) #/ s ∈ P 1 (G) % ! / - ! < / !": ;< ! , ( ( # (G) ! / # ! ! , # 2( 1 ,/ :5 # : 1 , T / 79Cc / ! Mc (G) = : -/ : ( ! T f = s ∗ f, ∀f ∈ Cc (G), ! 6 ! , ! ,# ! # : " /,: % FS ¦´¥ $»g« ~ ¥ ¦¿"¾±Ä"ų§¬ ¦´³#¥ #¥Ç¥ ! #¥± ³§¬$ ¥ ~%x ¥ "Å«$¬ ± «·" "Æ¥y! ¥ ¬$ ¥ µ¹CE±B" ¼± !y ¶+ ¬Å @]¥ ³´$j¥ !y $¥y ¬$ ¥y s $¥y" ·"¼ g ! ³´ D¦Á±®¬Å± ! ³§° ¥ ³´²Ä± !"¼ ŝ $W"S¦§±&¶+ ¬¦§³´! ª±Ä³§ ^«$!" ³§!#¥ D ¥ L1 (G) !y! $ ± s ŝ(f ) = s(fˆ). A ! $¬Å± ¥y#¥ ŝ ¥y" ·"¼ «$! ³§ Æ« ¬¬ÅÅ¥ !5$¦´$¬$!"® L∞(G) b µF&± °$¦´ JB½ $ °$¬)© µ ¯l»"¼ ¥y"Ð $¼±Ë"¼$¥y T C (G) ± ! M (G) « !"¼³´!¥ ¥ ¦§±~"¼ ¦§ ʳ§ b ² ±ÄÊ¥ Z M (G) «$¬¬®¥y"¼± !")±ª² «2" ¥y" c«$!² ¦§¥y" ³§!±ª² «2¥ ! Z¶+! «·"¼³´!< C (G) » c ²$ ³ Ð © µ¿¨ Tcf = hfˆ, ∀f ∈ Cc (G), h ∈ L∞(G) b µX^!y^ $ !"¾±Ä" ³§!Å ³§¬³§¦§± ³§ KÃÈy³´ " S± ¥y ³ ¥ g¦§ }¬&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y ¥ » p ≤ 1 < +∞ «·¶\µ ·µ Lp (G) |c c `bc a `_`bc a ` a Ec SY³ E ⊂ L1 (G) "^¥ !:$ ± «$ s ± ! ± «¾½1¼Ë± "¼³§W¶ ± ³§ ± !" I R ¨ ·» RH K·" RS K! ¥ O! " ! O«$ ¬Å¬ loc $«$ $¬¬$!" Mφ : f −→ f ∗ φ ¦ÍÇ ¼±Ä" $¥ O I«$ !#² ¦§¥l"¼³§ !~± «$³´&% φ ∈ C (G) µ>Q!T¥y"¼³´¦§³§ ± !"O $S± Ê ¥ ¬Å !"¼O c " » !1y"¼³´$!"O¦§¦´$¬¬ÅS ¥ ³¿² ±Ä!#"$µ © µ < E ⊂ L1 (G) !5 /,: , / /,: # B/ ) ) ) % * ,# - M ∈ M(E) 213loc4!5 6 ! <5 # / 4! (φ ) ⊂ C (G) -. = ( c ) M = lim M ,# . /( - 79 # 5 , ! # ( / 5 $# ! α ( ! ( 6+: ! / 5< / % 5 , φ < ) kM k ≤α CkM k, C M% φ c ]_³¿Èl! M ∈ M(E) " f ∈ E µ|AH!1y ! ³¿" T : Gb −→ E ± O¦Á±&¶+ ¬®¥ ¦´ α α f b Tf (χ) = χM (χf ), χ ∈ G. Uz¦E$W"S¶ ± « ³§¦´ I² ³§ #¥ T "^«$!"¼³´!#¥~ Gb ± ! ^>Bµ|^ " ! 1 ¦ÍÇ$¦´$¬$!"L! $¥l"¼ I b f G g µ l Y ´ ³ " » » 1 b ¥ ! Æ ¥ ³´" )Ê$!y$¼±Ä¦§³§ $"¼$¦´¦§¥y c b G (kα ) ⊂ L (G) kα ∈ Cc (G) kkα kL (G) b = 1 R " ¥ " ¥y"B² ³§ ³§!± Ê V y 1 µL¥ L±ª²!y k (χ) ≥ 0, lim k (χ)dχ = 0, 1 α α b G α χ∈V / lim(kα ∗ Tf )(1Gb ) = Tf (1Gb ) = M f, ∀f ∈ E. FS ! ³´ ! Y (f ) = (k ∗ T )(1 ) µ^¥yK± ¦´¦§[email protected]¬!"¼ $ #¥ H¦ $ ±Ä"¼ ¥ Y $W" ! ³ b α α f α ±ÄL¦§±®«$!² ¦§¥y" ³§!:±ª² $«I¦Á± #¥± G³§¬$ ¥y kcα µ µL¥ B ¬ ±Ä #¥ ! #¥Ç$!1Ê$!y$¼±Ä¦{¦§ Ä ¥y³´"@Ç¥ ! ¥ ± ³´¬Å ¥ ^±ª² $« F g » ¥ g ∈ L1 (G) b !y¥ g ! !y^¥ !y^±Ä¥y"¼ #¥±Ä ³§¬$ ¥ +« ¶\µ ·µUz« ³ α c c <k α µ, g >=< µ, kα g >, ∀g ∈ D(G). K? ¬¬ kc "¥ !yŶ+! «·"¼³´!%~ ¥ "&«$¬ ± «·"ª» ±ª² ! $» ¥ α f ∈ D(G) » x ∈ G » Yα f (x) = = Z b G = Z b G c kα µ "®¥y! $¥ ¬Å ¥ µL¥ kα (χ)χ−1 (x)M (χ−1 f )(x)dχ kα (χ)χ−1 (x) < µy , (χ−1 f )(y − x) > dχ Z b G kα (χ) < µy , χ(−y)f (y − x) > dχ =< µy , Z b G kα (χ)χ(−y)dχ f (y − x) > =< µy , (c kα )(y)f (y − x) >= (µc kα ∗ f )(x). FS ¦´¥ $» !y¥ Oy" $! !y^¦´«$!"¼¦§yI¦§±®! ¬Å D¦ $ ±Ä"¼ ¥ kYα f k = k(kα ∗ Tf )(1Gb )k = ≤ Z Z b G Yα µ>Q!Tj·"ª» kα (χ)χ−1 M (χ−1 f )dχ kα (χ)kχ−1 M (χ−1 f )kdχ ≤ kM kkf k. P~±Ä³§!"¼ !± !" ¥ L± ªÈl³§Gb¬$ M ± O¥ !y& ¥y³´"¼IÊ ! $ ± ¦§³´ $DÇ $ ±Ä"¼ ¥ ^ « !²¦´¥l " ³§! S±ª²$«y$H¶+! «·"¼³´! Sy C (G) ³§¦b ¥ Ð"± ªÈl³§¬$S¦´$^ ¼±Ä" $¥ Y µFH± !yH«$ ¥l"ª» ! ν = kcµ µAH!Å!y "¼ cV ¦ÍÇ ¼±Ä" $¥ Qy^« !²¦´¥y"¼³´! ±ª² $« ν µl?K! ³§ α ! @¥ ! ¥ ³´" IÊ $! ¼± ¦´³§ (hα ) ⊂ C (G) ∗ C (G) " $¦´¦§ #¥ Ð β c c ³ » ¥ ¼ " y ¥ " µ β β kL (G) = 1 ³´³ kh $ ± ! # ! ¥ ´ ¦ D $! 1 $½ O Ç ¥ !~« ¬ ± « " K yÈl µ hβ ³´³§³ h (x) ≥ 0 » ∀x ∈ G µ 1 β ³´²q_¥yO" ¥y"B² ³§ ³§!± Ê O 0 » lim R h (x)dx = 0 µ / NS¦§ $» h ∗ f « !²$ Ê ² $ f G± !y Eβ» x∈O ¥ "¼β ¥y" f ∈ E µqb ! β Vβ (f ) = V (hβ ∗ f ), ∀f ∈ E. ^¥ L±ª² ! lim kVβ f − V f k = 0, ∀f ∈ E. ± FS §¦ ¥ » ³§¦$W"L¶ ±Ä«$³§¦´S D« !"¼β ¦§$B¦§±®! ¬ÅHy Vβ µ>Q!Tj·"ª» ¥ " ¥y" f ∈ Cc (G) » ! Vβ f = V f ∗ h β " Z kVβ f k = khβ ∗ V f k = hβ (y)Sy (V f )dy G Z ≤ |hβ (y)|kSy (V f )kdy ≤ sup kSy k kV kkf k. NS³§!y ³Í»l! ¥ Ol"¼$!y! ^¥ L±ª² ! $ » ¥ G kVβ k ≤ CkV k, y∈K $ W L " ¥ ! D«$!y "¾±Ä!#" µ|F&± ¥y" ± W"ª» C Vβ f = (ν ∗ hβ ) ∗ f, f ∈ Cc (G). g ∈ D(G) » < ν ∗ hβ , g >=< ν, g ∗ hβ >=< νx , A!: ± $¦§¦´#¥ ν ∈ P (G) ·" Z (Sy hβ )(x)g(y)dy > . G Sy hβ ∈ Cc (G) ∗ Cc (G) µUz¦j$ "L¶ ± « ³§¦§S ²³´ ¥y Cc (G) ∗ Cc (G) ⊂ B(G). >g!j·"ª» ¥y » "« ¬¬Å F f, F g ∈ L2 (G) b »!± f, g ∈ Cc (G) f ∗ g = F −1 (F f F g) b µ|AH $ ² ! #¥ (F f F g) ∈ L1 (G) Z < ν ∗ hβ , g >= < νx , (Sy hβ )(x) > g(y)dy. A!: !T ¥ ³¿" #¥ D¦§± #¥±Ä ³§¬$ ¥ Cb±®¶+!y« "¼³´! $ "L« !"¼³§!#¥ D "L¦Á±®¶+!y« "¼³´!) ! ³§ ±Ä G ν ∗ hβ (ν ∗ hβ )(y) =< νx , hβ (x − y) >=< ν, Sy hβ >, ∀y ∈ G. b 3 χ −→ G : G 3 y −→ G ±! G b µ>Q!1 "ª» L1 (G) Z G b (Sy hβ )(x)χ(x)dx ∈ L1 (G) cβ (χ−1 ), G(y) : χ −→ χ(y)h ¥ B"¼ ¥y" y∈G µUz¦ "L¶ ± «$³´¦§SyI² ³§ #¥ ¥ b φ ∈ Cc (G) b 3 χ −→ χ(y)φ(χ) y −→ G ¦Á±&¶+ ! « " ³§! "Ы$ !#" ³§!#¥ ĵg?K¬ "¼ " $!#¥ ) )¦§±T ! ³¿"¼) C (G) b ± !y L1 (G) b !5y" ³§ !#" ¥y G c "L«$!" ³§!#¥ D«ª± B«Ä "L¦Á±®¦´³§¬³´" ^¥ !y³´¶+ ¬Å^¥y! D ¥ ³¿"¼ ¶+! ·« "¼³´! « !"¼³§!#¥ $µFS ! «D¦Á± ¶+!y« "¼³´! G 3 y −→ Sy hβ ∈ B(G) "«$!" ³§!#¥ : G ± ! B(G) µgAH! ! y$ ¥ ³¿" #¥ ν ∗ h β ¬Å ! " ¼±Ä" ³§!µ L¥ ^± ¦§¦´! K¬Å± ³´!#" $!±Ä!#" ¥y² $L¦§JB½ ° ¬ÅD© µ yµ ∈ Cc (G) µg?K «$³O± «¾½ °·²1¦Á± c c D]³¿Èl! M ∈ M(E) " θ ∈ Gf µ{Yl³¿" (φ ) ⊂ ¥ ! H ¥ ³´" SÊ ! $ ± ¦§³´ ^"¼ ¦§¦´ #¥ (M ) « !²$ ÊH² M ¥ }¦Á±D"¼ ¦§ ʳ§¶+W"¼ C (G) ~ $ ±Ä"¼ ¥ :·"1" $¦ # ¥ kM k ≤ CkM k µL^¥yT±ª² ! $» ¥ θ ∈ Gf " ¥ E c α φα » φ ∈ C (G) φα c −1 φ\ α θ (χ) = >"¾±Ä!#"L ! ! ¥y #¥ ³¿" "¼:%Z $¬ ¦Á±Ä«$$ Z E G b φα (x)θ −1 (x)χ(−x)dx, ∀χ ∈ G. −1 kφ\ α θ k∞ ≤ kMφα k ≤ CkM k, ± ¥ !y) ¥ \Âz ¥ ³´" )Ê$! ¼± ¦´³§ $Æ« !²$! ± ¦§Ä»Q! ¥ y"¼ ! ! #¥ (φ\ « !² Ê ¥ ¦Á±"¼ ¦§ ʳ§D¶ ± ³§y¦§ σ(L∞(G), −1 b L1 (G)) b ² $ D¥ !yж+ ! « " ³§! αθ ) b µFS ¦´¥ $» !~± khM,θ k∞ ≤ CkM k. ?K ¬Å¬ hM,θ ∈ L∞ (G) !:"¼ ¥y² lim α Z b G (φα ) −1 φ\ α θ (χ)g(χ)dχ = lim = Z α Z b G Z b G b hM,θ (χ)g(χ)dχ, ∀g ∈ L1 (G), −1 −1 \ [ φ α θ (χ)f θ (χ)g(χ)dχ b hM,θ (χ)f[ θ −1 (χ)g(χ)dχ, ∀f ∈ Cc (G), ∀g ∈ L2 (G). ?K ¦Á±Ð³§¬ §¦ ³ ¥y #¥ D¦GbÁ± ¥y³´"¼Ê $! ¼± ¦´³§ −1 [ −1 F ((Mφα f )θ ) = F ((φα ∗ f )θ ) = φ\ αθ f θ −1 « !²$ Ê ¥ B¦§±&"¼ ¦´Ê³§L¶ ± ³§y¦§H −1 b L2 (G) ² $ hM,θ f[ θ −1 lim(Mφα f )θ −1 = F −1 (hM,θ f[ θ −1 ), α µq±ÄL« ! #¥ $!"$» ± ¥: $!y^ D¦§±" ¦§Ê ³§O¶ ± ³§ ¦´H L2 (G) µF ± ¥l"¼ ±Ä " ± B« ! W"¼ ¥y« "¼³´!» lim k(Mφα f ) − (M f )k = 0, ∀f ∈ E "L!:y" ³§$!" ¥ » α g ∈ Cc (G) Z lim g(y)θ −1(y)(Mφα f − M f )(y)dy = 0. α A ! ¬ ± #¥ #¥ }¦§ b¶+!y« "¼³´! (M f )θ−1 " F −1(h f[ ! ³´ !"Q¦Á±O¬ ¬ÅQ¶+ ¬Å −1 M,θ θ ) ¦§³§!yª± ³´ S« !"¼³´!¥yI ¥ C (G) ¥ B«¾½±#¥ f ∈ C (G) µL¥ Oy" $! ! G c ^ " ! #¥ !1± q_ ! ¥y c (M f )θ −1 (x) = F −1 (hM,θ f[ θ −1 )(x), p.p., ∀f ∈ Cc (G). (M f )θ −1 = F −1 (hM,θ f[ θ −1 ) ∈ L2 (G) µjN^³§! ³Í» ¥ $#¥ "¼¥y" b» χ∈G F ((M f )θ −1 )(χ) = hM,θ (χ)F (f θ −1 )(χ), ∀f ∈ Cc (G). » f+ ·" ¥ $ #¥ " ¥y" χ ∈ G b M ∈ M(E) δ ∈ G E ^¥ L±ª² ! $ » ¥ "¼ ¥y" hM (δχ) = hM,δ (χ). f+ δ∈G E ·" ¥ $#¥ I" ¥y" b» χ∈G g M f (δχ) = hM (δχ)f˜(δχ), ∀f ∈ Cc (G). ?K$¦Á±« ¬ §¦ ° " ¦Á± $¥y² yI¦ ±Ä " ³§!:³ õ qg ¥y² ! L¬ ± ³´!"¼$! ± !"B³§³·µ Yl³§ !" U ¥y!5¥y² "Ð Cp " Π : U −→ Gf ¥ ! ƶ+ ! « " ³§!5±Ä!± ¦´¸#" ³#¥ µ?K¬¬Å E ¥y"¼¥y" λ ∈ U » Π(λ) ∈ GfE »! ¥ L ±ª²!y sup |Π(λ)−1 (x)| ≤ sup ρ(Sx ) ≤ sup kSx k < +∞, x∈K ¥y"¼¥y"O« ¬ ± « " x∈K K⊂G µq_¥ B" ¥y" x∈K b »¦Á±&¶+ χ∈G ! « " ³§! G × U 3 (x, λ) −→ φα (x)Π(λ)−1 (x)χ(−x) $ "Æ ± ¬Å !#":« !"¼³´!¥y ")¥ !y³´¶+ ¬Å ¬Å !#"Å| ! ")y! «Z«Ä ":¥ ! Z¶+ ! « " ³§!G¬·Â ¥ ± ¦§) ¥ G × U »²³´ ͵QYl³§ !" D , ..., D y³§ #¥ Ð¥y² " Å C "¼$¦´ #¥ µ q_¥ λ , j 6= i lÈy O 1" ¥ Kp" ¥y"B"¼ ³Á± !yʦ§ T ⊂ D » !1¥l"¼³§¦´³§ ± !"¦§ D1 × ... × Dp ⊂ U i "¼½ ° ¬Å I]¥ ³´! ³Í»l!T± j Z Z T φα (x)Π(λ1 , ..., λp )−1 (x)χ(−x)dλi dx G = Z φα (x)χ(−x) Z Π(λ1 , ..., λp )−1 (x)dλi dx. K? ¬¬ λ −→ Π(λG , ..., λ )−1(x) $WT"B± ! ± ¦´¸#"¼³ #¥ ^ ¥ " ½ $ °$¬D T P i ¼± #¥ 1D¦Á±®¶+p! «·"¼³§ ! Di 3 λi −→ "S± !± ¦¿¸" ³#¥ Ä» "O¦§±¶+ ! « " ³§! Z Ui »! ¥yKy" $! ! K± °$}¦§ φα (x)Π(λ1 , ..., λp )−1 (x)χ(−x)dx G U 3 λ −→ F (φα )Π(λ) −1 (χ) = Z "^ ± ¬Å !#"^± ! ± ¦´¸#"¼³ #¥ » y! «I± !± ¦¿¸#"¼³#¥ ¥ φα (x)Π(λ)−1 (x)χ(−x)dx G U µ|A! fα (Π(.)χ) ∈ H∞ (U ). b 3 χ −→ φ ∆α : G L¥ ^±¹² ! ¥ B"¼¥l" α » k∆α kL∞ (G,H b ∞ (U )) ≤ CkMφα k. CE±® ¥y³´"¼SÊ$! ¼± ¦´³§ $ (∆ ) ⊂ L∞(G, b H∞ (U )) $ "O¥y! ³´¶+ ¬Å ¬Å !"K| !y$ µ PT¥ ! ³´ ! U α ¦Á±T¬$ ¥ ÆC{$|$ Ê¥ ĵQ^¥y ¥y² ! ®³§y$!"¼³ $ ¦§ ¥± ¦@ L1 (U ) % L∞(U ) »_¦Á± ¥ ± ¦§³¿"¼ "¼± !"L !y³§ ± O¦§±&¶+ ¬®¥y¦§ < f, g >= CQÇ$ ± «$ Z U f (x)g(x)dx, ∀f ∈ L1 (U ), ∀g ∈ L∞ (U ). $W"^¶+ ¬ ¥ B¦Á±&" ¦§Ê³´ H∞ (U ) µ|AH! σ L∞ (U ), L1 (U ) ∞ H⊥ (U ) = {f ∈ L1 (U ) | < f, g >= 0, ∀g ∈ H∞ (U )}. AH! ¥y"L³§ !#" ³ B¦§ ¥ ± ¦{ ∞ H∗∞ (U ) := L1 (U )/H⊥ (U ) ±¹² $« H∞ (U ) µqb ! < P(f ), g >=< f, g >, ∀f ∈ L1 (U ), ∀g ∈ H∞ (U ), P : L1 (U ) −→ L1 (U )/H∞(U ) $ ³§Ê! Á¦ ±Ð ¥ $«·"¼³§ !~«ª±Ä! ! ³ ¥y µPT¥ ! ³´ ! S¬Å± ³´!l " $!± !" Gb y¦Á±Æ¬$ ¥y RS⊥± ± $µj^¥ ¥y² ! D³§ !"¼³ $H¦´ ¥± ¦y L1 (G, b H∞ (U )) ∗ ·" L∞(G, » Á ¦ ® ± ¥ Ä ± § ¦ ´ ³ " H ¼ " ± ! L " ! ! ± B Á ¦ & ± + ¶ ® ¬ ¥ ´ ¦ b H∞ (U )) < f, g >= Z b G AH! ¥y"QÃÈl" ¼± ³´ B b H∞ (U )), ∀g ∈ L∞ (G, b H∞ (U )). < f (χ), g(χ) > dχ, ∀f ∈ L1 (G, ∗ (∆α ) ¥ ! B ¥ ³´" BÊ$!y$¼±Ä¦§³§ $ #¥ ³y«$!² Ê ¥ ¦§±^"¼ ¦´Ê³´@¶ ± ³´ ¦§ b H∞ (U )), L1 (G, b H∞ (U )) σ L∞ (G, ∗ ²$ O¥ ! ¶+! «·"¼³§ ! ¦§³§ $ĵAH!T± b H∞ (U )) µ|AH!1!y HM,Π ∈ L∞ (G, lim α = Z Z "¼ ¼±Ð± ¥ ³ (∆α ) «$·" " I ¥ ³´" Ê$!y$¼±Ï < g(χ)(.), ∆α (χ)(.) > dχ b G b H∞ (U )). < g(χ)(.), HM,Π(χ)(.) > dχ, ∀g ∈ L1 (G, ∗ q_ ! L : H∞(U ) 3 F −→ F (λ) » ¥y ¼" ¥y" λ ∈ U µ^ " ! #¥ L ∈ H∞(U ) » λ λ ∗ ¥y λ ∈ U µ]³¿Èl!y g ∈ L1 (G) b "L ! ³´ ! G ∈ L1 (G, b H∞ (U )) ± B¦Á±®¶+ ¬®¥ ¦´ b G ∗ ¥y"¼¥y" G(χ)(λ) = g(χ)Lλ , λ∈U » ¥y $#¥ I" ¥y" b χ∈G µL¥ L±ª²! ¥ $ #¥ D"¼ ¥y" < G(χ)(.), ∆α (χ)(.) >= g(χ)∆α (χ)(λ), ∀λ ∈ U " A!: !T ¥ ³¿" #¥ b χ∈G » < G(χ)(.), HM,Π (χ)(.) >= g(χ)HM,Π (χ)(λ), ∀λ ∈ U. lim α Z b G fα (Π(λ)χ)dχ = g(χ)φ >g!º¥y" ³§¦§³´¼± !"S¦Á±) ! ³´" ³§! #¥ D"¼¥y" χ ∈ Gb » hM Z b G g(χ)HM,Π(χ)(λ)dχ. ! ! Å ± ! D¦Á± $¥y² Šг·»!2« ! «$¦´¥y" #¥ ¥y HM,Π (χ)(λ) = hM Π(λ)χ , ∀λ ∈ U. ^¥ Ol"¼$!y! #¥ ¥ " ¥y" λ∈U " ¥ $ #¥ I"¼¥l" b» χ∈G g M f Π(λ)χ = HM (χ)(λ)f˜ Π(λ)χ , ∀f ∈ Cc (G). c Cc (G) ^¥ S± ¦§¦´! O !y! $^¥ ! ®«$± ¼±Ä" $ ³§¼±Ë"¼³§ !T ³§¬ ¦§D H ! S ¥y"¼³´¦§³§ $}¦§¦§ ¬Å¬H ¥ ³´²Ä± !"ª»« $W"¾± ³´! $¬$!"O ³´$!:«$!y!¥jµ Cc (G) µ|^ ¥ ± ¦´¦§! G© µ"< G # "! < S ⊂ G # ! ! , 1 : <; ! & % "& ( ( ! / 5 ! ! = / 5 ! ) S ! # / ; ( : " /,% : < ) - ! ( <5 !, S% ! ! , <B /,: : / ( ( % ! c CQ³´¬ ¦´³§«$±Ä"¼³´! (i) ⇒ (ii) "E² ³ $g± ! " ¥y"E ± « @" ¦§Ê ³#¥ ±Ä µ lY ¥ ! # ¥ S ⊂ G ! " ±ÄS ¦Á±Ä" ³´² $¬$!"S« ¬ ± « "$µY³´" V = V ¥ !Z² ³§ ³§!±ÄÊ −1 « ¬ ± « "S &¦ ¥ !y³´"¼D±Ä! G µYl³´" x ∈ S µAH!~²Ä± « ! " ¥ ³´ ± ^ $« ¥ $! « ®¥ ! & ¥ ³¿"¼ 1 Ç $ § ¦ Å ¬ ! ¼ " ¼ " $¦ ´ § ¦ y ¥ » ¥ p 6= q µQYl¥ y ! #¥Ç! (xn )n∈N S xp V ∩ x q V = ∅ ±)«$ ! " ¥ ³¿"¥ ! ¥ ³´" ! ³§ (x , ..., x ) ² ³ ± !"D«$ " "¼ ³§ " µE?K ¬Å¬ S !Ç$W" ± n $¦Á±Ë"¼³´² $¬$!"H« ¬ ± « " ∪ x1 V 2 ! ®« !"¼³§ !" ± S µYl³´" x ∈ S "¼ ¦#¥ x ∈/ n≤m n m+1 m+1 » " Á ¦ 2 ± ¥ ¿ ³ ¼ " ! ³§ 2 2 µ}?K¬Å¬ ∪n≤m xn V xm+1 ∈ / xn V , xm+1 V ∩ xn V = ∅ ∀n ≤ m °$ :¦§± ³´ "¼)«¾½y$ «¾½y$ µ}AH! ¥y"Ð !y«)«$! W"¼ ¥ ³§ ± $«$¥y ! «$ (xn )n≤m+1 ¥ !y ¥y³´"¼ (x ) " $¦´¦§ #¥ x V ∩ x V = ∅ » ¥y p 6= q µbYl ³´" x ∈ G ·"I ¥ ! #¥ xp ∈ xV nµ NSn∈N¦´ x ∈ xpV » p« #¥ ³ q ¥y² #¥ xV «$!"¼³´$!"L± ¥ ¦´¥ B¥ !:"¼$ ¬ÅS D¦Á± ¥ ³´" (x ) "L«$·" " I ¥ ³´" D! ±·²l³´ $¬$!"^± ¥ « ¥ ! ³´!"O ±Ä«$«$¥y¬®¥ ¦Á±Ë"¼³§ !)± ! G µ CE± n n≥0 ³´" ³§!T ¥ ³¿² ±Ä!#" ¬Å !#" #¥ ¦´$«$ ! ³´" ³§!yH ¥ ³¿²Ä± !"¼$ » ²³´ $¬¬Å !" ¥  ± !"¼$L«ª±Ä¼± «·"¼$ ³§ $!"S¦´$B| !y$O "L¦§ B ¥ ³¿"¼ L« !² Ê !#" $S² $ 0 ±Ä! C (G) µ c 6 < © µ© 1) ! B ⊂ C (G) ! # ! ! ! c ( ( !>, 1 : <; ! ! / 5 ! ( # ) ∪ ( supp(f ) ! # / : " /,: f ∈B ( % < ) sup f ∈B kf k∞ < +∞ % 6 ! <- (φ ) ⊂ C (G) : # # ! 0 ,# / - , C (G) ! 2) ( - ( c ! 8! 213n4!n∈N - K ∈ Kc - = supp(φ ,# n ∈ N ( ( ( n) ⊂ K . c EYl³¿" "^«$!"¼³´!#¥ » Yl¥ ! #¥ lim kφn k∞ = 0. n→+∞ B ¥y! ± " ³§| !y$D Cc (G) µ?K¬Å¬¦Ídz§!y«$¦§¥y ³§ ! i : Cc (G) −→ Cb (G) Cb (G) $ ³§Ê!±Ä!#"L¦ ! ¬® ¦§y$O¶+! « " ³§!yB ! $ O ¥ G »!1± sup kf k∞ < +∞. f ∈B ◦ ∪f ∈B supp(f ) !Ç$ " ± O $¦Á±Ë"¼³´² $¬$!"O« ¬ ± « "$µ NS¦´ S = ∪f ∈B f −1 (C \ {0}) !Ç$W" ± ^ $¦§±Ä"¼³¿² ¬Å !#"S«$¬ ± «·"S ± ! G µjYl³´" (x ) ¥ ! & ¥ ³´" ® S #¥ ³!j ±Æ±Ä¥ «$¥ ! n n∈N ³§!"Bj ± « «$¥ ¬&¥ ¦Á±Ä" ³§!) ± ! G µ q_ ¥ "¼¥l" n ∈ N » ³´¦jÃÈy³´ " fn ∈ B "¼ ¦ #¥ fn(xn ) 6= 0 µ AH! p(g) = X n n∈N |g(xn )| , |fn (xn )| ¥y^"¼ ¥y" g ∈ Cc(G) µ{?K·" "¼ $ ³§® "D ³´$!Z ! ³´«ª± » (xn)n∈N ! ±ª¸± !"I± ¥ $« ¥y! ³´!#" ±Ä«$«$¥y¬®¥ ¦Á±Ë"¼³§ !»Ä¦ ³´!"¼$ « "¼³´! (x ) ±¹² $«O¦§K«$¬ ± «·" supp(g) $ " !y³§ ¥y"" ¥y" n n∈N | µ H A ) ! ² ¿ ³ " y ¥ ^ " ¥ ! D ¬Å³ Âz! ¬ÅĵYl ³´" K ∈ K "L ³´" g ∈ C (G) p c UK = {n ≥ 1 | xn ∈ K}. NS¦§ p(g) = X n |g(xn )| ≤ CK kgk∞ , |fn (xn )| n∈N ¥y g ∈ Cc(G) »H CK = Pn∈U n < +∞ ±¹² $«6¦Á± «$!² !"¼³§ ! CK = 0 ³ U $ "2²³§y µI?K$« ³ ¥l² #¥ ¦§±G|f (x¬Å³ Âz)|! ¬Å p $ "x«$ !#" ³§!#¥ ¥ ¦Á±8" ¦§Ä ʳ§ K C (G) ·"T¦ ! ¬® ¦´ {p(f )} "T ! Ä»B«$ ¥y³$ "~± ¥ y µLq± :« ! àc f ∈B ¥y$!"ª» ∪f ∈B supp(f ) $W"B ¦Á±Ä" ³´² ¬Å !"«$ ¬ ±Ä« "K " B ² $ ³ ¦§± ³´ " S³ õ Y³j¦Á±I ¥ ³¿"¼ «$!² $ Ê L² $ 0 »¦ ! ¬® ¦´ {φ } "}¥ ! ! O C (G) "@ ! « (φn )n∈N ⊂ Cc (G) $ "D« ¬ ± « "$µ{F&± ¥y" ± W"n«$ n∈N ¬Å¬®¦Ídz§! $« " ³§! C (G)c −→ C (G) K = ∪n≥0 supp(φn ) c b $ "L« !"¼³§!#¥ Ä» lim kφ k = 0. K n→+∞ Yl³´" n ∞ n n a Ea c Cn Y : Cn 3 z −→ (|z1 |, ..., |zn |) ∈ R+n . < 6 © µ ¯ 0, / <6 U ⊂ Cn !/ , / <6>, < / # ! ( Y −1 (Y(U )) = U. _q ¥ a = (a , ..., a ) ∈ R+n »|!Z! " D(a) ¦´ l ¥y³´"^± ! Cn $Hy³§ #¥ C D«$$!" 0 $ "L 1 D ±¹¸ n! a ¥ i = 1, ..., n µ i < 6© µÀ 0, / <6 , < / !/ # ( 5 : " ! ∗n 1) ⊂ U. $# z ∈ U ∩ C . ! : 21$ , ( 21$ D(Y(z)) : ! 2) ( {(log|z1 |, ..., log|zn |), z ∈ U ∩ C∗n } Rn % < 6© µ < ! ! ! , Cn < A ⊂ X G 687 / + X ( % (<( , 796:; ! : ;< ! ! ,# X ! % ! ( cG := {x ∈ X | |f (x)| ≤ sup |f (u)|, ∀f ∈ G}. A cG ! / ( ( ( G : 21$, A / ! X % A < <© µ ! X ! / G : 21$>! Kc ! : " /,: $# G ( ( - : " /,: K ⊂ X % u∈A HA !6 ³´" ¥j¥ ! ¥y² $W" X "¬! ¬³Á± ¦´$¬$!"ª» ¦¿¸l! ¬Å³§± ¦§ ¬Å !" ¥ ½y¦§¬ ½ ³¿Â #¥ ¬Å !#"« !²·Èl1 ³ X " G «$!²ÃÈl ¥ G $ $«·"¼³¿²$¬$!"$ʱ ¦%Z¦§±T¶ ± ¬³§¦§¦´Ð $ ¬! ¬$ zn ...zn , n ≥ 0, µ¿µ´µ , n ≥ 0 »E ¦¿¸l! ¬Å C[z , ..., z ] ¥x $D¶+! «·"¼³´! k 1 n ½ ¦§¬ ½ $1B ¥ kX µ 1 1 k µ <A= # X ⊂ Cn ! , / <6 # ,@! 2134!5 f ∈ H(X) = 6 ! # # 6 796:; # ( / 4!</ ( ( / : < , / 7 # # , X ( % ( 6© µ À < X , / <6 , < / # , Cn : -/ 0 ! C %&"( # !+: < ! ! / - ! ! = / - ! ( ( ! + / @: 21$ 1) X ( % ! + / : 21$ 2) X ( ,( = B: 2% 1$ ! 3) X ( , # / <6 : " : % 21$ ! 4) X , / <6>, : ( 6( : 6 ! % ; ! # # # % 5) X ( < © n C{D¦§ « "¼ ¥ ¥y"O"¼ ¥y² ^¥y! D $¬! W"¼¼±Ë"¼³§ !1y¥TJB½ ° ¬ÅD© µÀ± ! O¦´&?K½ ± ³´" ¨ ͵jYl³¿"S¬Å± ³§!" $!± !" ∗n ¥y!Z ¬Å± ³§!y ®ML ³§! ½±Ä y"$µAH!y" ³§ !#"S¥ !Z $ ¥y¦¿Â "¼±Ä"± !± ¦´Ê¥ Ʊ ¥6JB½ $ °$¬X:© ⊂µ ÀlCµ « $ "¼± ³§!y$¬$!"Ð ³§ !6«$! !#¥»g¬Å± ³§ #¥ T!y¥ !±ª²! ±Ä~"¼ ¥y² ± ! Z¦§± ¦´³´" " $ ±Ä"¼¥ )$!v $¬ ¦Á± ª± !"T¦§$~¬Å! ¬Å ± ~¦´$Z± ¦§³´«ª±Ä" ³§!y ¦´$ ¦´¸! ¬Å ± B¦§ ¦´¸! ¬$ByDCb±Ä¥ !#" (z1 , ..., zk ) −→ z1n ...zkn , n1 , ..., nk ∈ Z, ·"^¦§±«$! y³´"¼³´! \« ¬ ¦§·"B "L¦§ ÊÄÂz« !²·Èl ± B¦§±®«$! y³´"¼³´! \¦´ÊÄ«$!²ÃÈl ¾µCE± ¬Å!yW " ¼±Ä" ³§! ¥JB½ ° ¬ÅO© µÀ¥y"¼³´¦§³§ @³´ $ g $¬& ¦Á± y¦§$g%« $¦§¦´$Q ¥ ÊÊ $ K%¦ ± ¥l"¼$¥y ± !: ³§ $« " $¥ ByH"¼½ ° ¥ "¼± ¦´³§K¦ #¥ ³´²Ä± ¦´$! « H $L± ! ± ¦§Ê ¥ $y$B«$ ! ³´" ³§!yH¨ ·» ·"^© ¥yL¦´$B ¬ ± ³´! $By MO$³´! ½± y"O C∗n µ 1 k C 3 d Hf\d\e yk S d\h !"#%$ &('*),+.-0/21 -435-6,&798)-0:;7<)=8>:[email protected],[email protected]+.-107>AB/[email protected]),:HGI6JE [email protected]+KGLG)-035 [email protected] 4OQP##R.SUTWVXZYL[L\L]0"^#_\` acbL`JYI _\!dfeLgIh"ikjl1 &m1 -Q6,n*3c)@G)n)op,Uq"^L QairBZQs b,[email protected] yz {lN{|w} ZQtVXKYJYLuL]0 Jw~Zl/C&[email protected])@G1 -Q6,3),[email protected]<A*D"/[email protected][email protected]),::%+.31WAB/C3CDF3i-Q6,&I70n6,3i/2),&I7;N6-03;W 1Gm-107 1 &I36,3i/2),&[email protected])-Q1:%70[email protected],OP#q"e.MJ tV=ZYJ[L[L]s^ \JuJY,abL\s" z0LJ Iwx.N/C-0&U6+.:%0$ -0n/[email protected][email protected])=8[email protected]+.&UE3i/2),&I7l)&?oJ-Q)+KG70JBJM mMNIs4,OP#s>VXKYsL]0,^# bLa bLJY _!yzIIHw.ZJ$ [,% {l {|w;IwI~ &>7 [email protected] +I1&35/26n*[email protected]),&I,1 -5oL1 &[email protected] I*JI~ .HZZ4,OP#q"e.MxRmRWVXZYL["]0"^# LuJbaJ`s. 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