Текст статьи - Transactions of the TSTU

Математика. Физика
УДК 517.518
ЕXPONENTIAL METHODS OF SUMMATION
OF THE FOURIER SERIES
A. D. Nakhman1, B. P. Osilenker2
Departments: “Applied Mathematics and Mechanics”, TSTU (1);
“Higher Mathematics”, Moscow State University of Civil Engineering, Moscow (2);
[email protected]
Key words and phrases: convergence almost everywhere; convex; estimates of
Lp-norms; piecewise-convex sequences.
Abstract:
We
consider
the
semi-continuous
methods
Λ = {λ k (h), k = 0,1,...; h > 0 } of summation of Fourier series and conjugated Fourier
series, generated by exponential functions λ( х, h) = exp(−hu α ( х)), α > 0. The
estimates of Lp-norms of the corresponding maximal operators are obtained.
As consequence, we get some results about exponential method of summation of the
Fourier series almost everywhere and in Lp-metric.
Introduction
Consider f = f ( x ) ∈ L([− π, π]), and let
∞
∑ λ|k| (h)ck ( f ) exp(ikx),
U h ( f ) = U ( f , x; λ, h) =
~
~
U h ( f ) = U ( f , x; λ, h) = −i
(1)
k = −∞
∞
∑ (sgn k )λ|k | (h) ck ( f ) exp(ikx)
(2)
k = −∞
be the set of a linear means of Fourier series and conjugate Fourier series respectively.
In various questions of the analysis there is a problem of behaviour of (1) and (2)
when h → +0. Here ck ( f ) =
Fourier coefficients,
1
2π
π
∫ f (t ) exp(−ikt ) dt ,
k = 0, ± 1, ± 2, ... are complex
Λ = {λ k (h), k = 0, 1, ...}
(3)
−π
is infinite sequence defined by the values of parameter h > 0. This sequence defines
so-called semi-continuous method of summation. Тhe regularity conditions of such
methods will be the following [1, p. 79]:
λ 0 (h) = 1, lim λ k (h) = 1, k = 0,1, ...;
h →0
(4)
∞
∑ | Δ λ k (h) | < ∞.
h >0
sup
(5)
k =0
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The similar problems for (1) have been studied by L. I. Bausov [2] in case of
discrete h.
We consider the semi-continuous methods of summation corresponding, basically,
to the case of
λ 0 (h) = 1, λ k (h) = λ( х, h) | x = k , k = 1, 2, ...,
where
λ( х, h) = exp(−hϕ( x)),
(6)
and function ϕ( x) ∈ С 2 (0, + ∞). Note that if λ k (h) = exp(−hk ) we get Poisson-Abel
means [3, vol. 1, p. 160 – 165].
1/ p
⎞
⎛π
Let || f || p = ⎜ ∫ | f ( x) | p dx ⎟
⎟
⎜
⎠
⎝ −π
1
L = L ; || f ||=|| f ||1 and
be a norm in Lebesgue space L p ( p > 0 ;
)
~
1
f ( x) = − lim
2 ε → +0
∫
ε ≤|t |≤ π
t
f ( x + t ) ctg dt
2
be a conjugate function; this function exists almost everywhere for each f ∈ L [3,
vol. 1, p. 402]. Define
~
~
~
U * ( f ) = U * ( f , х; λ ) = sup | U ( f , х; λ, h) | ; U * ( f ) = U * ( f , х; λ ) = sup | U ( f , х; λ, h) | .
h >0
h >0
Estimates of Lp-norms
The
sequence
(3)
is
called
a
convex
(concave),
if
2
= Δ(Δλ k (h )) = λ k (h ) − 2λ k +1(h ) + λ k + 2 (h ) ≥ 0 Δ k ≤ 0 , k = 0,1,K The sequence
(3) is piecewise-convex, if Δ2k changes sign a finite number of times, k = 0,1, ...
Theorem 1. If the sequence (3) is a convex (a concave) and
(
Δ k2
)
λ k (h) ln k = O(1), k → ∞,
(7)
for each h > 0 then the estimates
~
|| U * ( f ) || p + || U * ( f ) || p ≤ C p, Λ || ( f ) || p , p > 1;
(8)
~
|| U * ( f ) || + || U * ( f ) || ≤ CΛ (1+ || f (ln + | f |) ||);
(9)
~
|| U * ( f ) || p + || U * ( f ) || p ≤ C p, Λ || ( f ) ||, 0 < p < 1
(10)
hold.
Here C will represent a constant, though not necessarily one such constant.
The estimates (8) – (10) remain valid, if a piecewise-convex sequence (3) satisfies
to the condition (7) and there is constant С = CΛ , such, that
| λ k (h) | +k | Δλ k (h) | ≤ CΛ
(11)
for all h > 0, k = 1, 2, ...
Proofs of both statements will be based on the Abel transform of sums (1), (2) and
on the estimates of Fejér means [3, vol. 1, p. 148] by maximal operators
1
f * = f * ( x) = sup
h >0 h
102
x+h
∫ | f (t ) | dt
x−h
and
f (x + t)
dt .
h > 0 h ≤|t |≤ π 2 tg t
2
f * = f * ( x) = sup
∫
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~
Thus the inequalities (8) – (10) occur when || U* ( f ) || p + || U* ( f ) || p is replaced by
~
|| f ∗ || p + || f ∗ || p , [1, vol. 1, p. 58–59, 404].
Theorem 2. Let the sequence (3) be a convex (a concave) and at every h > 0
satisfies the conditions (7) and (4). Then relations:
lim U h ( f ) = f ;
(12)
~
~
lim U h ( f ) = f
(13)
h →0
h →0
hold almost everywhere (a.e.) for every f ∈ L and in the metrics Lp for every p > 1.
The statements remain valid for every piecewise-convex sequence (3), satisfying
to conditions (4), (7), (11).
Besides, under the formulated conditions the relation (12) holds in each point of a
continuity of function f. The relation (12) holds uniformly over x for everyone
continuous f. Last statement does not extend, generally speaking, on a case (13).
Auxiliary statements
Lemma. If a piecewise-convex sequence (3) satisfies to conditions (7) then the
following relations hold:
U * ( f , х; λ) ≤ C f * ( x)
∞
∑ (k + 1) Δ2λ k (h) ;
(14)
k =0
∞
~
~
U * ( f , х; λ) ≤ C f * ( x) + f ∗ ( x)
(k + 1) Δ2λ k (h) .
(
)∑
(15)
k =0
Proof. We shall prove (15); the relation (14) one can deduce exactly in a similar way.
According to the integrated form of Fourier coefficients and Abel transform we obtain
~
~
1
U h ( f ) = U ( f , x; λ, h) = lim
N → +∞ π
=
π
⎧⎪ N
⎫⎪
f (t )⎨ λ k (h) sin k ( x − t )⎬dt =
⎪⎩k =1
⎪⎭
−π
∑
∫
π
π
⎧⎪
~
~
1
lim ⎨λ N (h) f ( x + t )DN (t ) dt + NΔλ N −1 (h) + f ( x + t )FN −1 (t ) dt +
π N → +∞ ⎪
⎩
−π
−π
∫
+
∫
N −2
∑
k =0
(k + 1) Δ2 λ k (h)
π
~
⎫⎪
∫ f ( x + t )Fk (t ) dt ⎬⎪.
−π
(16)
⎭
Here
k
~
Dk (t ) =
sin ν t =
∑
ν =1
1
cos⎛⎜ k + ⎞⎟t
2⎠
⎝
−
;
1
1
2 sin t
2 tg t
2
2
1
~
~
~
1 k ~
1
Fk (t ) =
D(t ) =
− Fk (t )
∑
1
k + 1 ν =0
2 tg t
2
are conjugate Dirichlet and Fejer kernels, respectively, and
~
~
cos(k + 1)t
.
Fk (t ) =
21
2(k + 1) sin t
2
Further, we shall establish the following estimates for the integrals containing in
the right part of (16):
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π
~
∫ f ( x + t )Dk (t ) dt ≤ C f
*
( x) ln k , k = 2, 3, ...;
(17)
)
(18)
−π
π
∫ f ( x + t )Fk (t ) dt ≤ С ( f
~
*
~
( x) + f * ( x) ,
k = 0,1, ...
−π
For this purpose we shall use the obvious inequalities:
~
~
Dk (t ) + Fk (t ) ≤ C (k + 1), 0 ≤ t ≤ π, k = 0, 1, K;
1
~
Dk (t ) ≤ C , 0 < | t | ≤ π,
|t |
~
~
Fk (t ) ≤ C
1
(k + 1)t 2
,
k = 0,1, ...;
(20)
k = 0,1, ...
0 < | t | ≤ π,
and choose a natural number S = S (k ), k = 0,1, ...,
(19)
(21)
2 S −1
2S
≤π<
.
k +1
k +1
such that
According to (19), (20), we have
⎞
⎛
⎟
⎜
S
⎟
⎜
k +1
~
f ( x + t )Dk (t ) dt ≤ С ⎜ (k + 1)
| f ( x + t ) | dt +
| f ( x + t ) | dt ⎟ ≤
j −1
j =1 2
⎟
⎜
1
−π
2 j −1
2j
|t |≤
⎟
⎜
≤t ≤
k
+
1
⎠
⎝
k +1
k +1
π
∫
∫
∑
∫
≤ С (1 + 2S ) f * ( x) ≤ Cf * ( x) ln k , k = 2,3,...
Further, we shall prove (18). In view of relations (19) and (21) we obtain
⎛
⎜
f (x + t)
~
⎜ (k + 1) | f ( x + t ) | dt +
+
≤
(
)
(
)
f
x
t
F
t
dt
С
dt +
k
∫
∫
∫
⎜
t
1
1
tg
2
−π
⎜
≤|t |≤ π
|t |≤
2
⎝
k +1
k +1
π
⎞
⎟
⎟
~
k +1
+
| f ( x + t ) | dt ⎟ ≤ C ( f * ( x) + f ∗ ( x )).
j −1 2
) j −1
j =1 (2
⎟
2
2j
⎟
≤t ≤
⎠
k +1
k +1
The statement (15) is now a direct consequence of equality (16), estimates (17), (18),
1
relations (7) and Δλ k (h) = O⎛⎜ ⎞⎟, k → ∞. The last relation is valid [3, vol. 1, p. 156] for
⎝k ⎠
any convex or piecewise-convex sequence.
S
∑
∫
Proof of the theorems 1, 2
We consider a case of a piecewise-convex sequence Λ. Then Δ2 λ k (h) keeps the
sign for m ≤ k ≤ n, where m and n are some natural numbers. By Abel summation
formula we obtain
п
∑ (k + 1)Δ2λ k (h) = λ т +1(h) − λ п +1(h) + (т + 1)Δλ т (h) − (п + 1)Δλ п +1(h).
k =т
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(22)
∞
Hence,
∑ (k + 1) Δ2λ k (h)
is equal to finite number of sums, each of which looks
k =0
like (22); if п → +∞ then λ п +1 (h) + (п + 1)Δλ п +1 (h) → 0 [3, vol. 1, p. 155–156]. Now,
using relations (11) (14), (15), and (22) we obtain the second statement of the theorem 1;
the first statement can be received by similar arguments.
Тhе statements of the theorem 2 (convergence a.e. and in metrics Lp ) follow from
(22) and (11) by standard arguments [3, vol. 2, p. 464–465]. It is necessary to notice,
that the conditions of regularity of Λ -method are valid; the validity of (5) follows [4,
p. 748] from (22) and (11).
The convergence (12) in points of continuity and uniformly over x follows from
(14) and Banach-Shteinhaus theorem. To use this theorem it is enough obtain the
boundedness of Lebesgue constants of summation method. In turn, it will be follow
from (14), (22), (11) for f ≡ 1 if to notice that Fk (t ) ≥ 0.
The last statement cannot be extended to a case (13) because the conjugate
~
function f can lose a continuity in points of continuity f [5, p. 554].
Convex and piecewise-convex еxponential summarising sequences
We shall address to consideration of a case (6). It be required to us
′ ( х, h) = h exp(−hϕ( x))(h(ϕ′( x)) 2 − ϕ′′( x)). (23)
λ′х ( х, h) = −h exp(−hϕ( x))ϕ′( x), λ′хх
Let restrict oneself, basically, to consideration of functions
ϕ( х) = u α ( х), α > 0.
Theorem 3. Let u ∈C 2 (0,+∞), u > 0, u′′ < 0 ( x ∈ (0,+∞)),
0 < α ≤ 1,
λ( х, h) = exp(−h u α ( х)),
(24)
and
exp (−h u α ( х)) ln x = O(1), x → +∞
(25)
for every h > 0. Then the estimates (8) – (10) are valid and the relations (12), (13) hold
a.e. for every f ∈ L and in the metrics L p , p > 1. These assertions remain valid if
V = V ( x ) = αh u α (u′) 2 − (α − 1)(u′) 2 − u u′′, α > 0
(26)
has on (0, + ∞) finite number of zeros, the conditions (25) is satisfied and there is
constant C = Cu ,α , such that for all h > 0, x ∈ (1,+∞)
xh exp(−h u α ( х))u α −1 ( х) u′( х) ≤ Cu , α .
(27)
Proof. We shall apply the results of theorems 1, 2. The condition (7) is satisfied by (25).
It is remain to prove that (24) is convex for 0 < α ≤ 1. According to (24), (23), (26) we
have
′ ( х, h) = αh exp(−hu α ( x))u α − 2 ( x)V ( x).
λ′хх
(28)
′ ( х, h) < 0 for u ′′( x) < 0 and 0 < α ≤ 1 as it was required to obtain.
Then λ′xx
Further we shall notice that the formulated condition on function V (x) in (26)
provides a piecewise-convexity of sequence (6), defined by (24). Really, let, for
example, V (x) is a function of constant sign for m ≤ x ≤ n + 2 (т and п are some
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105
non-negative integers). We shall apply to λ ( x, h) (as functions of х) the Lagrange
theorem twice (on [k, k+1] and on [k + θ1 , k + 2 ] respectively):
Δλ k (h) = −λ′x (k + θ1, h);
′ (k + θ1 + θ2 , h),
Δ2λ k (h) = (1 − θ1 )λ′xx
(29)
where θ1 = θ1 (k ), θ 2 = θ2 (k ), θ1 , θ2 ∈ (0,1). Let θ = θ1 + θ 2 . If m ≤ k ≤ n, then
2
m < k + θ < n + 2, such that Δ λ k (h) is sequence of constant sign by (29).
Since a number of intervals (with the integer ends) on which V (x) is a function of
constant sign, is finite, then Δ2 λ k (h) has finite number of changes of a sign. It remain
to note the validity of (11), if condition (28) holds.
Theorem 3 is proved.
Examples
1. Let u ( x) = ln x, then
(
)
λ 0 (h) = 1, λ( x, h) = exp − h ln α x , x > 0, α > 0.
(30)
It is evidently that (25) holds. For 0 < α ≤ 1 the sequence (6), defined by (30), is
convex.
If α > 1, then function (26) vanishes once; hence, the summarising sequence is
1
h exp − h ln α х ln α −1 х =
piece-convex. It is remain to note that
ln x
(
)
(h lnα х)exp(− h ln α х) ≤ Cα at all α > 1 and x > 2.
So, the statements of theorem 3 hold for a case (30) at all α > 0. In particular
(α = 1)
∞
~
1
1
c0 ( f ) + ∑ h ck ( f ) exp(ikx) → f ( x) and − i ∑ (sgn k ) h ck ( f ) exp(ikx) → f ( x)
k
k = −∞
1≤|k |< ∞ k
a.e. for everyone f ∈ L and in metrics L p ,
p > 1.
2. Let u ( x) = x, then
(
)
λ 0 (h) = 1, λ( x, h) = exp − hх α , x > 0, α > 0 .
(31)
It is evidently that (25) holds. For 0 < α ≤ 1 [6] the sequence (6), defined by (31), is
onvex. If α > 1, then function (26) vanishes once; hence, the summarising sequence is
piece-convex. It is remain to note that
(
)
h exp − h х α х α ≤ Cα at all α > 1 and x > 0.
So, the statements of theorem 3 hold for a case (31) at all α > 0 . In particular
1
⎞
⎛
⎜ α = 1, h = ln , 0 < r < 1⎟ we obtain the convergence of Poisson-Abel means
r
⎠
⎝
U r ( f , x) =
106
∞
~
∞
∑ r |k |ck ( f ) exp(ikx) and U r ( f , x) = −i ∑ (sgn k )r |k |ck ( f ) exp(ikx).
k = −∞
k = −∞
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3. Consider a method of summation defined by the function
λ( x, h) = exp(−h Рп ( х)), x > 0,
where Рп ( х) = an x n + an −1x n −1 + ... + a0 ,
(32)
a = an > 0 is any polynomial, п = 1, 2, ... By
′ (h, х) = exp(−hPn ( х))hΨ (h, х), where
relation (23) λ′xx
Ψ (h, х ) = h( Рп′ ( х)) 2 − Рп′′ ( х) .
(33)
The right hand part of (33) is a polynomial of degree of (2п – 2), so it has no more
(2п – 2) changes of signs. Hence, the condition of piecewise-convexity of sequence (6)
is satisfied.
Verify a condition (11). The production k Δλ k (h) is a value of function
π( х) =
xhPn′ ( x)
hPn ( x) Qn ( x)
=
,
exp(hPn ( x)) exp(hPn ( x)) Pn ( x)
where
Qn ( x) = xPn′ ( x).
Then | π( х) | is bounded, since
(34)
Qn ( x )
→ n ( х → +∞ ), and the first fraction
Pn ( x)
hPn ( x)
t
in (34) looks like
, t > 0.
exp t
exp(hPn ( x))
So, the statements of theorem 3 hold for a case (32) at all п = 1, 2, ...
Operator of the translation type
1. Let f ∈ L and
τ h ( f ) : f ( x ) a f ( x + h) ~
∞
∑ ck ( f ) exp(ikh) exp(ikx)
(h > 0)
k = −∞
is a translation operator. Using the integral form of Fourier coefficients ck ( f ) we have
π
⎛1 π
⎧⎪ ∞
⎫⎪ ⎞
⎧⎪1 ∞
⎫⎪
1
f ( x + h) ~ ⎜ ∫ f (t )⎨ + ∑ cos kh cos k ( x − t )⎬dt − ∫ f (t )⎨∑ sin kh sin k ( x − t )⎬dt ⎟ .
⎜π
π
⎪⎩k =1
⎪⎭ ⎟⎠
⎪⎭
⎩⎪2 k =1
⎝ −π
−π
(35)
~
By analogy with (35) we will consider two summarising sequences Λ, Λ and operator
of translation type
~
~
~
τ h ( f ) = τ h f ; Λ, Λ; x : f ( x) a U ( f , x; λ, h) − U ( f , x; λ, h);
(36)
~
~
denote τ* ( f ) = τ* f ; Λ, Λ; x = sup τ h f ; Λ, Λ; x .
(
(
)
)
h >0
(
)
Applying to (36) theorems 1, 2, we obtain the following statements.
~
Theorem 4. If the elements of each sequence Λ and Λ satisfy to condition (7) and
both sequences have certain character of convexity, then the estimates (8) – (10) hold
~
with replacement || U ∗ ( f ) || p + || U ∗ ( f ) || p on || τ∗ ( f ) || p . The statement remains valid
~
for any piecewise-convex sequences Λ and Λ which elements satisfy to conditions of
a kind (7), (11).
If the condition (4) is besides, satisfied, then the relation
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107
~
lim τh ( f ) = f − f
h →0
holds almost everywhere for everyone f ∈ L and in metrics Lp at any p > 1.
2. The result of theorem 4 can be applied to the operator τh ( f )
f ( x ) a τ h ( f ; u , α; ω, β; x) =
+i
∞
∑ exp(−hu α (| k |))ck ( f ) exp(ikx) +
k = −∞
∞
1
(sgn k ) exp⎛⎜ − ωβ (| k |) ⎞⎟ck ( f ) exp(ikx).
⎠
⎝ h
k = −∞
∑
Let u (x) be one of the following functions: u ( x) = ln x, or u ( x) = x, and ω( х) = ln x
or ω( х) = x. Then for every α > 0, β > 0 the relation
lim τ h ( f ; u , α; ω, β; х) = f (x)
h →0
holds a.e. for every f ∈ L and in the metrics Lp ( p > 1).
References
1. Kuk R. Beskonechnye matritsy i prostranstva posledovatel'nostei (Infinite
matrices and sequence spaces), Moscow: GIFML, 1960, 471 p.
2. Bausov L.I. Matematicheskii sbornik, 1965, vol. 68(110), no. 3, pp. 313-327.
3. Sigmund A. Trigonometricheskie ryady (Trigonometric series), 2 vol., Moscow:
Mir, 1965.
4. Efimov A.V. Mathematics of the USSR - Isvestiya, 1960, no. 24, pp. 743-756.
5. Bari N.K. Trigonometricheskie ryady (Trigonometric series), Moscow:
Fizmatlit, 1961, 936 p.
6. Nakhman A.D. Transactions of the Tambov State Technical University, 2009,
vol. 15, no. 3, pp. 653-660.
Экспоненциальные методы суммирования рядов Фурье
А. Д. Нахман1, Б. П. Осиленкер2
Кафедры: «Прикладная математика и механика», ФГБОУ ВПО «ТГТУ» (1);
«Высшая математика», ФГБОУ ВПО «Московский государственный
строительный университет» (2); [email protected]
Ключевые слова и фразы: выпуклые, кусочно-выпуклые экспоненциальные суммирующие последовательности; оценки Lp-норм; сходимость почти
всюду.
Аннотация:
Рассматриваются
полунепрерывные
методы
Λ = {λ k (h), k = 0,1, ...; h > 0 } суммирования рядов Фурье и сопряженных рядов
(
)
Фурье, порожденные экспоненциальными функциями λ( х, h) = exp − hu α ( х) ,
p
α > 0. Получены оценки L -норм соответствующих максимальных операторов.
В качестве следствия приводятся некоторые результаты об экспоненциальных
методах суммирования рядов Фурье почти всюду и в метрике L p .
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ISSN 0136-5835. Вестник ТГТУ. 2014. Том 20. № 1. Transactions TSTU
Список литературы
1. Кук, P. Бесконечные матрицы и пространства последовательностей :
монография / Р. Кук. – М. : ГИФМЛ, 1960. – 471 с.
2. Баусов, Л. И. О линейных методах суммирования рядов Фурье /
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Exsponentialmethoden der Summierung der Reihen von Fourier
Zusammenfassung: Es werden die halbununterbrochenen Methoden
Λ = {λ k ( h), k = 0,1,...; h > 0 } der Reihen von Fourier und der verknüpften Reihen von
Fourier, die von den Expontionalfunktionen λ( х, h) = exp(−hu α ( х)), α > 0. erzeugt
wurden. Es sind die Einschätzungen von Lp-Norm der entsprechenden maximalen
Operatoren erhaltet.Als Untersuchung werden einige Ergebnisse über die
Expontionalmethoden der Summierung der Reihen von Fourier fast überall und in der
Metrik L p gebracht.
Méthodes exponentielles de la summation des séries Fourier
Résumé:
Sont
examinées
les
méthodes
semi-continues
Λ = {λ k ( h), k = 0,1,...; h > 0 } de la summation des séries de Fourier et des séries
conjuguées générées par les fonctions exponentielles λ( х, h) = exp(−hu α ( х)), α > 0.
Sont reçues les estimations Lp-normes des opérateurs correspodants. En qualité de
conséquence sont cités les résultats sur les méthodes exponentielles de la summation
des séries Fourier presque partout et dans la métrique L p .
Авторы: Нахман Александр Давидович – кандидат физико-математических
наук, доцент кафедры «Прикладная математика и механика», ФГБОУ ВПО
«ТГТУ»; Осиленкер Борис Петрович – доктор физико-математических наук,
профессор кафедры высшей математики, ФГБОУ ВПО «Московский государственный строительный университет», г. Москва.
Рецензент: Куликов Геннадий Михайлович – доктор физико-математических наук, профессор, заведующий кафедрой «Прикладная математика
и механика», ФГБОУ ВПО «ТГТУ».
ISSN 0136-5835. Вестник ТГТУ. 2014. Том 20. № 1. Transactions TSTU
109