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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
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H ¥ ³´" $B ¥y Z »l² $ ³ Â
"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 %ž ¬ ± ³´!
$!y 1›± ! E µQ^¥yб l« ³§! ®%" ¥y"ž¬&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y ¥ E ¥ ! ƶ+ ! « " ³§!5 WÂ
F (Z)
◦
$!" ³§$¦´¦§ ¬Å !#"T­| ! ¥ spec(S) " #¥ ³ $W" ¦§¥ T½ ¦§ ¬Å ½ º ¥ spec(S)
»L ³
◦
µ€q_¥yL" ¥y" k ∈ Z »€! ¥ S± ¦§!y e ¦§±Ð ¥ ³´" D ¥ Z »|y!"L" ¥ L¦§ L« lžÂ
spec(S) 6= ∅
«$³§ !"¼O !"L!#¥ ¦§L%ž¦ÍÇ·Èl "¼³´!: e (k) #¥ ³E$ "L Êk± ¦{%:¨µ€AH!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¦§±ž¶+ ¬®¥ ¦´
Nv"¼¥y"
f(z) =
M
X
c(n)z n , ∀z ∈ C.
M
» ! $¥y"^± ¥y ³E±Ä Ž«$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 µšL¥ 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 µš^¥ S± ¦§¦´! L± ¥ ³_ $¬!"¼ $
›±Ä! ¦§&?K½›± ³¿"¼ ž #¥ ž ³ E "¥ ! ±Ä«$ž žsK± !›± «¾½Z² $ ³ €± !"¦´$H« ! ³¿"¼³§ ! R ¨ û
RH
S·" RS ^¦§®JB½y$ °$¬1¨ µ $ " ž² ±Ä¦Á± ­ ¦´ µšFH± ! D« ž«ª± ! ¥ ²$ !y #¥ ž ³ S "
­| !y® " S −1 !j " ± S­š ! »|!2± ρ(S −1) = +∞ µšFS®¬ $¬& ³ S −1 "D­| !y& " S
!Ç$W" ± g­š ! » !ű ρ(S) = +∞ µ#^ ¥ g $¬Å± Ž¥y$ ! Ž¥yL ³›¦§ @ ¼±Ä" $¥ S " S −1
! K !" ± _­š ! Q ¥ E »Ä¦´KJB½ ° ¬ÅS¨µ L! " ±Ä•²Ä± ¦Á± ­y¦§g$!žÊ ! $ ± ¦ÍµÄC{$Q¬ " ½ ly$
¬ ¦´¹¸$ ›± ! ^¦§ «ª± Hy$H¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ B ² !"I± ¥ ³Q%Ʊ Ž«$³´$S ± ! S¦´ž?K½›± ³´" ž%&" ¥y"O $ ±Ä"¼ ¥ Jbl ¦§³¿"¼Ì^¥ !) ¸Ž¬®­|¦§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«$! ³§ _ $ ± « $ }s± ! ± «¾½
R ¨
! ›±
⊂ E ⊂ L1loc (G)
! C c(G)
µ
E
RHq_ ¥ "¼¥y"
µ
K⊂G
RS Bq_ ¥ B"¼¥y"
E
¼±Ä" ³§ ¶ ±Ä³§¼±Ä!#"_¦§ _« ! ³¿"¼³´! E ¥ ³¿²Ä± !"¼$}
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»
"
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§ !#" ­š ! $ ¥ E Kµ A!
± y ¦§¦§¬&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y} ¥ E " ¥y"O $ ±Ä"¼ ¥ B­| ! M : E −→ E
"¼$¦ #¥
Sx M = M Sx , ∀x ∈ G.
Cg± ¦´Ê°$­ ž¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ D ¥ E $¼±! Ä"¼$ M(E) µYl³´" Ge ¦§ÅÊ ¥ $ž¬WÂ
½ ³§ ¬Å «$ !#" 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$S¥y!T ¥ W†$! $¬®­y¦§ Gf Ge "¼ ¦ #¥ ¥yO" ¥y" M ∈ M(E) " ¥ "¼¥y" f ∈ C (G) ¦§±Ð¶+ ! « " ³§! (M f )θ−1 ±Ey ± " ³§ ! ! % L2(G) » ¥ O"¼¥y" θ ∈ Gf µj?K$« 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 "ž³´ ¬ ½ %T¦ÍÇ$! $¬®­ ¦´ ! !“²l³´ y$®«$± ¼± «·"¼° $žy
¦Í± ¦§Ê° ­ A(E) µ›A! 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
µ€X^! ¶+! «·"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 µŽAH!ž! "¼ d ¦§±S¬Å ¥ B ³§ «$ ° "¼B ¥ Gf+ µ^¥y­y" $! [email protected]¦´K $ ¥y¦´"¾±Ë"
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¨ 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%
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,
6
9
7
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f
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U #
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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¥ ž«$ ! $«·"¼¥ ! #¥ )¦Á±~«$± ¼±Ä« "¼ ³´¼±Ä" ³§! $« $ !"¼:$W"ž² ±Ä¦Á± ­ ¦´ ¥ &" ¥y"žÊ ¥ Cb?KN&µŽFH±Ä! }¦´$Q"¼ ³§Q«¾½ ± ³´" }! ¥ @± ¹Èl³´¬Å[email protected]¥y!Ŭ®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ ¥ g¦Á±S"¼ ¦´Ê³´
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! ¥ ³¿"¼ ›± ! L¦§«$± SÇ¥ !TÊ ¥ ICb?KN Ê $! ¼± ¦E !#"L"¼ °$L ³š$ $!"¼ S I«$ ¦§¦´$O¥y"¼³´¦§³´ $ ›±Ä! D¦§ž?K½ ± ³´" lµj>Q!ºj·"D›± ! ¦´Ð?K½›± ³´"¼ Ðl»š! ¥ ! ¥ ² ! j¥ !yž¬Å·"¼½ Ž
« ! " ¥ «·"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¿Èš»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± !" ž
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) µ›q_¥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"O­š ! ĵ›AH!:! "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 µ€CQ± ¦§Ê °$­ M $^¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ K ¥ L2 (R) $ "
­ ³´$!1«$!y!Ž¥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¶+ ¬$ ž]›¥y ³§ ^j¥ !y&¶+! «·"¼³§ ! f ∈ L2 (R) µšCE±Å¶+! «·"¼³§ ! h $ "
fˆ
± y ¦§$S¦§L ¸Ž¬®­|¦§^ 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¸Ž¬®­š ¦§žÇ¥ !º¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ S !" « ! !#¥ I›± ! D¦´ž«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) < +∞
$ "^± Ž«$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 ¦§® ± ! µšFHρ(S
± ! −1H)¦´ «$± H« !"¼³§!#¥Zy$H·"¼¥ ± ¶+!y ³§
{z ∈ C |
$ ³§Ê!›±Ä!#"}¦§L ½y³´¶
"¼ ° S $«$ ¬Å¬$!" ±
¥Æ ¸Ž¬®­|¦´^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 ¸Ž¬®­|¦§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"ž¬®¥ ¦¿"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
µ›qb¥ x→+∞
a∈R
"
1
1
Rω− = lim ω̃(−x)− x = lim k(S−1,ω )n k− n =
» !1 ! ³´"O¦§±ž¶+ ! « " 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 Ž $B " Í»l¬ ± ³´
¦´Å«$± &« !"¼³§!#¥ $!" ω) Å $ ³§$¥y $& ³ «$¥y¦´"¼ ¥ ¦´$¬$!"¾±Ä³§ ± & ± " ± ¥x«ª± ³´ « "$µ>Q!< š "$»K ³ ω $W")¥ ! ³§y #¥ $¦´«$! Ž¥y »³´¦L!j " ± ¥8"¼ ¥y") ²Ž³´ $!" #¥
» ¥y M ∈ M , a ∈ I ·" f ∈ C ∞(R) µ^ ¥ H"¼ ± ³´" ! S ! ¬Å³´$
\
(M
f )a ∈ S(R)0 ¦´1«$± Ð $Ð $ ±Ä"¼ ¥ Ð 1«$ !#² ω¦§¥l"¼³§ ! ±ª² ω$«1¥ ! )¶+! c«·"¼³§ ! y C ∞(R) µQJ_ ¥y" ±Ä­š »
›±Ä! ^¦§± ± " ³§ ž D«$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 "ž¦§±1¦´³§¬³´"¼ ¥ ¦Á±
M ∈ Mω
c
" ¦§Ê³´I¶+ "¼& $D $ ±Ä"¼ ¥ ž¦n §n∈N
± ¥y³´"¼ (M
»j M : f −→ f ∗ φ $W"D¦§
φ )n∈N
¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ } ¥ L2 (R) ± Ž«$³§I% φ µ›FS ¦§¥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" $» %ž¦§±& « "¼³´!T¯l»y¦´S"¼½y$ °$¬«$! « $ !›± !"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 $W"Å¥y! ³§ ž ¥ R+ = [0, ∞[ ³ δ "Å¥ ! )¶+! «·¼" ³´!6¬Å ¥ ± ­ ¦§Ä»Q " ³§«·"¼ ¬Å !#"
³¿"¼³¿² ¥y R+ ²$ 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§! $± ³§ H ! ³§ ± B 䦧±®¶+ ¬®¥y¦§
< f, g >:=< f, g >δ =
A!: ›! ¿³ "ª» ¥ »
a≥0
Z
R+
f (x)g(x)δ(x)2 dx, ∀f ∈ L2δ (R+ ), ∀g ∈ L2δ (R+ ).
Ua,δ : L2δ (R+ ) −→ L2δ (R+ )
± B¦Á±&¶+ ¬®¥ ¦´ž
Ua,δ f (x) = f (x − a), p.p pour x ≥ a, Ua,δ f (x) = 0, pour 0 ≤ x < a.
A!: ›! ¿³ "ª» ¥ ± B¦Á±&¶+ ¬®¥ ¦´ž
q_ ! "
a>0
»
Va,δ : L2δ (R+ ) −→ L2δ (R+ )
Va,δ f (x) = f (x + a), p.p. pour x ≥ 0.
+ (a) = kU k, ∀a ≥ 0
δf
a,δ
+ (a) = kV
δf
−a,δ k, ∀a < 0.
V¥ ± ! ³§¦›! ¸± ± }y^ ³§ #¥ L L«$!l¶+¥ ³´!»! ¥y} «$ ³´ !y U + $ µ
µ
·" U $ µ V B± ¥:¦§³´$¥: U + $ µ V ·µ€AH!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±&¶+ ¬®¥ ´¦ ž 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^ ¶ž ¥ 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 š "$» ¥ "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¸Ž¬®­š ¦§ T » "¼ ¦§¦§ Ž¥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 › ͎µ´¨ õ
>Q!:"¼$! ± !"^« ¬ "¼D L ³§¬³§¦´³´"¼¥y [email protected]$!"¼ ¦´$B $ ±Ä"¼ ¥ O É-³´$! δWÂ\RL ¶_ "O¦´$B¬®¥ ¦¿"¼³ Â
¦´³§«$±Ä"¼ ¥ »³§¦ "I!›±Ä" ¥ ¦Q ž«$ ! $«·"¼¥ $ Ž¥yÐ" ¥y"D $ ±Ä"¼ ¥ D žÉ-³§ ! $\Â\R^ ¶}± ¬ "
¥ !yS !#"¼±Ä"¼³´!)± ! ± ¦§Ê ¥ L% ͎µ 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»š!~! & ± ³´" ± S ³b" ¥y"S ¼±Ä" $¥ ^ É-³´$! WÂzR^ ¶ ¥ L2 (R+) $W"¦Á± WÂ
" ³§«·"¼³´!ž% 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 Ž $Oω ·²$¦´ $ ¥ O¦§ B¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ } ¥ ¦´$ ± « $LR % ³§ ¥y² !" ·"¼ D± ›± " $ O±¹² $« #¥ ¦ #¥ $O¬ly³ ›«$±Ä"¼³´! }%®¦Á± ³´" ¥›±Ä"¼³´!
ž $¼±Ë"¼$¥y ® É-³§ ! $\Â\R^ ¶\µ•L¥ ®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$ #¥ $¦ Ž¥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"
µ›Uz«$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 < +∞
ȴ
» ¥ ^" ¥y" x ∈ R µAH!2± kS k = +∞ » ¥ ^" ¥y"
a,ω
@
µ
E
C
2
±
+
¶
y
!
«
¼
"
´
³
!
±
Å
Á
¦
±
$
«
!
´
³
"
§
³
!
͎µ´¨ õ@?K $! ± !"ª»K ± ! ¦ÍÇ$ ± «$
a 6= 0
"¼¥yH¦§ H¬®¥ ¿¦ "¼³ ¦§³§«$±Ä"¼ ¥ B !""¼ ³´²Ž³Á± ¥ŽÈj»|³ÍµÇ µ|$ʱ ¥lÈZ% ¥ ! ž« ! "¼± !"¼®¶+³´S¦ ³´ $!lÂ
L2ω (R)
"¼³´" «·¶\µ ·µ
ω(x) = ex
! 1²$ ³ ›
ω
2
c a c LFH± ! @«$·" " ^ $« " ³§!j»!Ŭ!"¼ Ž¥y^"¼ ¥y" ³´ ω "}$!ж ± ³´"
#¥ ³´²Ä± ¦´$!"K%&¥ ! ³§y ω # ¥ ³€²$ 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§±¹²Ž³´! ± ! 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·"® ¥ WÂ
± ³¿"¼³´² µ›F^ ¦§¥ »›$¦§¦´H$W"L¬Å ¥ ± ­ §¦ ĵ€?K¬¬Å¦ ¥y! ³§!)y$L$! $¬®­ ¦´$
$ " R »›¦Á±Ð¬Å ¥ Ž¥y
En = {t ∈ R | b(t) ≤ n}
En
$ "^ " ³´« "¼ ¬Å !" ³¿"¼³¿² ¥ n
E = {t ∈ R | b(t) ≤ M }
± Ì Ê ± ! µšYl ³´"
M >0
"¼ ¦
" )¬$ ¥y )! !6!#¥ ¦´¦§ µVH¥ ³´" "¼Æ% $ ¥ ³´ E »! $¥y"ž ¥ Ž ¥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 Ž¥yIy E µ€C{ ¥ 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"ž¦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!1²Ž³´ 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
" $¦ #¥
µ€AH!: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±ž³§¬ §¦ ³ #¥ #¥ ¥ #¥ D"¼¥y" x ∈ [−a, a] » !T± |γ(x + t) − γ(x)| ≤ M , ¥ a
$Ž¥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´!Ž¥jµAH! >g!1 š "$» 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´ " ž
y∈K
ω0
FS ¦§¥ »›¦§ ³§y
lim sup ω˜0 (y) = 1.
n→+∞
ω0
"L#¥ ³¿² ±Ä¦§$!"^± ¥ ³§ ω µ >g!1 š "$»›!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)
µ€AH!~±
βω = exp
Z
1
2
− 12
sup essx∈R (γ(x + u) − γ(x)) du
= exp
AH!1 $¬Å± Ž¥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 ± °$ ͎µ´¨¨ ·» !:­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§­ ¦´ »Ž¦Á±I! ¬Sy
T
¼±®! " $
kT k.
AH!Æ $¬Å± #¥ #¥
βω −2 kT kBω ≤ kT kBω0 ≤ βω 2 kT kBω .
?K ¦Á±Ð³§¬ §¦ ³ Ž¥y
·"^¦´$O­›± !y $
µ$q_¥y
Aω
"
Rω+ = Rω+0 , Rω− = Rω−0
A ω0
±Ä Ž«$³´$ S± Ž¥ È ³§ ω
"
ω0
!"L$ʱĦ§$ µ
bq ¥ g $¬!"¼ $Q¦ ÃÈy³´ " $! « B ¥Ð ¸Ž¬®­|¦§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¦´¥lÂ
"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§! µ›Yl ³´"
ω ∗ (x) =
"
ω
¥ ! ³´ B ¥y R µ€AH! 1
, ∀x ∈ R
ω(−x)
[f, g] := [f, g]ω =
Z
f (x)g(−x)dx,
¥y f ∈ L2ω (R) " g ∈ L2ω (R) µ•C{$ $¥ŽÈ ¦´$¬¬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 šYl³¿" (hn)n∈N ¥ ! H ¥ ³´" S $$¦´¦§ Ž¥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±ž$!"¼ ±ÄÎÁ!y
n→+∞
lim
S−hn f − f
− f0
hn
lim
n→+∞
Z
+∞
=0
M (S−hn f ) − M f
− M (f 0 )
hn
?K¬Å¬DP$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¿" Ž¥y¦§±Æ $ ³´² $
± ¥: $!y^ ^y³§ " ³´­ ¥y"¼³´! " M (f 0 ). loc
Mf
Ǝµ 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±)­|¥ ¦§ž¥y! ³´" ž R µEYŽ³´" g ¥ ! ж+! « " ³§!y
« ¦Á± C ∞ »E$ʱ ¦§Å%º¨ ± ¥2²³´ ³§! ± ÊÐ Æ: "®%1 ¥ " ›± !y B µ_Yl³ f ∈ C ∞(R) »E!“±
µ€F^! « gM f "^$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 Ž¥y$S¦´&¦´$¬¬ÅI ÐYl ­š¦´ 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±ž¬Å !#" #¥Ç³§¦ÃÈl³§ " D¥ ! D«$!y "¾±Ä!#"
|(M f˜)(x)| dx
2
21
+
Z
"¼ ¦§¦§ Ž¥yÅ
C
12 0
2
˜
|(M f ) (x)| dx
!1­y"¼³´$!"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´!µ›AH!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« ¬ ± « "Ž¥ ± ! 2¦Á±Æ ³´ " ³§­y¥y"¼³´! µ ı Ž«$³´$Ð%
ω
M
$
^
"
ž
%
¥
L
"
$
«
¬
±
·
«
ª
"
µ
M
< 8Žµ < M ∈ M ! # : " /,: / # /+! >, M
%
(<(
(
/ 796: µ̂ 8 ! $# C / # ω (
M
C{D ¸Ž¬®­|¦§y
St
$W"
µ̂M (s) = hµM , e−isx i.
µ|Yl³ φ ∈ C ∞(R) »›¦§W¸l¬&­š¦´
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 ¦ Ž¥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±ž !#" ¼±ÄΧ! ž
"
»›!1±
[M f, h] = ((µM ∗ f ) ∗ h)(0) = (f ∗ (µM ∗ h))(0) = [f, M h]
M f = M ∗f
» ¥ " ¥y"
f ∈ Cc∞ (R)
µ|YŽ³´"
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
< ’Žµ´¨ 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"
µ•Uz¦K $ ¥ ¦´" ¥“" ½ $ °$¬Æ Æ« !² Ê ! «$: ¬³§t!y$ #¥ (V ) $W"ž¥ !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 ¥ µ€q_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´! $ʱÄ−∞
¦§³´" ^y&Ñ−∞! Z
! %ž¦Á±®¬$ ¥ γ (y)dy "L%ž¦§±&¶+! « " 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´!#" $ $ $•%L¦§±Oy³§WÂ
(Yn )n∈N
" ³§­y¥y"¼³´!®± Ž«$³´$B%n Y µYl ³´" f ∈ C ∞(R) "Q ³´" n ∈ N µ>Q!ž !›± !"Q¦ ± Ê¥ ¬$!"_ ¦Á±
n
$¥y² S ¥C{$¬¬Llµ´¨ »!ЬŠ!#" #c ¥ L¦Á±H ³¿²$O M (f˜gn) ± ¥Ð ! @ $g ³´ "¼ ³§­ ¥l"¼³§ ! " M (f˜g )0 µYŽ³´" g ∈ C ∞(R) ¥y! K¶+! «·"¼³´!®$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 ­š¦´ 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 Ž¥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 $! Ž¥y1y¥ « ¬ ± « " K ⊂ R µQA! «$ ! $« ¦´¥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$¥yB ¥ L2 (R)
± l« ³§Å% φ µ•c AH! ! Ä"¼ C ∞(R) ¦ÍÇφ$ ± «$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"Ž¥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 »Ž?K½›± ³¿"¼ L õ#^¥y}±ª² ! $»#¦§±¶+ ¬®¥ ¦´ ¥y³´²Ä± !"¼
Z
lµ´¨¹
¯
Mφ =
φ(x)Sx dx.
R
>Q! j·"ª» ³´" K ¥ ! ¥ W†$ ± «$L«$ ¬ ±Ä« "@ R µŽA!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
sKŽ«¾½ ! ¥y C ∞(R)
%I² ±Ä¦§x$¥y @›± ! K C ∞
µy?K¬Å¬O¦§ K³§!" $Ê ± ¦§ } SsKŽ«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 ·µŽAH!Ð ¬ ± #¥ Ž¥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±ËÂ
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´ "^%ž ¥ "L›± ! [− 1 , 1 ] µ›NS¦´ » ¥ 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) µCE±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 $ °$¬D I]€¥y­ ³§! ³ »y!)" ¥y² Å
≤
kMn f k2ω
≤
Z
sup kSy k
2
kM f k2ω , ∀n ∈ N, ∀f ∈ L2ω (R).
q•± O«$ ! Ž¥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
Ž¥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!: $¬Å± #¥ #¥
x→+∞
ω(y) − x1
ω(x + y) x1
−
, Rω,1 = lim sup
,
sup
x→+∞ y≥0 ω(x + y)
ω(y)
y≥0
sup
+
Rω,1
y≤0
ω(y) x1
ω(−x + y) − x1
−
, Rω,2
.
= lim sup
x→+∞ y≤0
ω(−x + y)
ω(y)
−
≥ Rω,1
"
+
−
Rω,2
≥ Rω,2
·"^!y¥ O ! ³§ ! −
+
Iω,1 := [ln Rω,1
, ln Rω,1
], Aω,1 := {z ∈ C | Im z ∈ Iω,1 },
q_¥ O·"¾± ­ ¦´³§
−
+
Iω,2 := [ln Rω,2
, ln Rω,2
], Aω,2 := {z ∈ C | Im z ∈ Iω,2 }.
(2.16)
»y! ¥ S±ª² ! O­š ³´!Ty §¦ ¥ ³§$¥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 Ž¨
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 ±ž¬!"¼ $ #¥
−
k Sgn −
=k
n+1
X
p=1
λgn k2ω
"L«$ ¦Á±ž³§¬ §¦ 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§¦š·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±ž³§¬ ´¦ 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!yž
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¿¨ õ€AH! ¥y"^± ¦´³#¥ $B¦´DC{$¬¬ÅIlµÐ± ¥ ³§ n∈N
Pn
K? ¬¬ lim
"I«$ ¬Å¬ž¦§ 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±ËÂ
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 =
" !ž$!ž ¥ ³´" #¥
Ž¥y
R
Vp
ω
kS 1 hα,k,p − e−i p hα,k,p kω = 0.
p
µšYl 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 µ€AH!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!ž ¬ ± #¥
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 − . YŽ³´" (f ) ⊂
α,k k∈N
³§·"¼ (2.20)ω,1" Ílµ l¨ p ¥ αρ,1·"L¦§ ³§y ρ µ›NS¦´ 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 jYŽ³´" α ›± ! B "} ³¿" p ∈ N yÈlĵ›Yl³¿" (h
² $ ³ € ± !"D¦§ ³´ " $ lµ L·" Ílµ ¥y p! µqb¥ H"¼ ¥y"
k→+∞
−
ω,1
∗
¥ ! ^ ¥ ³¿"¼
" $¦ # ¥ q ≤ p !
q ∈ N∗
⊂ L2ω (R)
α,k,p! )k∈N
±:
p!
α
α
p!
k S 1 hα,k,p! − e−i q hα,k,p! kω = k (S 1 ) q hα,k,p! − (e−i p! ) q hα,k,p! kω
q
p!
Y
α
α
−i p!
≤
k k S 1 hα,k,p! − e−i p! hα,k,p! kω .
k S 1 − ue
p!
C{ Ž ¥ ³¿"
u∈C,
p!
u q =1,
p!
u6=1
Y
α
k S 1 − ue−i p! k
p!
"^¬ ± ± B¥y! D«$! 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·ÈŽ"¼ ± « " ³§!1 ³Á±ÄÊ!›± ¦´ »Ž! ¥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§! µ›FS!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¬HŽµ 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± Ž¥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
µ€AH!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
&": ( # <; $! # ! / α5 ∈! Bω,2
(
ω
Ílµ
i) kz k = 1, ∀k ∈ N.
k=0
∗
α,k ω ∗
Ílµy¨
ii) lim kSt,ω∗ zα,k − e−itα zα,k kω∗ = 0, ∀t ∈ R.
c {YŽ³´"
k→+∞
−
α ∈ Bω,2
.
A!T±
1
+
− = ln Rω,1
Rω,2
Im(−α) ≤ ln
µF&± °$¦§“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
µ€AH!T !T ¥ ³¿" Ž¥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* $#
α ∈ Bω,1
6 ! <- (vα,k )k∈N ⊂ L2ω (R) 5 (<( = . (
Ílµ
kv k = 1, ∀k ∈ N, lim kM ∗ v − φ̂(α)v k = 0, ∀φ ∈ C ∞ (R).
∗
α,k ω ∗
k→+∞
φ
α,k
α,k ω ∗
c
3)B*
lµ Ä©
,# 5 +
α ∈ Bω,2
.
2 134!5 6"! <5
(
(yα,k )k∈N ⊂ L2ω (R)
- >
= (<( kyα,k kω = 1, ∀k ∈ N, lim k Mφ yα,k − φ̂(α)yα,k kω = 0, ∀φ ∈ Cc∞ (R).
4B
) * , # 5 lµ ¯
kz
k→+∞
2 134!5 6"! <5
−
α ∈ Bω,2
. (
α,k kω ∗
(zα,k )k∈N ⊂ L2ω∗ (R)
= 1, ∀k ∈ N, lim kMφ∗ zα,k − φ̂(α)zα,k kω∗ = 0, ∀φ ∈ Cc∞ (R).
k→+∞
c {AH!yÈl α ∈ B µ€Yl ³§$!" φ ∈ D (R) "
² $ ³ € ± !"L¦§ ³§·"¼$ lµ Ë© } " lµ ¯·µ€AH!:­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" Ž¥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¸ÂWYŽ«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)
YŽ³´" φ ∈ C ∞(R) yÈl$ĵQAH!“ ¥ #¥ α ∈ A T B − µ•Yl³¿"
− 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
ȴ
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 ± ¥ž¬³´! _¥y! B $• $ ³§ Pn e2k Im z ω(−k)2 " Pn e−2k Im z
! « A ⊂ B − S B + µ u « K± ¥ŽÈ k=0
³§·"¼ Í Žµ © {·" 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
} 
#!
|φ̂(α)| ≤ k Mφ k, ∀α ∈ Aω .
c BuÄ«$1%¦Á±MO$¬Å± #¥ Tlµ¿¨» !8± y ¦§³ Ž¥ ± !"± ¥ ³´ ž ³´ «$ "­y" $!#¥8 !
$W"¼ $³§Ê !›± !" ω % Z ¥ ! ³§ " "¼± ! › ± : $O ½y³´¶ "¼K­ ³§¦§±Ä"¼ ¼± ¥ŽÈ)± Ž«$³´$O%ž $ 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§ Ž¥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 ͎µ ÀK¦Ídz§! ʱ ¦§³¿"¼I
|φ̂(α)| ≤ kMφ k, ∀φ ∈ Cc∞ (R), ∀α ∈ Aω .
µ
` a Ec bc
HFH± ! H« " "¼& $«·"¼³§ !»š!y¥ H±Ä¦¿Â
¦´! O $¬!"¼ $B¦´JB½ $ °$¬ l µ´¨ » $! ! «$I›± ! O¦ ³´!#" Ž ¥ «·"¼³§ !µ
L2ω (R)
c €c lYl³¿" ω ¥ ! ³§ b ¥y R µYŽ³´" M ∈ M µ AH!&²Ä±S¥y" ³§¦§³´ ¦´ ³§ ω » # ¥ ³g±Æ " ®³§!"¼ ly¥ ³´"H± ¥ $­y¥y"D ž«$ž«¾½ ± ³´" ĵEAH!¼± $¦´¦§ Ž¥yž¦§$ 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 ž ¼±Ä" $¥ ž "ž" $¦§¦´
#¥ ¥ &" ¥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
Ž
!
¥
»
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
±
ž
¥
!
1
¥
W
†
Â
¥
¿
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¥ ! ž¶+!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 !›± ­ ¦´ »›! $¥y"O ¥ φ
a
Ž¥y (Mφ f )a «$ !#² $ ʶ ±Än∈N
³§­ ¦´$¬$!"›± !y L2 (R) ²$ ¥y! ž¶+! « " 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.
Ž¥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
Ž ¥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·"_š 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 ¦§¦´ Ž¥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± ¥ŽÈž $ ±Ä"¼ ¥ yÉ-³§$!y$WÂzR^ ¶€ ¥ µ›FH± ! 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 Ž $g$¬ ¦´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§! žy C ∞(R) % ¥ "›± !y ]0, +∞[ µJb¥y"Ðj ± ­| »
C0∞ (R+ )
$¬Å± Ž¥y! #¥ j ± °$O¥ !1¼±Ä³§ ! ! $¬$!"O"¼ °$O ³´¬Å³´¦Á± ³´ ^%« $¦´¥ ³{·È I›± ! O¦§±Ð ¥ \Â
$«·"¼³´!Tlµ Ž»›¦§ ³§ δ $ "L Ž¥y³´²Ä± ¦§ !"L± ¥ ³´ B«$!" ³§!#¥ δ y ›! ³ ± O¦Á±&¶+ ¬®¥ ¦§
FS §¦ ¥ »
δ1
δ1 (x) = exp
$W"L" $¦ #¥
ln δ1
lim
K⊂R
q•± O«$ ! Ž¥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±ž$!" ¼±ÄΧ!
n→+∞
" ¥ "¼¥l"O«$¬ ± «·"
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´" Ž¥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|YŽ³´" 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 õFS 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± !" » Ž¥y³}! Å $!y Ž¥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 $W"¥ ! } ³§W"¼ ³§­ ¥y" ³§!jµÏF& ±Ä¥y"¼ ± W "ª» ¥ ! ¥ L ±ª²!y ¥ z > Ty y≥0
(T f )(y) = (S−y T f )(0) = (S−y S−z T Sz f )(0)
= (S−z (S−y T Sy )S−y Sz f )(0) = (S−z T S−y Sz f )(0)
= (T Sz S−y f )(z).
q•± O«$ ! Ž¥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 #¥ &« $¦§¦´ #¥Ç!± %Ŭų´ I$!ZŽ$¥y²Ž ¥ ­l"¼$!y³§b¦Í± ¹Èl³´¬ ±Ä" ³§!y$b¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ µ¹>"¾± !"_ ! ! #¥ #¥ ¦#¥ $_¬ÅŽ ³ ›«ª±Ä" ³§!y
" $«¾½ ! ³ #¥ $O$dz§¬ !#"$»y! ¥ B ! !y! B¦§ ·"¾± ³´¦§$µyAH!)! " T ¦ÍÇ $¼±Ë"¼$¥y SÉ-³´$! WÂ
RL ¶b ! ³ ± K¦Á±&« !²¦´¥y"¼³´!)±ª² « µ ¥ f ∈ C ∞(R+) µ›Yl³ 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 Ž ¥ ± !"ž¦§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¦§±ž ¥ \Âzδ $« " ³§!lµ lnµ›NS¦§ ¥yn f ∈ L2(Rn+)n∈N
»€! ¥ δ
­l"¼$!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 $!" Ž¥y C̃(K)
K
µ€q_± B«$!y #¥ !"ª»
∞
|(T Sz (f˜gn ))(z)| ≤ C̃(K)(kf k∞ + kf 0 k∞ ), ∀z ≥ zK , ∀f ∈ CK
(R)
"L!:«$!y«$¦§¥l" Ž¥y
µT g n
›! ³ ±Ä
< µT gn , f >= lim (T Sz (f˜gn ))(z)
$ "L¥ !yIy³§ " ³´­ ¥y"¼³´!)Ç y Шµ›F&± ¥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±ž«$ ¬ ¦´° " ¦Á± ¥y²Äµ T n
6 < ¢Žµ ¯ < 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 •YŽ³§ !#"
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 µšA!~ ¥ #¥ µ
¥y! ž ¥ 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§!:Žµ© µ Ec c €` FH±Ä! ž «$·" "¼) ¥ \Â
± " ³§Ä»¦Á± ³´! «$³ ± ¦§By³ «$¥ ¦¿"¼O²Ä± " ^ ± ¦§³ # ¥ $g¦§ }¬ " ½ Ž [email protected] % ·È $ ›± !yK¦§
«ª± O L¬&¥ ¦´" ³ ¦´³§«ª±Ë"¼$¥y K !T« !"¼¥y !›± !"L¦§H¶ ±Ä³´" # ¥
`_c a 1
n
|` # cÏa
U1,δ V−1,δ 6= I.
A! δ ∗ (x) = δ(−x)−1 , ∀x ∈ R− .
^¥ O³´!#" Ž ¥ ³´ ! B¦ ±Ä«$
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±ËÂ
k→+∞
" $¥ +« ¶\µ€C{$¬¬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´ " ž¥ ! Ð ¥ ³¿"¼
³§·"¼[−a,a]
$ lµ K " lµ ·µ€AH!:­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© " ͎µ© ¨ { ! ­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 $¬Å± #¥ Ž¥y) ± ° ž¦Ídz§! ʱÄ"¼¦´³´" y:?± ¥ «¾½¸ÂWYl½ K± " Ì »g!5­l"¼³§ !"
Ž¥y ¥ z ∈ C ± ¥Ð¬Å³´! ¥ !yLy$g ³§ P∞ e−2k Im z δ(k)2 " P∞ e y³´² ÊO·"
δ(k)
!б Ω ⊂ B − S B + . YŽ³´" φ ∈ C ∞(R) µYŽ¥ k=0
!y Ž¥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 Žµ ·» !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" $¥ O " #¥ ³² ³ n
]•³¿Èl! + (y)
kTφn k ≤ kn kT k, avec kn = sup δf
a ∈ Jδ
µ ^¥ L±ª² ! 1
0≤y≤ n
[
c
|(φ
n )a (x)| = |φn (x + ia)| ≤ kTφn k ≤ kn kT k , ∀x ∈ R.
L¥ ¥l²! L·ÈŽ" ¼± ³´ ((φ
¥ !yI ¥ \Âz ¥y³´"¼Ä» Ž¥y³{«$!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$¬Å± #¥ ! Ž¥y ¥ Rg ∈ L2 (R) »
n→+∞
lim
n→+∞
Z R
1
0≤t≤ n
f ∈ Cc∞ (R)
»
d
d
[
(φ
)
(x)
(f
)
(x)
−
ν
(x)
(f
)
(x)
g(x) dx = 0.
n a
a
a
a
AH!)«$ ! «$¦´¥y" #¥ » ¥y f ∈ C ∞(R) » (φ
« !² Ê ¥ ¦§±I" ¦§Ê³´L¶ ±Ä³§­ ¦´
d
[
)
(f
)
n a
a
c
L2 (R) ² ν (fd) µ?K¬¬Å}!б (T f ) = Pn∈N
+
d
[
)a ) = P + F −1 ((φ
n )a (f )a ),
¦§±^ ¥y³´"¼ ((T f )a) a «$ !#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
¥
"
Ž
y
¥
(T f )
a
n
a
d
(T f )a = P + F −1 (νa (f
)a )
HF ±Ä! ®¦Á±1 ¥ ³¿"¼Ä»_! ¥ Ž¥y J◦ 6= ∅ µ_>"¾± !"žy! ! #¥ (φc) $W"ž¥ ! ¥ ³¿"¼
δ
n n∈N
H¶+ ! « " ³§! K½ ¦§ ¬Å ½ K¥ ! ³¿¶+ ¬$¬$!"K­š ! $ 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´!
◦ µŽNS³§!y ³Í»
¥ g"¼¥l" a ∈ Jδ »Ž¦Á±D ¥ ³¿"¼ (φbn(. + ia))n∈N « !² ÊS² $ ν(. + ia)
ν ∈ H ∞ (Ω δ )
± ¥) $! By$ ³´ " ³§­y¥y"¼³´! $µŽF&Ç¥ !:± ¥l"¼ ^« " »l¦Á± ¥ ³´" ((φ
«$ !#² $ ÊH² ν ¥ [
n )a )n∈N
a
¦§±ž" ¦§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 ¦ Ž¥y lim
»{!x ­y"¼³´$!" kνk ≤ kT k µbYl³§!y!2! ¥ e+
n→+∞ sup0≤t≤ δ (t) = 1
±¹² ! kνk ≤ c kT k »š c $ "D¦§± $« ! "¼± !"¼Ð ! ³§® ± ! ¦Ídz§!"∞ Ž ¥ « " ³§!jµj?K ¦Á±Æ« ¬ÐÂ
∞
δ
δ
¦§°·"¼¦Á± $¥l² µ 1
n
C
?RLNHqQUzJOM^> Sf†ild
Hf\d jkilh hji6e ykiyh Sh eBh Hf\d†i 8f\h6h Hk j h ^h
3 k Sk
Sh S
d†ilh
c C{$: $ ¥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 $I ¥y³´"¼ y !"I" ¥ D¦§ D«$Ž «$³´$!"¼I¼± ¥l¶¥ !x! ¬®­ › ! ³Q !#"
!#¥ ´¦ $µ€^ ¥ O ¥ ! Ž¥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´!Ž¥y »l± ¦§ g¦§O"¼½ ° ¬ÅL ¥ Ê ¼± y½ L¶+ ¬^! ¥ K ! ! Ž ¥y ψ "}¥ !Æ $ ±Ä"¼ ¥ ­| !yž ¥ E µFH± ! D«$ž«¾½ ± ³´" ž!2« ! ´³ °$ ž $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" $¥ %ºy¬ ±Ä³§! E µ}Yl³ S !Ç$ " ± ­š ! spec(S) ³§Ê ! T¦´: $« " T S Q» ¦§±2¶+ ¬·Â
S
"¼¥ S| µjML± $¦´! Ž¥y S $W"H¦§± ¦´¥ ·"¼³´" ÃȎ"¼$!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 #¥ ! $¥y"_± Ž«$³§ b%O" ¥y"E¬®¥ ¦¿"¼³ ¦§³´«ª±Ä" $¥ ◦
¥ E ¥ ! 2¶+! « " ³§! L∞ ¥ spec(S) Ž¥y³$ "T$! ¦§¥y1½y¦§¬ ½ ¥y spec(S)
»B ³
◦
µOq_ ¥ )"¼¥y" k ∈ Z »B! ¥yT± ¦§!y e ¦§±“ ¥ ³¿"¼2 ¥y Z »B !#":" ¥ :¦§ spec(S) 6= ∅
k
«$Ž «$³´$!"¼Ð !#"!#¥ ¦´Ð%2¦ ÃÈl«$ "¼³§ ! y e (k) #¥ ³B$W"Å
ʱ ¦B%5¨µ•Uz¦$W"Å« ¦Á± ³´ #¥ )"¼ ¥y"
k
¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ • $ " ³§!"% F (Z) $ "}¥ !Å ¼±Ä" $¥ @ L«$!² ¦§¥y" ³§!jµ#>g! š "ª» ¥ M ∈
"
» !T± ¥y N ± $ÌIʼ± !y
M(E) a ∈ F (Z)
"
Sx = y
Ma = M
N
X
NS³§!y ³Í»l!:²³¿" Ž¥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 ) µ›AH! ¥y"L± Ž«$³§ %®«¾½›±Ž¥yI¬&¥ ¦´" ³¿Â
¦§³´«ª±Ä" $¥ ¥y! D $ ³§D ICE± ¥ $!"O¶+ ¬$¦´¦§ M
f ! ³§ ±ÄL¦§±&¶+ ¬&¥ ¦§
f(z) =
M
^¥ ^± ¦´¦§! O± §¦ M
f ¦§D ¸Ž¬®­|¦§I
$ ³§H¶+ ¬Å ¦§¦´H$! ¼± !"
ã(z) =
X
n∈Z
c(n)z n , ∀z ∈ C.
M
M
X
n∈Z
µ|Nr"¼¥y"
a∈E
a(n)z n , ∀z ∈ C.
C!
»€! ¥y"H±Ä¥ ³•± Ž«$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 ± ¥ $!"L¶+ ¬$¦´¦§$} !"L¦§ B«$Ž «$³´$!"¼L !"L ^ ¥ ³´" $O›± ! E µ€AH!~±
ga(z) = M
f(z)ã(z), ∀z ∈ C.
M
L¥ Æ«¾½ «¾½ ! )¦§ Ųı ¦§ ¥ Å r ¥ ¦´$Ž¥y$¦§¦´$Å! $¥y")y! ! ¥ !< $! ± " ¥y"
¥ #¥ ± W"¼¥l"Æ% M
f(reiθ ) µ@C{$Æ $ ¥ ¦´"¼±Ä"¼Å % ­y"¼ !Ž¥yƛ± ! Å«$Zy¬ ±Ä³§! ¥ O$ ± «$ ^ DsK± !›± «¾½ ± W"¼³§« ¥ ¦§³´$ $»Ž! ¥ O± ¬°$!y$!"L%ž«$! « " ¥ #¥ D¬ $¬H ¥ B! ½¸ Ä"¼½ ° $K¦§B ¸Ž¬®­|¦´B B"¼¥y"@¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ b$W"@¥ ! O¶+! « " ³§! L∞ ¥ @¦Á±S¶+ !"¼³§° B ¥
« "¼ ž S ·"½ ¦´¬Å ½ D­| !y$ž% ¦ ³´!"¼$ ³§ ¥ ^y¥Z $«·"¼ žy S ³_«$® ! ³´$^! "
±Ä ²Ž³§ ĵb^¥ &± ¦§¦´! ¬Å± ³§!" $!›± !"I !y! $ ¥ !“± $$¥“y$& $ ¥ ¦´"¼±Ä"¼I«$!y!Ž¥y$µ_FH± ! ¦§
«$± ± " ³§«$¥y¦§³§ E = lp(Z) »›! ¥y^±ª² ! L¦§ ¥ ¦¿"¾±Ä"O« ¦Á± ³ Ž¥yI ¥ ³´²Ä± !"$µ
6 < yµ´¨< E = lp(Z) 1 ≤ p < +∞ # ! $# 5 M ∈
.
% &"( .
f ∈ L∞ (T) /
M(E) . M
fkL∞ (T) ≤ kM k.
kM
HA !Z ± $¦§¦´ Ž¥y®›± ! ^«$ «ª± ^³§¦E$W"H­ ³´$!~« ! !#¥ #¥ spec(S) = T. q_ ¥ p = 2 »|!
±~·²Ž³§ ¬Å !#" kM
fkL (T) = kM k µ•C{ ®¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ !#"®± ¥ ³K·"¼Æ·"¼¥ y³§$ ›± ! &¦§$
± « $L% ³§ µ AH!:± y ¦§¦§ ³§ } ¥ Z "¼ ¥y"¼^ ¥ ³¿"¼HW"¼ ³´« " $¬$!" ³¿"¼³´² µyAH!:± l« ³§
%Ð¥ ! ³´ ω ¦§ L ±Ä«$$
∞
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 < +∞ µ€Yl½y³§$¦´ O±ž $¬!"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«$¦´¥y"º¥ ! 6¦Á± ʓ« ¦Á± 6Ç$ ± «$ x s±Ä!›± «¾½µDq_± 6= ∅
ÃÈy ¬ ¦§O›± ! Q¦´B«ª± ±Ä "¼³´«$¥ ¦´³§ lp(Z) ¦ ³´!#" $ ³§$¥y ¥Ð $«·"¼ O S $ "g²Ž³§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 Ž¥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¿²Ž³Á±ËÂ
¦§$¬$!"H² $ ³ › ś±Ä! ¦§ $ ± «$ & Ðs±Ä!›± «¾½º !"I¦Á±Æ! ¬Å&$ "I ›! ³´ ± ¥ ! Ð ³´
«$!² Ê !"¼ µ|VH¥›±Ä!#"B%®¦Íǽ#¸ "¼½y°$ RS ûl³´¦š ¥ 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´! $»Ž³§¦|·Èl³´ "¼D±Ä¥ ³j B$ ± «$ Lys± !›±Ä«¾½)›± ! B¦´$WÂ
Ž¥y$¦§ k(x(n))n∈Z k 6= k(|x(n)|)n∈Zk " #¥ ³² 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 ¥ ÅJbŽ ¦´³´"¼Ì
¥ y$B$ ± « $B sK± !›± «¾½) S ¥ ³¿"¼$B ¥ Z+ = N µ›AH!)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 ž ¥ ³¿"¼$& ¥ Z+ + µ Z− û• !#" "¼¥y®¦§ «$Ž «$³´$!"¼O¼±Ä¥y¶¥ !1! ¬&­ ›!y³{ !"^!Ž¥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´ $ $H¦´$^$ ± «$ žs± ! ± «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 #¥ ! Ž¥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»
!ž ­y"¼³´$!" #¥ S| »Ä¦Á±^ $W"¼ ³´« " ³§!®y S % E + $W"Q¥ !ž $ ±Ä"¼ ¥ ­| ! K E + ›± ! E + µ
L¥ @ ³§ ! #¥ SE + $ µ S  "}­š ! Ž¥ ± ! 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§!" Ž ¥ 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" $¥ $»Ž! ¥ O ›! ³§ !yO¶+ ¬$¦§¦´$¬$!"
Te(z) =
X
n∈Z
Tb(n)z n , ∀z ∈ C.
C2C
+
A! ± $¦´¦§ Te ¦§ ¸Ž¬®­|¦§Å T µ_>g!x"¼$! ± !"®«$ ¬ " & ³§¬³§¦§³¿"¼¥ y$$!" Ʀ´$&¬&¥ ¦´" ³¿Â
¦§³´«ª±Ä" $¥ S "D¦´$H ¼±Ä" $¥ žJ_Ž ¦§³´" Ì®³§¦b$W"D¦§Ê ³#¥ &yž$±Ä" " $! %Æ«$ #¥ Te ³´"
¥ ! O¶+! « " ³§! L∞ ¥ @¦Á±H¶+ !" ³§°$ B spec(S) ∩ (spec(S ))−1 µUz¦›$W"K ʱ ¦´$¬$!"@!›±Ä"¼¥y $¦
Ы$! « " ¥ #¥ Te " ½ ¦´¬Å ½ & ¥ D¦Ídz§!" $ ³´$¥ S−1 spec(S) ∩ (spec(S ))−1 » ³
«$ž ! ³´$H!Ç$W" ± S²Ž³´ µUz¦_$W"D«$¦Á±Ä³§ Ž¥y ³ M $W"D¥ !2¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ B ¥y E−1− ⊕ E + »
E − " E + !"®y$& ±Ä«$$& Æs±Ä!›± «¾½“ ¥y³´"¼ & $ $«·"¼³¿²$¬$!"ž ¥ Z− " Z+ »
± ¦§ P +M "¥y!Z $¼±Ë"¼$¥yS J_Ž ¦´³´" Ì ¥ E + µ?K ! ›± !"ª»š¬ ±Ä¦§Ê I¦§±Å ³§«¾½ žy
¦Á±T¦´³´" "¼$ ±Ä"¼¥y ¥ ®¦´$ ¼±Ä" $¥ ® ÆJbŽ ¦´³´"¼ÌÄ»E³§¦@ $¬&­ ¦§Ž¥j³´¦K!± ± & "¼ "¾±Ä­ ¦§³ #¥
"¼¥y"@ $ ±Ä"¼ ¥ g OJ_Ž ¦§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_Ž ¦´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 " Ž¥y S !j " ¦§¥ O¥ !1 $ ±Ä"¼ ¥ B³§!² ³§­ ¦´
³´!²$ S µ›?K ! ›± !"ª»y! ¥ B±Ä¦§¦§ ! }!y¥ B ²Ž³§K ¥Æ¶ ± ³´" #¥ S S = I µ›AH!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 WÂ
±Ä«$$L Is± ! ± «¾½ #¥ ³² $ ³ › !#"^¦§ B½¸ " ½ °$ $ R ¨ û RSK " R^ ·µ
` ¥ ! ³´ ¥y Z µ{Uz¦ "D"¼ °$¶ ± « ³§¦§& ž²³´ Ž¥y¦´$ ±Ä«$$ lp (Z) ¥ 1 ≤
³ $!"H! S½¸ " ½ °$ $ µE^¥ S± ¦§¦´! O !y! $ Ž¥y$¦#¥ «ª± ω± W"¼³§« ¥ ¦§³´$ #¥ ³
³´¦§¦§¥y "¼ $!"H¦´$H ³´" ¥›±Ä"¼³´! L ¥ ³¿²Ä± !"¼$ž S ·" S −1 !"D­| !y$H·"¦§® « " ž S $W"D ³¿"
¥ !yž«$¥ ! ! žj³´!#" $ ³§$¥yH! !Z²Ž³§y® ³¿"¥ !« $ « ¦§Ä»¥ ! I ¼±Ä" $¥ S ·" S −1 ! "
±Ä­š ! ^¥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´"
µŽNS¦´ S " S −1 !"­| ! " kSk = e » kS −1k = e−1 »
³@¦§ $«·"¼ ) S $W"ž¦§«$ « ¦§Æ ż±ª¸ ! e "ž³´¦g$ "
dz§!" $ ³´$¥ H²Ž³´ µEFH±Ä! D«$Ыª± ¦´ž ¸Ž¬®­|¦§žyž«¾½›±#¥ ¬®¥ ¦´" ³ ¦´³§«$±Ä"¼ ¥ ^ "I¥ ! ž¶+!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 ¸Ž¬®­|¦´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± Ž¥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
Ž¥y³š "@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¦§žJB½ ° ¬Åyµ¿¨Ð$ " #¥›± !"I¬ ¬Å®² $ ³ ›Äµ•AH!2 ¬ ±Ä #¥ #¥
·" S 2 "­| ! µšq•± 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!#"
«$ " "¼ž 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 ! ¦ " ± $µ€q_± «$! #¥ !#"$»l³´¦y "}«$¦Á±Ä³§ #¥ spec(S) = C µ#YŽ³›¦§BJB½ ° ¬ÅLyµ¿¨O "¾±Ä³´"Q²Ä± ¦Á±Ä­ ¦§Ä» ¦§K ¸Ž¬®­|¦§
S 2 #¥ ³Q$W" z2 ± ¥y¼± ³¿"D "¼Ð­| !y ¥ C µjUz¦Q$ " ²Ž³´ $!" #¥ Å«Ä " ³§¬ ³´­ ¦§ÄµšN^³§! ³Í»
! ¥ « ! "¼±Ä"¼ ! Ž¥y!y "¼ ž $ ¥ ¦´"¼±Ä"! $¥y" ± " ÐÊ$!y$¼±Ä¦§³§ ¥ ¦§ $ ± «$ y
s± !›± «¾½1 ¥yL¦´$Ž¥y$¦§O! ³ S ! ³ S −1 ! D !"^­š ! $ 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´!Ž¥y) "ž« 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§! }­š$± ¥ « ¥ ) ¦´¥ K« ¬ ¦§³#¥ $}«$¬¬Å ± K·Èl$¬ ¦§ xp+sin(log(− log(x)) , p >
√ ² $ ³ ›$!"S± ¥ ³{¦§ B«$!y ³´" ³§! B #¥ ³´ $ µšYl³´"
¥ ! ³§ B ¥ Z µ€q_ ! 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» ž$ " ¥ !G ± « ~
s±Ä!›± «¾½ ² $ ³ €± K,ω
!" ! I½#¸ ¼" ½y°$ R ¨ " R^ ·µEqb¥ Ž ¥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! ž ¥ ¿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± !›± «¾½žj ± °$ Í» µ Uz¦y "g·²l³´ $!" #¥Ç³§¦Ž² ³ ! ½¸ "¼½ ° R ¨ H " RS õEYl³
¥ S −1 $W" |­ ! Ð ¥ l{q} (Z) ¦´žJB½ $ °$¬yµ´¨ µ$W"I²Ä± ¦§± ­ ¦´®›± ! S
µ
l{q} (Z)
` FS ³´Ê! ! ± C ¦ÍÇ$ ± «$) ®¶+! «·"¼³§ ! &« !"¼³§!#¥ $» 2π  $ ³§Ž ³#¥ &·"Ð%~²Ä± ¦§ ¥ « ¬ ¦§·Èl H ¥ R šµ [0,2π]
q_¥y f ∈ C »|!Z!y "¼ fˆ ¦Á±Å ¥ ³´" ® H «$Ž «$³´$!"¼S ž]›¥ ³§ ^
[0,2π]
µf €qb ! ·"
C = {fˆ | f ∈ C[0,2π] }
µCQǽ#¸ "¼½y°$ RI¨ ^
"²Ž¼± 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´® ž]›¥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 ·µ€qb¥ A!:« ! "¼±Ä"¼ #¥
Ž¥y³$W"ž›± ! 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«$Ž «$³´$!"¼L I]›¥ ³§$K D¦Á±®¶+! «·"¼³§ !
0
t −→ f (t + α),
µ Uz¦}$W"Ð ²Ž³§ !" Ž¥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 $¥ŽÈЭš ! $µŽ^ " B"¼½y$ °$¬L ± ¦§³ #¥ B ! «O›±Ä! ¦ÍÇ$ ± «$ C » Ž¥y³ ¥ "¼± !"®!± ±Ä® Æ­›±Ä y:Yl«¾½›± ¥y $ž·"®!Ç$!"¼ ) ! « ± ® ± ! ®¦´
«ª± y$L$ ± «$ H s± !›± «¾½T " ¥ ³§ ± Hu$¦´¦Á± µ L¥ O¶ ± ³§ ! L $¬Å± Ž¥y$L± ¥~¦´$«·"¼$¥y
Ž¥yI ± ! C ³§¦ÃÈl³§ " D $O ¥y³´"¼ a "¼ ¦§¦§ Ž¥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§ $µš?K $!y›± !"ª»¦´$^ ¥ ¿³ "¼$ ›! ³§ S !"­ ³§ !
$! $L›± ! C µ
n=−k
+` a
bc
Yl³´" E ∗ ¦ ± « ^y¥›± ¦› E µŽ^ ¥ 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 ± #¥ Ž¥yI ³{! |||x||| = sup kψ (x)k » ! ¥ Ok­l"¼$!y! ¥ ! ®! ¬® ¥ E #¥ ³¿² ±Ä¦§$!" ž%ƦÁ± !y ¬ k.k µšCg±ª²Ä± !"¾± Ê&z∈Tž¦Á± z !y ¬ |||.||| " #¥
! ¥ L±ª²!y
sup |||ψz (x)||| = 1, ∀z ∈ T.
q•± O«$ ! Ž¥y$!"ª»š¼± ! $ " y Ê ! $ ± ¦§³¿"¼Ä» ! ¥y"Sy !›±ª² ±Ä!#"^«$ ! ³´ $ $ Ž¥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
µ€CE±ž¶+ ! « " ³§!
x∈E
Ψx : T 3 z −→ ψz (x) ∈ E
"Æ« !"¼³´!Ž¥y µ}>g!8 š "ª» ¥ x ∈ F (Z) Á¦ ±2«$!"¼³´!#¥ ³´" T "Æ·²Ž³§ !#" ~·" ¥ x ∈ E
#¥ ¦§«$ ! #¥ :$¦§¦´Æ «$¥ ¦´Æ³§¬¬$ ³§±Ä"¼ ¬Å !#" ½¸ "¼½ ° R ¨ û RS ·" SR ·µL¥ ³´!#" Ž ¥ 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§±ž«$!"¼³´!#¥ ³´" L¥ H±Ä¦§¦§ ! B $«
µ€q_ ¥
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) µ›Uz¦ "L¶ ± «$³´¦§HyD² ³§ #¥
yµ c) ∗ x, ∀z ∈ T.
Mx (z) = (ψz ◦ M ◦ ψz )(x) = ψz (M
>g!1 š "$» ¥ 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#¥ Ž¥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´" Ž» ! ¥ K± ¦§¦´! @! ¼" M »l¦ÍÇ ¼±Ä" $¥ @ H« !²¦´¥y"¼³´! ± µ€^ ¥ L¹± ² ! L­š ³´!: T
¥ ¦´$¬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´¬ ¦§žÇ­y" $! ³´I¦´Å« !"¼¦§Ð ¦Á±:! ¬ÅÐ
$¥y² Æ¥y" ³§¦§³´ ž¦´$
± Ê¥ ¬$!"¼D Q± ›± " $I%)! " Ð ³¿"¼¥›±Ä" ³§! #¥ ³Q "I­|ª± ¥ « ¥ º ¦§¥yÊ$! ¼± ¦´ µ{L¥ ± ¦´¦§!yB! ¥ O ²Ž³§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.
µš>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 $»l!: ¬ ± #¥ Ž¥y ¥ Z
T
c
gk (z)ψz −1 (M )dz ∗ x.
»› !T±
|n| ≤ k
Z
Z
c
c(n)dz
gk (z)ψz −1 (M )dz (n) = gk (z)z −n M
T
± ¦´ #¥ ¥ |n| > k
T
c(n),
c(n) = 1 − |n| M
= gbk (n)M
k
+
1
»y!~±
Z
CQÇ$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´¦ ²Ž³´$!"O¶ ± «$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
»š!y¥ S ­y"¼ÃÂ
≤
"L¦Á±® ! ³¿"¼D
Yl¥ ! #¥
¦ÍÇ ¼±Ä" $¥ F (Z)
›± !y
S −1
Z
T
|gk (z)| kψz −1 kkM kkψz k kakdz
≤ kM kkak, ∀a ∈ F (Z)
E
! ¥ O ! !
!Ç$ " ± b­š ! g"¾± !y ³§ #¥
Mk =
$ " |­ ! Ä»{³§¦Q " «$¦Á±Ä³§ #¥
! ¥ Lª± ²!y
S k−1 Mk = 1 −
Ž¥y
kMk k ≤ kM k, ∀k ∈ N.
k X
S
Mk
"•­š ! 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
$ "K­š ! ^ " Ž¥y S −1 !Ç$ " ± }­š ! »Ž! ¥yK«$ ! «$¦´¥ ! Ž ¥y M
c(−k) = 0. FSH¬ $¬ »#$!
«$¬ ¼± !" M " S p » ¥ p = k − 2, k − 3, ...., 1 »l ! ¬ !"¼ #¥ M
c(−n) = 0 » ¥ k
µYl³ S −1 $ "Э| ! Ʊ ¦´ #¥ S ! " ± ®­| ! Ä» ±Ä&¦§Å¬ ¬Åż± ³´ !y! $¬$!"ª»
n > 0
! ¥ ¥y² ! O«$¬ ¦§·"¼ B¦Á± $¥y² & ¥1¦§ ¬Å¬ µ µ
n=−k+1
lµ© <
!5 /,: , / /,: # B/ ! : < !
(
<
=
) % ! φ ∈/ -F (Z)
( φ! ∗E8⊂- E.! ! , 1 ! / !+ /
!
1)
#
,# S S −1
(
# . ( #
(
E
e
|φ(z)|
≤ kMφ k, ∀z ∈ Ω :=
2)
S
! /! # / 4 ! = .
3)
S
! # / 4 ! = .
n
o
1
≤
|z|
≤
ρ(S)
.
ρ(S −1 )
! # !
φ ∈ F (Z− ) .
n
1 o
e
|φ(z)|
≤ kMφ k, ∀z ∈ O := z ∈ C | |z| ≥
.
ρ(S −1 )
S −1
! /! # !
φ ∈ F (Z+ ) .
n
o
e
|φ(z)|
≤ kMφ k, ∀z ∈ U := z ∈ C | |z| ≤ ρ(S) .
S −1
) . ) / # ! /
(
6 >
! / !
c gYl¥ ! I ±Ä­š #¥
·"
S
S −1
µ€FS$¥ŽÈ1«ª± O $ $!"¼ !#"&
spec(S)
¨µ›Uz¦{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)
ȴ
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 $« ¥ ¦´ #¥
± $ÌIÊ ± ! » !~±
p→+∞
e
e
e
|φ(z)|
= kφ(z)h
p k ≤ kφ(z)hp − φ ∗ hp k + kφ ∗ hp k,
e
|φ(z)|
≤ lim kφ̃(z)hp k ≤ kMφ k.
P~±Ä³§!"¼ !›± !"L ¥ ! Ž¥j!1$p→+∞
W"S ± ! O¦§ ¥lÈl³§° ¬Å«$± $µšAH!1±
Mφ∗ (y) =
X
φ(n)(S ∗ )n (y)
n∈Z
=
·"^³´¦$W"L ²Ž³§ !" #¥
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´ ĵ•YŽ³ S
!Ç$W" ± }­š ! Ä»y± ¦´ # ¥ S −1 "K­š ! »Ž! ¥yK¥y" ³§¦§³´ [email protected]¦´^ « " Sy S "K¦§[email protected]¬ $¬$
± Ê¥ ¬$!"¼ µ›NS³´! 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| ! ${·"_ ¥ ! Ž¥y 1 < ρ(S). ]_³¿Èl! φ ∈ F (Z) µ•Yl³´$!" 0 < R1 < R2 y$¥lÈ $¦§I"¼ ¦§ Ž¥yƦ§ «$$ «$¦´$ρ(S C ) " C ±ª¸!T $ « " ³´¶ R " R ± ± W"¼³´$! ! !"H% spec(S) µš?K¬¬¦Á±
2
¶+! « " ³§! φRe $W"½ R¦§¬ ½ B ¥ C\{0} " 1|φ(z)|
» ¥ z ∈ C ∪ C »± ° e
≤ kMφ k R
R
¦§ ³§! « ³ H ¥1¬ ±ËÈl³´¬®¥ ¬1»l!1­y" ³§ !#"
−1
1
2
1
2
n
o
e
|φ(z)|
≤ kMφ k, ∀z ∈ ΩR1 ,R2 := z ∈ C | R1 ≤ |z| ≤ R2 .
>Q"¼± !"L ! ! #¥ D¦§ O«$$ «$¦´$
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 lµ€ML± $¦´! #¥
€c yµ
lim kMk x − M xk = 0, ∀x ∈ E
k→+∞
"
k, ∀k ∈ N.
DkM
W" kk#¥ ≤ kM
q_
M k = M φk .
A! yÈl r ∈ [ 1 , ρ(S)] µ€F
ρ(S −1 )
q_¥yH± ¦´¦§$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 ! Ž¥j!ž!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«$ ! Ž¥y$!" ¥ a ∈ F (Z) » (φ
«$ !#² $ Ê ¥ O¦Á±ž"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$!"¼ ±ÄÎÁ!
k→+∞
k→+∞
(M a)r (n) = (νbr ∗ (a)r )(n), ∀n ∈ Z, ∀a ∈ F (Z).
N^³§! ³Í» !:¬!"¼ #¥
c)r ∗ (a)r = νbr ∗ (a)r , ∀a ∈ F (Z)
(M
·"
c)r = νbr .
(M
q_± O«$! #¥ !#"$»š !~±
f(rz) =
M
X
c(n)r n z n =
M
X
νbr (n)z n = νr (z), ∀z ∈ T.
?K¬¬ kν k ≤ kM k »€³§¦ !~ «$¥ ¦´ Ž¥y M
f $W"S $!"¼³´$¦§¦´$¬$!"L­š ! ± r ∞
" ¥y"L«$ «$¦´D Ω µ|Yl³ ρ(S) = 1 »›³´¦j$W"L ²Ž³§ !" #¥ 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 $ °$¬Ð PT!"¼ ¦Í»{! ¥y ¥y² ! I$! ÃȎ"¼¼±Ä³§ Ð¥ !yÅ ¥ W† ¥ ³¿"¼ #¥ ³g«$!² $ Ê ¥ ! ³ Â
¶+ ¬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¥ ! Ž¥j³´¦H·Èl³§W"¼ C ⊂ Ω "¼ ¦
r
Ž¥y Cr !Ç$W" ± 1³§! « ¦§¥ )›±Ä! spec(S) µOYŽ³´" α ∈ Cr "¼$¦ #¥ 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± $± !"
a
±Ä
(S − αI)a
» !1" ¥y²
gr (z) = νr (z)F ((S − αI)a)r (z)
(a)
gr (z), ∀z ∈ T, ∀a ∈ F (Z),
= νr (z)(rz − α)(a)
H" !± (rz − α)ν (z) = 1 µjA!Z± α = rz , z ∈ T µjqb¥ S" ¥y" > 0 »|³§¦E·Èl³§W"¼ z ∈ T
"¼$¦ Ž¥y |rz − rz |r ≤ " |ν (z )| ≤ kν k0 µb0?K ¦Á±Æ³´¬ ¦´³#¥ Ž¥y 1 ≤ kν k "D ! ¥ ± ­š ¥y"¼³´ ! K% ¥ ! 0S« !"¼ ± ³§«·"¼r³´! µŽ^¥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$!1 ¬ ¦§± ª±Ä!#" Ω ± O ¥ U µšqb¥ O¦§ O ›!y³´"¼³´! B O ·" U ¦´
¦§$«·"¼ ¥ $¥y"B W"¼ O± ¥:C{$¬¬ÅSyµ©yµ›AH!) ¬ ± #¥ #¥ #¥›± ! spec(S) = O »l! ¥ $ ¥y³§ ! ρ(S) = +∞ » ± ¦§ #¥ Ž¥ ± ! spec(S) = U »›! ¥ L±ª²!y ρ(S −1 ) = +∞ µ ` #cÏa E c
` HF ± ! }«$ " "¼ ± " ³§Ä»Ž!y¥ @ $¬!"¼ ! K¦´^JB½y$ °$¬Hyµ lµŽq•± @¦§L¬ $¬L ± ³§ ! ! ¬Å !"
Ž¥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 + »š! ¥ ! Ä"¼! T ¦ÍÇ ¼±Ä" $¥ ^ ¥ E +
›!y³ ± T x = P +(φ ∗ x), ¥ x ∈ F (Z+). >Q!x± ¦´³#¥›± !"φ¦´$D¬ ¬Å ¬Å·"¼½ Ž Ž¥yI« $¦´¦§$ByI¦φ §±ž $«·"¼³´! «$$y$!"¼Ä»š! ¥ ^ ­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 !"I­š ! $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¥ ! žy®«$ ³§·"¼ $ "^²Ž¼± ³´I³´¦{ ¥ Ð"S &¬!"¼ $ Ž¥y lµ ¯·" lµ ÃÈy« ¦§¥ !"¬®¥y" ¥ $¦´¦§$¬$!"$µ|Yl¥ ! Ž¥y&! ¥yH±ª¸!yS ³´¬®¥ ¦¿"¾± ! ¬Å !" yµ ¯O·" yµ ·µšYl³¿" a ∈ E +∗ \{0} " $¦ #¥ S∗ (a) = λa
"L ³¿" b ∈ E +\{0} " $¦ #¥ S b = λ−1b µ€q_ ¥ 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 µ€AH!T±
?K¬Å¬
E+
·"
b(n)λn = b(0), n ≥ 0.
b 6= 0
uk =
»b! ¥yž±ª²! k
X
n=0
»
a(0) 6= 0 b(0) 6= 0
µqb¥ 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 Àž·"^!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 "ž­| ! »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} " $¦ Ž¥y S∗ c = λc µšFS ¬ ¬Å #¥ ®›± !yS¦Á± ¥y²n& ¥C{$¬¬Å yµÇ© »
!1¬!"¼ #¥
e
|φ(λ)|
≤ kTφ k, ∀φ ∈ F (Z+ ), ∀λ ∈ spec(S).
$W"Эš ! Ä»•! ¥yž¥y"¼³´¦§³§ ! ¦§ « " : S Ð
" ¦Á±T¬ ¬ÅƬ " ½ Ž µ•FH± ! ž¦´$
−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´! $» #¥ $¦ Ž¥y$1¬ÅŽ ³ «ª±Ä" ³§! Å$dz§¬ $!":·":! ¥ ± ¦´¦§!y ! !y$L¦§$ ³§!y«$³ ± ¦§ B "¾± $µ€F&± ­| »› ¥ ! #¥ S ·" S !"O"¼¥ O¦´$
¥lÈÅ­š ! $µlYl³´$!" T ¥ ! $ ±Ä"¼ ¥ g LJ_Ž ¦´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.
¦§)ǷȎ" ¼± ³´ : (φ
¥ ! : ¥ W† ¥ ³¿"¼ #¥
]
)
k r
zU ¦B$ " ! « ³´­
³B«$ !#² $ Ê ¥ ž¦§±
"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¿"¼ž 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 «$Ž « ³§$!" ]›¥ ³§ @y ν µyCE± " ¼± ! r W¶+WrÂ
¬ ±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 + µ€q•± 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 Ž¥y#¥ :¦Á±
¶+! « " ³§! Te $ "L !"¼³§ ¦§¦´r$¬$!"O­š ! ± kT k ¥ O«¾½›±#r¥ I«∞$$ «$¦´I³´! «$¦´¥ ›Ä± ! Ω µ
Yl³j! ¥ ¥ ! # ¥ ρ(S) > 1 »l ¬ $¬ # ¥ ± ! ¦§± ¥y²y¥)JB½ $ °$¬
ρ(S )
y µ´¨ » !: $ ¬!"¼ Ž ¥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 jkilh \f h5h Hk jh ^h 3 k ^k S h ¾e |ild\e e h[f\e š[email protected]\h h}i še H k |i k { fWd\h
a `b` Ec c b`
a `_a
HA !) ³¿" #¥Ç¥ !:Ê ¥ L"¼ ¦´Ê³#¥ L± ­|$¦´³§ ! G $W"B¦§Ž«ª± ¦´$¬$!"}«$¬ ± «·" ³ G !Æ"¾± !"
#¥Ç$ ± «$^"¼ ¦§ ʳ#¥ "}¦§Ž«ª± ¦´$¬$!"g«$ ¬ ±Ä« "ª»« #¥ ³€ Ž¥y³´²Ä± ¥y"}± ¥¶ ± ³´" #¥ L¦ÍÇ¥ ! ³¿"¼B
° O¥ !ž²³´ 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 ¥ žCb?KN ¥ H¦Á±"¼ ¦§ ʳ§I ®¦§± «$!² $ Ê $! « Å¥ !y³´¶+ ¬ÅI ¥ " ¥y"D«$ ¬ ±Ä« "ªµq•± ÃÈy ¬ ¦§ »j!º± Rb = R » Zb = T ·"D¦§žÊ ¥ ® ¥›±Ä¦Q ®"¼¥l"DÊ ¥ ³´ « "@ "}«$¬ ± «·"ªµFS ¦´¥ $» !ű Gbb = G " G\
c c ³ G1 » G2 ·" G !#"
1 × G 2 = G1 × G2 ,
^Ê ¥ $LCb?KN&µ€L¥ L±ª²!y^¦´H"¼½ ° ¬Å DW"¼ ¥y« "¼¥y I ¥ ³¿² ±Ä!#" ²³´ µ¿¨¨¹ õ
G© µ¿¨ ) # / : / : " /,: G ! , /+79 # / # = : ;( ( ! ( ! # +: " /,: H 5 = ( G /H
Rp × G 1 . G1 (
1
! < 4! : # < ) # % / : / : " /,:" : 621$ ! , / 79 # Rp × H : "
( /,:( ( : 621$
(
.
!
H
# %
AH!®! Ä"¼ C(G) µ C (G) E¦ÍÇ$ ± «$} $_¶+! « " ³§!y_« !"¼³§!#¥  ¥y G $ µ ¦ÍÇ$ ± «
&¶+! «·"¼³´! &« !"¼³´!Ž¥y$® ¥ 0 G #¥ ³K« !² Ê !#"ž² $ 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%Эy" $! ³´B¦Á±
dχ
¶+ ¬®¥ ¦´³´!²$ 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±
€` µ›q_¥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
µ€qb ! Cc (G) = ∪K∈K CK (G)
kf k∞ = max |f (x)|, ∀f ∈ Cc (G).
NS¦§ (C (G), k.k ) "L¥ ! I± ¦´Ê°$­yx∈G
Ds±Ä!›± «¾½ ¥y"¼¥y" K ∈ K µ€AH!T¬&¥ ! ³´" C (G)
K
∞
®¼±Æ" ¦§Ê³´D³§! ¥y« "¼³¿²®¦´l«$± ¦§ ¬Å !"S« !²·Èl!›±Ä" ¥ ¦§¦§Ä»š«Ä "I%Æ ³´ ž¦Á±Å" ¦§cÊ ³§
¦§Ž«ª± ¦´$¬$!"« !²·Èl ¦Á± ¦§¥y ›! ¥ O¦Á±#¥ ¦§¦§S¦Ídz§! « "¼³´!
$ "&«$!" ³§!#¥ ¥ I"¼¥l"
± ¦´³§«ª±Ë"¼³§ !Ʀ§³´! ª ± ³´
iK : CK (G) −→ Cc (G)
K ∈ K
µQ?K·" "¼"¼ ¦´Ê³§ž "ž«$± 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 $W"& ³¿"¼Ð­| ! Å ³
¥yD" ¥y"D² ³§ ³´!›± Ê V 0 ³´¦Q·Èl³§W"¼ λ > 0 "¼ ¦ #¥ λB ⊂ V µ•YŽ³ E ·" F !"&y$¥lÈ
$ ± «$ L² $« " ³´$¦´O" ¦§Ê ³#¥ $ »y!: ³¿" #¥¥y! &± ¦§³´«ª±Ä" ³§! ¦§³§!yª± ³´
K
φ : E −→ F
" ­š ! $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"B­š ! 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 $« ³ #¥ ^ !"^" ³¿²l³§± ¦§ ¬Å !"
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6 < º© µ¿¨ < E !5 /,: : / : 21$ ! < φ : C (G) −→ E
6 / : / ; < / # # ! ! : <;( !"( ! / 5 ! ! . = / 5 ! c
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(
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!( # (
φ
)
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c jAH!Æ ± ³´" #¥ \¨ g·" Í g !" «$!y #¥ ! «$ ^y ·µlYl¥ ! #¥ \¨
"@²$ ³ ›ÄµlYl³´" K ∈ K µŽ?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› »š± ¦§ φ ◦ i $ "^ #¥ $!"¼³´$¦´¦§$¬$!"S«$!"¼³´!#¥ »| ! «I«$ !#" ³§!#¥ ¥yL" ¥y"
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­| !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¶+ "
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#¥ f |K ³´"г´!#" $Ê ± ­ ¦´ ¥ ЦÁ±Z¬$ ¥y 1 1RS±± ž ¥ м" ¥y"Ы ¬ ± « " K ⊂ G µgq_¥ »y ³¿" S ¦ $ ±Ä"¼ ¥ B ›! ³E ¥ L1 (G) ± x∈G
x
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b »›!:!
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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 µ€FS&¬ $¬ »› ³ Γ (E)x ⊂ E »|±ª² « χ ∈ Gb »›¦ ÃÂ
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R ¨
$! ±
⊂ E ⊂ L1loc (G)
! C c(G)
µ
E
RH Hq_¥ " ¥y"
µ
K⊂G
RSq_¥ B" ¥y"
»•¦´$® ¥lÈ6³´! «$¦´¥ ³´! &·"¾± !"Ы !"¼³§!#¥ $»·"
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x ∈ G Sx (E) ⊂ E
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·"
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supχ∈Gb kΓχ k < +∞.
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! ¬H E ·"O¼± ! $Wχ∈
"¼Gb DÊχ ! $ ± ¦§³¿"¼ »Ž! $¥y"L«$ ! 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 ¦ Ž¥y loc
Z
© µ¿¨
|f (x)|dx ≤ CK kf kE , ∀f ∈ Cc (G),
±¹² $« C > 0 ¦§±Z«$! y³´"¼³´!5$W" ·²Ž³§ ¬Å¬$!"Ð! «$$ ¼± ³´ ·µ}>g! š "¥ ! ± Ê¥ ¬$!"
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q_¥ L"¼¥l" K « ¬ ± « "^ G »€³§¦EÃÈy³´ " x , ..., x ∈ K "¼ ¦§ Ž¥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
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|(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»¦ !lÂ
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AH! ! Ä"¼ σ({S } ) ¦§ $« " x x∈G
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x x∈G
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\
σ({Sx }x∈G ) = {(γ(Sx ))x∈G , γ ∈ B(E)}.
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¥ 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).
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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 ž«$!" 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 Ž¥y µ|AH!~ $¬Å± #¥ Ž¥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
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¥[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 ¥ W†$ ± «$ Gf Ge " $¦ Ž¥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!ž"¼ !›± !"Q« ¬ "¼ g± Ê ¥ ¬Å !"¼ ·" ͻĥ !ž«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 DYŽ$« " 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 ! Ž¥y1 ³ GE $W"Å¥y! Ê ¥ «$¬ ± «·"ª»±Ä¦§ ®¦Ídz§¬Å± ÊE |θ|E$W"Ц§Æ ¥ \ÂzÊ ¥ )«$¬ ± «·"®"¼ ³´²Ž³Á±Ä¦} R+ ¥y&"¼¥l"
fE "} ! « G
fE = G
b µyA!Å ¬ ± #¥ #¥ ^ ³ x1 , ..., xn ∈ G !" z³§! ! ›± !" ¾»#³ µÄµ
θ∈G
³y¦´}W¸Ž "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
$ "•¥ !ž ¬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§ Ž¥y Gf = Gf+ Gb µ
E
E
E
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9
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G 3 x −→ φ(x)λ ψ(x)1−λ ∈ R
/ / # C ,# - λ ∈ [0, 1]
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^¥ ¥y² $ ! ^¦´H"¼½ ° ¬Å ¥ ³´²Ä± !"$µ
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(
.
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!
/
!
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f = {θ ∈ G
e | |θ −1 (x)| ≤ sup η(x)−1 , ∀x ∈ G}.
G
E
A!)²Ä±ž²³´ Ž¥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 ²Ž³§ !" #¥›± !y G $ "« ¬ ± « "$µQ^¥y« ! « "¼¥y ! #¥ :¦Á±~¶+ ¬®¥ ¦´
© µÇ© K$W"O²l ± ³§ ¥ " ¥y"LÊ ¥ DC_?KN µ
Yl³´"
U
¥ !1¥y² "^
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Cp
U
µ€X^! ¶+! «·"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/ ! ! ! ) )
(
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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%
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,
6
9
7
6
:
/
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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 yµEqbÄÂ
]•³ Èl! φ
"
L = {z ∈ C | Re z ∈ [0, 1]}.
f+ µ|Yl¥ y ! #¥
ψ∈G
E
φ 6= ψ
µ›CE±ž¶+ ! « " ³§!
Π : L 3 λ −→ φλ ψ 1−λ
"1± !›±Ä¦´¸#"¼³ Ž¥y~ ¥y L◦ µB?K$«$³S¬Å!" #¥ ¦´ZJB½ ° ¬ÅZ©yµ “³´³ ž ! ! y$ ³´ " $
± !›±Ä¦´¸#"¼³´«$³´" ® h y°$ #¥ Gf+ ! " ± D
¥ !º ³´! ʦ´ "¼ !¬ ¬Åž ³ Gf ±:¥ !2³§!"¼ ³§ ¥ E
M
E
²Ž³§yI ± ! Ge µ€q_ ! AH!:²³¿" Ž¥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)
µ€Yl³
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±ž !#" ¼±ÄΧ!
"L³§¦š$ "L·²l³´ $!" #¥
ψ(x) = e− ln ρ(S−1 )x , ∀x ∈ R.
h
Sφ,ψ ≈ ln
i
1
, ln ρ(S1 ) ,
ρ(S−1 )
n
h
Ωφ,ψ ≈ z ∈ C | Im z ∈ ln
io
1
, ln ρ(S1 ) .
ρ(S−1 )
•q ± Ы$ ! Ž¥y$!"ª»@¦§)JB½ ° ¬Å:© µ³§¬ ¦§³ #¥ +¬ ± ³´ž :¬ ±Ä! ³§° Æ! ! « ! " ¥ «·"¼³¿² ž¦§
"¼½ ° ¬Å®y® $!"¼±Ä"¼³´!2 $¬®¥y¦´"¼³ ¦§³´«ª±Ä" $¥ O ¥ L2 (R) ÃÈ ž›± ! ¦´ž?K½›± ³´"¼
ω
lµ
` =Z
HA ! e = (e G, ...,
±ª²$«
ej,k ),
j
j,1
AH!1 $ ¥y³´"O ©yµ© #¥
k
ej,i = 0
¥y
i 6= j
» "
ej,j = 1
µ€AH!: ›! ³´"
S j = S ej .
fk ≈ σ(S , ...., S ).
Z
1
k
E
?K½›± Ž¥y φ ∈ Zfk $W"S I¦Á±ž¶+ ¬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
›
»
µ€Eq_ ! 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± Ž¥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◦ $W"^¥y!1y¬ ±Ä³§! ^ IML ³§! ½›±Ä y"B¦´ÊĆ«$!²ÃÈlI ³ F◦ 6= ∅ µ€AH!) ¥ 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 µ€FS ›! 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¦§±ž¬Å ¥ g
k+
Z
× Tk
E
χ−1 =
¼" ¦#¥ θ (z) = h (φ ) µ µ} ¥ M
M
z
µ€L¥ 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¥ ! Ž ¥y Rf◦k 6= ∅ µ€AH! E
A!: ± $¦§¦´#¥ I¦ÍÇ$! $¬®­ ¦´
E
UE = E
◦
g
k+
R
E
−1
◦
◦ g
k
f
k
RE ≈ R + iRk+
E .
"L¦§ ÊÄÂz« !²·ÈlĵjYŽ³´"
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 Ž¥y1 ¥ ]•³¿Èl! M ∈ M(E) µ€>Q!~± y ¦§³ Ž¥ ± !"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 µ€AH!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 ` EYŽ³´" ω ¥ ! ¶+! «·"¼³´!:¬Å ¥ ± ­ ¦§ ³´"¼³¿² ¥ G µ›qb¥ 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´" $¬$!"ªµ€AH! 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!: ¬ ± #¥ Ž¥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
◦
◦
"B²Ž³§ ^­ 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
` {YŽ³´" ω ¥ ! ³§ B« !"¼³§!#¥1 ¥ G µ€q_ ! 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" Ž¥y ω ± ³´"∞¦Á± ³´ " ¥y Ž¥y
C0,ω (G)
±Ä"¼³´ ¶ ± ³¿"H¦§ ω(x + y)
< +∞, ∀y ∈ G,
ω(x)
x∈G
0 < sup
² $ ³ › RS·µ
` YŽ³´" 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¥ ¬ÐÂ
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$žy GfE #¥ ³² !#" ¥ $ž¥ !6¦§Æ³´¬ W"¾± !"® ± ! ЦÁ± $¥l²1 ¥
³´! «$³ ± ¦@ « )«¾½›± ³¿"¼ ĵAH!5 $¬Å± #¥ Ž¥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 2sKŽ«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> ! š "$» ³¿" K ¥ ! ¥ W†$ ± «$^«$¬ ± «·"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
sKŽ«¾½ ! ¥y C (G) % ²Ä± ¦§ ¥ @›± !y C
µy?K¬¬ÅL¦§ K³´!#" $Ê ± ¦§ }ysKŽ«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 õyYŽ³´"
=
·"B¦Á±I $! ³´" H C (G) ›± ! E
¦Í± ½ $!y«$H›±Ä! A(E)
c
&¦ ±Ä¦§Ê° ­ D$!yÊ$! y $ ± {M }
µšAH!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 ¦ Ž¥y V + V ⊂ F + V µ
AH! c
AH!1 $¬Å± Ž¥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!l†³§° ¬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§«$± ¦§Äµ›AH!T $¬Å± #¥ #¥ S ◦ M = M » ¥ B"¼ ¥y" φ ∈ C (G) µ›F^! «
x
φ
S (φ)
c
»
y
¥
¼
"
y
¥
"
·"
¥
"
y
¥
"
|
µ
l
Y
´
³
"
µ›A!T±
[
R ◦ T ∈ A(E)
R ∈ B(E)
T ∈ A(E)
γ ∈ A(E)
x
¥y
γ(R ◦ T1 )
γ(R ◦ T2 )
=
,
γ(T1 )
γ(T2 )
»
PT± ³§!"¼ !›± !"® !yÈl
R ∈ B(E) T1 , T2 ∈ A(E) \ Ker(γ).
"L!1 ›!y³´"
Mφ ∈
/ Ker(γ)
∆γ : B(E) −→ C
± B¦Á±&¶+ ¬®¥ ¦´
© µ
∆γ (R) =
Uz¦{$ "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!: ¬ ± #¥ Ž¥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 $ °$¬B B]›¥ ­ ³´! ³Ž¥yB«$ " "¼BÃÈEl" $! ³§θ!Ð "g¥ !ž ¦§$¬$!"Qy A(E)
[ µ ^¥yQ± ¦§¦´! ¥l²$
#¥
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·µ›A!: $¬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±ž ¬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 ¥ µ€qb¥ 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 Ž¥y$!"¼³´$¦§¦´$¬$!"D«$ !#" ³§!#¥ ž·" y! «ž«$ !#" ³§!#¥ ž
¦´³§«$±Ä"¼³´!
x −→ Sx (φ)
Cc (G)
›± ! A(E)
µE?K¬¬®¦Í± Â
$ "L« !"¼³§!#¥ D
G
±!
Cc (G)
!)²³¿" #¥ ¦ ± ¦§³´«ª±Ä" ³§!
x −→ Sx ◦ Mφ = MSx (φ)
$ "H«$!"¼³´!#¥ ž G ›± ! A(E) µAH!$! ¥ ³´"Ž¥yž¦Á±¶+ ! « " ³§!
A!)²Ä±Ð¬ ± ³´!"¼$! ± !"B¬Å !#" Ž¥y θ ∈ Gf µ€q_ ! γ
Cg± ¦§³§«$±Ä"¼³´! η $W"¥ ! @¶+ ¬Åg¦§³´! ª±Ä³§ Q« !"¼³´!Ž¥yK ¥y
•«·¶\µ »|?K½›± ³¿"¼ D" $¦´¦§#¥
µ
?K$¦Á±ž¬!"¼ Ž¥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«$ ! Ž¥y$!"ª»|!:«$!y«$¦§¥l" Ž¥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§!#¥ ž ¥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 µ›Yl³
θ0 ∈ G
θ −1 (x)φi (x)dx < ,
G
.
(1 + supi=1,...,k kφi k∞ )m(∪i supp(φi ))
"^«$!"¼³´!#¥ µšAH!1²Ä±Ð¬ ±Ä³§!"¼ !›± !"O¬Å !#" #¥ R "L«$ !#" ³§!#¥µšYl³¿"
[ ¥ !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 ) µ€AH!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
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R(γ)
R
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¬³§!›±Ä³§ $µ
6 < © µ < G # +5 = ! < φ : G −→ R #
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[−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±ž !#" ¼±ÄΧ! #¥ φ $W"^« !"¼³´!Ž¥1$! 0 "L !y«I« !"¼³´!Ž¥1 ¥y G µ x∈V
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/
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A!1±y» ¥ K = K −1
¥ !:²³´ ³§! ± Ê«$ ¬ ±Ä« "L
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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 ·µšAH!1¼± $¦´¦§ #¥ I«$·" "¼I«$ ! ³´" ³§!:³´¬ ¦´³#¥
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fE , ∀χ ∈ G.
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¬Å!y¬Å³§± ¦§ ¬Å !"g«$!²ÃÈlI ³j "B $¥ ¦´$¬$!"O ³š« Ç$ "¥ !) ¬Å± ³§!yS¦§ ÊÄÂz« !²·ÈlH« !"¼ !›± !"
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½›± l" X " $¦ #¥ 0 ∈ X $W"Ƭ! ¬³Á±Ä¦§$¬$!"ž« !²ÃÈy2 ³^·"Æ ¥ ¦§ ¬Å !"Æ ³ X ")¦´
¬Å± ³§! & Å« !² Ê ! «$Æj¥ !yÅ $ ³´Ð$!"¼³´°$ «·¶\µ ·µbCE± $¥l² y¥ JB½ ° ¬ÅЩ µ Ž¥y³ ¥ ³¿" $ $!"¼Ð D± !›± ¦´Ê³§ H±ª² «ž¦Á± $¥l²Ð ® $ ¥y¦´"¾±Ë"¼ ³´¬Å³´¦Á± ³´ «$ "¼± ³§! ¬Å !"
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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!" ! ±Ä­š # ¥ Gf " «$¬ ± «·"ªµ}^¥ ± ¦§¦´! ±Ä­š :¬Å!" $ # ¥ Gf "L# ¥ ³´«$!"¼³´!#¥T ¥ G µ›Yl³´" θ ∈ Gf µ|AH!~±
E
+
E
+
E
+
E
1
≤ θ −1 (x) ≤ ρ(Sx ), ∀x ∈ G.
ρ(S−x )
q•± O«$ ! Ž¥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 ¦ Ž¥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 µ€NS³§!y ³{³§¦{ "
« ¦Á± ³´ #¥ Gf+ "D#¥ ³´«$!" ³§!#¥2$! 0 "Dy! « Gf+ "D#¥ ³´«$!" ³§!#¥2 ¥ G µEδAH!2±Æ²Ž¥ #¥
E
E
$
S
"
š
­
!
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 Ž¥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 #¥
± Ž¥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$¦´ Ž¥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±ž¶+! « " 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¿¶+ ¬$¬$!"_­š ! $K "Q ! «¬$ ¥y¼± ­ ¦´ ¥y G×L ·µ
AH!1 $ ¥y³´"^± ¦§ L" ½ $ °$¬$O PT $ ±ž "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
µŽF&± ° K¦´ ³§!y«$³ O ^qg½y¼± ʬ$!ŽÂ\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±ž³´¬ ¦´³#¥ Ž¥y
fE , ∀λ ∈ L.
θλ ∈ G
A !)²Ä±Ð¬ ± ³´!"¼$! ± !"B¬Å !#" Ž¥y ¦ !1±
n
© µ¿¨Ï¯
f = θ∈G
e | |θ −1 (x)| ≤
G
E
Yl³´" θ ∈ Gf+ "¼ ¦ Ž¥y
© µ¿¨¹À
o
sup |η −1 (x)|, ∀x ∈ G .
g
η∈G
E
f+ µ›N^¦§ B³§¦ÃÈy³´ " φ ∈ C (G) " > 0
θ∈
/G
c
E
Z
φ(x)θ −1 (x)dx > kMφ k + .
"¼ ¦§ #¥
Yl³´" K = supp(φ) µŽ?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 }
"@¥ ! $« ¥y²Ž $¬$!"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 } µ›F^ ›! 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«$! #¥ !#"$»š !~±
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 »
∆
µ›Uz¦{ $ ¥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= ∅
µšYl 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 )|.
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"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 µ›XL! 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
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BG,µ = {U ∈ BG | sup µi (U ) < +∞, i = 1, ..., 4}.
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ML³´$ ̺¬!"¼ #¥ «¾½ ±#¥ ¶+ ¬Å2¦§³´! ª±Ä³§ « !"¼³´!Ž¥y ¥y C (G) $¥y" "¼ $ !lÂ
c
"¼$ Ƭű ! ³§° Å¥ ! ³#¥ ¥ ž¦§±T¶+ ¬Å L(f ) = R f (x)dµ (x),
µ ∈ M (G) µQLÄÂ
L
G
"¼! M (G) ¦ÍÇ$!y $¬&­ ¦§1 $ŬŠ¥ $Æ ¥ G %2²Ä± ³§±Ä"¼³´!6­š ! $Ä»g¬®¥ L! ³L 1¦Á±x!y ¬
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Ž¥y³ $¥l" ·"¼ ³´ $!" ³ ›Ð«$ ¬Å¬ž«$³ Âz ¥y&±ª² «Å¦´ž ¥›± ¦Qy Cc(G) µ
kµkM (G) = |µ|(G),
^¥ g ³´Ê! $ ! ± M (G) ¦ ! ¬®­ ¦§[email protected]¬Å ¥ $}%I ¥ "@«$ ¬ ±Ä« "@ ¥ G µ#CQ WÂ
c
Ž¥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± ¥lÂ
¸)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 ·µš^ ¥ ±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© »Ž¦Á±I "¼ ³§«·"¼³§ !)% C (G) Ç¥ !)¬®¥y¦´"¼³ ¦§³´«ª±ËÂ
" $¥ 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! ! ž¥ !" ½ $ °$¬®
$ !"¾±Ä" ³§!1² ±Ä¦Á± ­ ¦´ ¥ O¦§ B¬®¥ ¦¿"¼³ ¦§³§«$±Ä"¼ ¥ } C (G) ›± !y M (G) µ›qb !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) µ€L¥ 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± !"¼ž ŝ $W"S¦§±&¶+ ¬¦§³´! ª±Ä³§ ^«$!" ³§!#¥ D ¥ L1 (G) !y! $ ± s
ŝ(f ) = s(fˆ).
A ! $¬Å± Ž¥y#¥ ŝ ¥y" ·"¼ «$! ³§ Æ« ¬¬ÅÅ¥ !5$¦´$¬$!"® L∞(G)
b µF&± °$ž¦´
JB½ $ °$¬)© µ ¯l»•"¼ ¥y"Ð $¼±Ë"¼$¥y T C (G) ›± ! M (G) « !"¼³´!Ž¥ ¥ ž¦§±~"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 µŽX^!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 SYŽ³ 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§ !~± Ž«$³´&% φ ∈ C (G) µš>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$¦´¦§Ž¥y c
b
G
(kα ) ⊂ L (G)
kα ∈ Cc (G) kkα kL (G)
b = 1
R
"
¥ " ¥y"B² ³§ ³§!›± Ê V y 1 µ›L¥ 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 ) µŽ^¥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 µ›L¥ B ¬ ±Ä #¥ ! #¥Ç$!1Ê$!y$¼±Ä¦{¦§ ÄÂ
¥y³´"@Ç¥ ! Ž¥ ± ³´¬Å ¥ ^±ª² $« F g » ¥ g ∈ L1 (G)
b !y¥ g ! !y^¥ !y^±Ä¥y"¼ #¥›±Ä ³§¬$ ¥ +« ¶\µ ·µ›Uz« ³
α
c
c
<k
α µ, g >=< µ, kα g >, ∀g ∈ D(G).
K? ¬¬ kc "ž¥ !yŶ+! «·"¼³´!“%~ ¥ "&«$¬ ± «·"ª»
±ª² ! $» ¥ α f ∈ D(G) » x ∈ G »
Yα f (x) =
=
Z
b
G
=
Z
b
G
c
kα µ
"®¥y! $¥ ¬Å ¥ µL¥ 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¦´¥lÂ
" ³§! S±ª²$«žy$H¶+! «·"¼³´! Sy C (G) ³§¦b ¥ Ð"± ªÈl³§¬$S¦´$^ ¼±Ä" $¥ Y µFH± !yH«$
­ ¥l"ª» ! ν = kcµ µŽAH!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 µ›qb ! β
Vβ (f ) = V (hβ ∗ f ), ∀f ∈ E.
^¥ L±ª² ! lim kVβ f − V f k = 0, ∀f ∈ E.
±
FS §¦ ¥ » ³§¦š$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)
µ›Uz¦j$ "L¶ ± « ³§¦§S ²³´ Ž¥y
Cc (G) ∗ Cc (G) ⊂ B(G).
>g!j·"ª» ¥y
»
"« ¬¬Å F f, F g ∈ L2 (G)
b »š!±
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 š "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
µ›Uz¦ "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§ !#" Ž¥y G
c
"L«$!" ³§!#¥ D«ª± B«Ä "L¦Á±®¦´³§¬³´" ^¥ !y³´¶+ ¬Å^¥y! D ¥ ³¿"¼ ¶+! ·« "¼³´! « !"¼³§!#¥ $µ€FS ! «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 ! ! Ž¥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,
± ž¥ !y) ¥ \Âz ¥ ³´" )Ê$! ¼± ¦´³§ $Æ« !²$! ± ­ ¦§Ä»Q! ¥ ž ­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 µ›FS ¦´¥ $» !~± 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 Ž¥y #¥ D¦GbÁ±ž ¥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
µ›q•±ÄL« ! #¥ $!"$»
± ¥: $!y^ D¦§±ž" ¦§Ê ³§O¶ ± ³§­ ¦´H
L2 (G)
µ€F ± ¥l"¼ ±Ä " ± B« ! W"¼ ¥y« "¼³´!»
lim k(Mφα f ) − (M f )k = 0, ∀f ∈ E
"L!:­y" ³§$!" ¥ »
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´!Ž¥yI ¥ C (G) ¥ B«¾½›±#¥ f ∈ C (G) µ€L¥ 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» ¥ $#¥ ž"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±ž« ¬ §¦ ° " ¦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
µ€q_¥ 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´!Ž¥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± !›± ¦¿¸Ž" ³#¥ Ä» "O¦§±ž¶+ ! « " ³§!
Z
Ui
»Ž! ¥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
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n
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