Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents
In the present work, we study the estimates for the periodic functions of linear operators constructed on the basis of their Fourier series in weighted Lebesgue spaces with variable exponent and Muckenhoupt weights. In this case, the obtained estimates depend on the sequence of the best approximatio...
Збережено в:
Дата: | 2014 |
---|---|
Автор: | |
Формат: | Стаття |
Мова: | English |
Опубліковано: |
Інститут математики НАН України
2014
|
Назва видання: | Український математичний журнал |
Теми: | |
Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/166117 |
Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
Цитувати: | Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents / S.Z. Jafarov // Український математичний журнал. — 2014. — Т. 66, № 10. — С. 1348–1356. — Бібліогр.: 32 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraineid |
irk-123456789-166117 |
---|---|
record_format |
dspace |
spelling |
irk-123456789-1661172020-02-19T01:28:15Z Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents Jafarov, S.Z. Статті In the present work, we study the estimates for the periodic functions of linear operators constructed on the basis of their Fourier series in weighted Lebesgue spaces with variable exponent and Muckenhoupt weights. In this case, the obtained estimates depend on the sequence of the best approximation in weighted Lebesgue spaces with variable exponent. Вивчаються оцінки періодичних функцій лінійних операторiв, що побудовані на основі рядів Фур'є у зважених просторах Лебега зі змінним показником та вагою Макенхаупта. В даному випадку отримані оцінки залежать від послідовності найкращого наближення у зважених просторах Лебега зі змінним показником. 2014 Article Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents / S.Z. Jafarov // Український математичний журнал. — 2014. — Т. 66, № 10. — С. 1348–1356. — Бібліогр.: 32 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166117 517.5 en Український математичний журнал Інститут математики НАН України |
institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
collection |
DSpace DC |
language |
English |
topic |
Статті Статті |
spellingShingle |
Статті Статті Jafarov, S.Z. Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents Український математичний журнал |
description |
In the present work, we study the estimates for the periodic functions of linear operators constructed on the basis of their Fourier series in weighted Lebesgue spaces with variable exponent and Muckenhoupt weights. In this case, the obtained estimates depend on the sequence of the best approximation in weighted Lebesgue spaces with variable exponent. |
format |
Article |
author |
Jafarov, S.Z. |
author_facet |
Jafarov, S.Z. |
author_sort |
Jafarov, S.Z. |
title |
Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents |
title_short |
Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents |
title_full |
Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents |
title_fullStr |
Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents |
title_full_unstemmed |
Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents |
title_sort |
linear methods for summing fourier series and approximation in weighted lebesgue spaces with variable exponents |
publisher |
Інститут математики НАН України |
publishDate |
2014 |
topic_facet |
Статті |
url |
http://dspace.nbuv.gov.ua/handle/123456789/166117 |
citation_txt |
Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents / S.Z. Jafarov // Український математичний журнал. — 2014. — Т. 66, № 10. — С. 1348–1356. — Бібліогр.: 32 назв. — англ. |
series |
Український математичний журнал |
work_keys_str_mv |
AT jafarovsz linearmethodsforsummingfourierseriesandapproximationinweightedlebesguespaceswithvariableexponents |
first_indexed |
2025-07-14T20:46:56Z |
last_indexed |
2025-07-14T20:46:56Z |
_version_ |
1837656720593125376 |
fulltext |
UDC 517.5
S. Z. Jafarov (Pamukkale Univ., Denizli, Turkey; Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku)
LINEAR METHODS OF SUMMING FOURIER SERIES AND APPROXIMATION
IN WEIGHTED VARIABLE EXPONENT LEBESGUE SPACES
ЛIНIЙНI МЕТОДИ ПIДСУМОВУВАННЯ РЯДIВ ФУР’Є ТА НАБЛИЖЕННЯ
У ЗВАЖЕНИХ ПРОСТОРАХ ЛЕБЕГА ЗI ЗМIННИМ ПОКАЗНИКОМ
In the present work, we study estimates for the periodic functions from the linear operators constructed on the basis of
their Fourier series in weighted variable exponent Lebesgue spaces with Muckenhoupt weights. In this case, the obtained
estimates depend on the sequence of the best approximation in weighted Lebesgue spaces with variable exponent.
Вивчаються оцiнки перiодичних функцiй лiнiйних операторiв, що побудованi на основi рядiв Фур’є у зважених
просторах Лебега зi змiнним показником та вагою Макенхаупта. В даному випадку отриманi оцiнки залежать вiд
послiдовностi найкращого наближення у зважених просторах Лебега зi змiнним показником.
1. Introduction and the main results. Let T denote the interval [0, 2π], C complex plane, and
Lp(T), 1 ≤ p ≤ ∞, the Lebesgue space of measurable complex valued functions on T.
We consider the sequence of the functions {λk(r)} defined in the set E of the number line,
satisfying the conditions that
λ0(r) = 1, lim
r→r0
λν(r) = 1
for an arbitrary fixed ν = 0, 1, 2, . . . .
Let
a0
2
+
∞∑
k=1
Ak(x, f), Ak(x, f) := ak(f) cos kx+ bk(f) sin kx (1.1)
be the Fourier series of the function f ∈ L1(T), where ak(f) and bk(f) are Fourier coefficients of
the function f. The nth partial sums of the series (1.1) is defined
Sn(x, f) =
a0
2
+
n∑
k=1
Ak(x, f).
Let us denote by ℘ the class of Lebesgue measurable functions p : T → (1,∞) such that
1 < p∗ := ess inf
x∈T
p(x) ≤ p∗ := ess
x∈T
sup p(x) < ∞. The conjugate exponent of p(x) is shown by
p′(x) :=
p(x)
p(x)− 1
. For p ∈ ℘, we define a class Lp(.)(T) of 2π-periodic measurable functions f :
T→ C satisfying the condition ∫
T
|f(x)|p(x)dx <∞.
This class Lp(.)(T) is a Banach space with respect to the norm
‖f‖Lp(.)(T) := inf
λ > 0 :
∫
T
∣∣∣∣f(x)
λ
∣∣∣∣p(x) dx ≤ 1
.
c© S. Z. JAFAROV, 2014
1348 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
LINEAR METHODS OF SUMMING FOURIER SERIES AND APPROXIMATION . . . 1349
A function ω : T→ [0,∞] is called a weight function if ω is a measurable and almost everywhere
(a.e.) positive.
Let ω be a 2π-periodic weight function. We denote by Lpω(T) the weighted Lebesgue space of
2π-periodic measurable functions f : T→ C such that fω1/p ∈ Lp(T). For f ∈ Lpω(T) we set
‖f‖Lpω(T) :=
∥∥∥fω1/p
∥∥∥
Lp(T)
.
L
p(.)
ω (T) stands for the class of Lebesgue measurable functions f : T → C such that ωf ∈
∈ Lp(.)(T). L
p(.)
ω (T) is called the weighted Lebesgue space with variable exponent. The space
L
p(.)
ω (T) is a Banach space with respect to the norm
‖f‖
L
p(.)
ω (T)
:= ‖fω‖Lp(.)(T) .
Its known [17] that the set of trigonometric polynomials is dense in Lp(.)ω (T), if [ω(x)]p(x) is integrable
on T.
We suppose that for an arbitrary fixed r ∈ E and for every function f ∈ Lp(.)ω (T), the series
Ur(x, f, λ) =
a0
2
+
∞∑
k=1
λk(r) Ak(x, f) (1.2)
converges in the space Lp(.)ω (T).
For each linear operator Ur(x, f, λ) we set
Rr(f, λ)
L
p(.)
ω (T)
:= ‖f − Ur(x, f, λ)‖
L
p(.)
ω (T)
. (1.3)
Let B be the class of all intervals in T. For B ∈ B we set
pB :=
1
|B|
∫
B
1
p(x)
dx
−1 .
For given p ∈ ℘ the class of weights ω satisfying the condition [1]∥∥∥ωp(x)∥∥∥
Ap(.)
:= sup
B∈B
1
|B|pB
∥∥∥ωp(x)∥∥∥
L1(B)
∥∥∥∥ 1
ωp(x)
∥∥∥∥
B(p′(.)/p(.))
<∞
will be denoted by Ap(.).
We say that the variable exponent p(x) satisfies Local log-Hölder continuity condition, if there is
a positive constant c1 such that
|p(x)− p(y)| ≤ c1
log
(
e+
1
|x− y|
) , (1.4)
for all x, y ∈ T.
A function p ∈ ℘ is said to belong to the class ℘log, if the condition (1.4) is satisfied.
We denote by En(f)
L
p(.)
ω (T)
the best approximation of f ∈ Lp(.)ω (T) by trigonometric polynomials
of degree not exceeding n, i. e.,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1350 S. Z. JAFAROV
En(f)
L
p(.)
ω (T)
= inf{‖f − Tn‖Lp(.)ω (T)
: Tn ∈ Πn},
where Πn denotes the class of trigonometric polynomials of degree at most n.
In this study we obtain estimates of the deviation of the periodic function from the linear operators
constructed on the basis of its Fourier series in variable exponent Lebesgue spaces with Muckenhoupt
weights. In particular, we obtain the general estimate for the deviation Rr(f, λ)
L
p(.)
ω (T)
of the function
f from its Abel – Poisson means Ur(x, f, λ) (0 ≤ r < 1, λν(r) = rν , ν = 0, 1, 2, 3, . . .) in variable
exponent Lebesgue spaces with Muckenhoupt weights. Note that the estimate obtained in this study
depends on the rate of decrease of sequences of the best approximation En(f)
L
p(.)
ω (T)
.
Lebesgue spaces with variable exponents have been investigated intensively by many authors
(see, for example, [21, 22, 24, 25, 27]).
The approximation problems in nonweighted and weighted Lebesgue spaces with variable expo-
nents were studied in [1, 2, 11, 16 – 20, 28].
Similar problems of the approximation theory in the different spaces have been studied by several
authors (see, for example, [4 – 10, 12 –15, 23, 26, 29 – 32]).
Our main results are as follows.
Theorem 1.1. Let {λν(r)} be an arbitrary triangular matrix (r = 0, 1, 2, 3, . . . , λ0(r) =
= 1, λν(r) = 0, ν > r). If p ∈ ℘log, ω−p0 ∈ A(p(.)/p0)′ for some p0 ∈ (1, p∗), γ := min{2, p∗}
and f ∈ Lp(.)ω (T), there exists a positive constant c2 depending on p such that
Rr(f, λ)
L
p(.)
ω (T)
≤ c2
m∑
µ=0
δγ2µ(r) Eγ2µ−1(f)
L
p(.)
ω (T)
1/γ
+ Er(f)
L
p(.)
ω (T)
,
where 2m ≤ r < 2m+1 and
δ2µ(r) =
2µ+1−1∑
ν=2µ
|λν(r)− λν+1(r)|+ |1− λ2µ+1(r)|.
Corollary 1.1. 1. Let r = 0, 1, 2, 3, . . . ,
λν(r) =
1− ν
r + 1
, 0 ≤ ν ≤ r,
0, ν > r.
Then for the Fejér means, the estimate
Rr(f, λ)
L
p(.)
ω (T)
≤ c3
r + 1
{
r∑
ν=0
(ν + 1)
γ−1
Eγν (f)
L
p(.)
ω (T)
}1/γ
holds with a constant c3> 0 depending on p.
2. For r = 0, 1, 2, 3, . . . ,
λν(r) =
1− νk
(r + 1)k
, 0 ≤ ν ≤ r,
0, ν > r.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
LINEAR METHODS OF SUMMING FOURIER SERIES AND APPROXIMATION . . . 1351
Then for the Zygmund means, the estimate
Rr(f, λ)
L
p(.)
ω (T)
≤ c4
(ν+1)k
{
r∑
ν=0
(ν + 1)
γk−1
Eγν (f)
L
p(.)
ω (T)
}1/γ
holds with a constant c4 > 0 depending on p and k.
Theorem 1.2. Let 0 ≤ r < 1, λν(r) = rν , ν = 0, 1, 2, 3, . . . . If p ∈ ℘log, ω−p0 ∈ A(p(.)/p0)′
for some p0 ∈ (1, p∗), γ := min{2, p∗} and f ∈ Lp(.)ω (T), then for the Abel – Poisson means the
estimate
Rr(f, λ)
L
p(.)
ω (T)
≤ c5(1− r)
{ ∞∑
ν=0
rν(ν + 1)γ−1Eγν (f)
L
p(.)
ω (T)
}1/γ
holds with a constant c5 > 0 depending on p.
In the proof main results we need the following theorems [18].
The following Littlewood – Paley type theorem holds:
Theorem 1.3. Let
∑∞
ν=1
Aν(x) be the Fourier series of f ∈ Lp(.)ω (T). Under the conditions of
Theorem 1.1 there are constants c6, c7 > 0 such that
c6
∥∥∥∥∥∥∥
∞∑
µ=1
∆2
µ
1/2
∥∥∥∥∥∥∥ ≤ ‖f(x)‖
L
p(.)
ω (T)
≤ c7
∥∥∥∥∥∥∥
∞∑
µ=1
∆2
µ
1/2
∥∥∥∥∥∥∥
L
p(.)
ω (T)
, (1.5)
where
∆µ := ∆µ(f, x) =
2µ−1∑
ν=2µ−1
Aν(x, f), µ = 1, 2, . . . .
The following theorem is true:
Theorem 1.4. Let {λk}∞0 be a sequence of numbers such that
|λk| ≤ c8 and
2m−1∑
k=2m−1
|λk−λk+1| ≤ c8, (1.6)
where c8 > 0 does not depend on k and m. Suppose that the conditions of Theorem 1.1 are satisfied.
If f ∈ Lp(.)ω (T) has the Fourier series
a0
2
+
∞∑
k=1
Ak(x , f ),
then there exists a function F ∈ Lp(.)ω (T) with the Fourier series
λ0a0
2
+
∞∑
k=1
λkAk(x , f ),
and
‖F‖
L
p(.)
ω (T)
≤ c9‖f‖Lp(.)ω (T)
·
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1352 S. Z. JAFAROV
2. Proofs of theorems. Proof of Theorem 1.1. Let f ∈ L
p(.)
.ω (T) and 2m ≤ r < 2m+1.
According to reference [11] the following inequality holds:
‖f (x )− Sr (x , f )‖
L
p(.)
ω (T)
≤ Er (f )
L
p(.)
ω (T)
. (2.1)
Using (2.1) we get
Rr(f, λ)
L
p(.)
ω (T)
=
∥∥∥∥∥ f(x)−
r∑
ν=0
λν(r)Aν(x, f)
∥∥∥∥∥
L
p(.)
ω (T)
≤
≤
∥∥∥∥∥
r∑
ν=1
(1− λν(r))Aν(x, f)
∥∥∥∥∥
L
p(.)
ω (T)
+
∥∥∥∥∥
∞∑
ν=r+1
Aν(x, f)
∥∥∥∥∥
L
p(.)
ω (T)
≤
≤
∥∥∥∥∥
r∑
ν=1
(1− λν(r))Aν(x, f)
∥∥∥∥∥
L
p(.)
ω (T)
+ Er(f)
L
p(.)
ω (T)
. (2.2)
According to inequality (2.2) and (1.5) we obtain
Rr(f, λ)
L
p(.)
ω (T)
≤
∥∥∥∥∥∥∥
m∑
µ=0
∣∣∣∣∣
2µ+1−1∑
ν=2µ
(1− λν(r))Aν(x, f))
∣∣∣∣∣
2
1/2
∥∥∥∥∥∥∥
L
p(.)
ω (T)
+
+Er(f)
L
p(.)
ω (T)
. (2.3)
Use of Abel’s transformation leads to
σr,µ(x) =
2µ+1−1∑
ν=2µ
(1− λν(r))Aν(x, f) =
=
2µ+1−1∑
ν=2µ
{Sν(f, x)− S
2µ−1
(f, x)}{λν+1(r)− λν(r)}+
+{1− λ2µ+1(r)}{S2µ+1−1(f, x)− S2µ−1(f, x)}. (2.4)
Using (2.1), (2.4), Minkowski’s inequality and monotonicity of the best approximation sequence we
reach
‖σr,µ‖Lp(.)ω (T)
≤
2µ+1−1∑
ν=2µ
‖Sν(f, x)− S2µ−1(f, x)‖
L
p(.)
ω (T)
|λν+1(r)− λν(r)|+
+|1− λ2µ+1(r)|‖S2µ+1−1(f, x)− S2µ−1(f, x)‖ ≤
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
LINEAR METHODS OF SUMMING FOURIER SERIES AND APPROXIMATION . . . 1353
≤ c10E2µ−1(f)
L
p(.)
ω (T)
2µ+1−1∑
ν=2µ
|λν+1(r)− λν(r)|+ |1− λ2µ+1(r)|
=E2µ−1(f)
L
p(.)
ω (T)
δ2µ(r).
(2.5)
In addition, the following inequality holds:∥∥∥∥∥∥∥
m∑
µ=0
|σr,µ(x)|2
1/2
∥∥∥∥∥∥∥
L
p(.)
ω (T)
≤ c11
m∑
‖
µ=0
σr,µ‖γ
L
p(.)
ω (T)
1/γ
. (2.6)
Use of (2.3), (2.6) and (2.5) gives us
Rr(f, λ)
L
p(.)
ω (T)
≤ c12
m∑
µ=0
‖σr,µ‖γ
L
p(.)
ω (T)
1/γ
+ Er(f)
L
p(.)
ω (T)
≤
≤ c13
m∑
µ=0
Eγ2µ−1(f)
L
p(.)
ω (T)
δγ2µ(r)
1/γ
+ Er(f)
L
p(.)
ω (T)
.
Theorem 1.1 is proved.
Proof of Theorem 1.2. Let f ∈ Lp(.)ω (T) and λν(r) = rν , 0 ≤ r < 1, ν = 0, 1, 2, . . . . We have
Rr(f, λ)
L
p(.)
ω (T)
=
∥∥∥∥∥ f(x)−
∞∑
ν=0
rνAν(x, f)
∥∥∥∥∥
L
p(.)
ω (T)
=
=
∥∥∥∥∥∥
∞∑
ν=0
(1−rν)Aν(x, f)
∥∥∥∥∥∥
L
p(.)
ω (T)
. (2.7)
Selecting m such that 2m ≤ n =
[
1
1− r
]
< 2m+1 (here, [β] denotes the integer part of a real
number β) from (2.7) we get
Rr(f, λ)
L
p(.)
ω (T)
≤
∥∥∥∥∥∥
2m+1−1∑
ν=0
(1− rν)Aν(x, f)
∥∥∥∥∥∥
L
p(.)
ω (T)
+
+
∥∥∥∥∥
∞∑
ν=2m+1
(1− rν)Aν(x, f)
∥∥∥∥∥
L
p(.)
ω (T)
= I1 + I2. (2.8)
According to (1.5) we obtain
I1 ≤ c14
∥∥∥∥∥∥∥
m∑
ν=0
∣∣∣∣∣
2ν+1−1∑
µ=2ν
(1− rµ)Aµ(x, f)
∣∣∣∣∣
2
1/2
∥∥∥∥∥∥∥
L
p(.)
ω (T)
(2.9)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1354 S. Z. JAFAROV
and therefore
I1 ≤ c15
m∑
ν=0
∥∥∥∥∥
2ν+1−1∑
µ=2ν
(1−rµ)Aµ(x, f)
∥∥∥∥∥
γ
L
p(.)
ω (T)
1/γ
. (2.10)
Using Abel’s transformation and (2.1), we find that∥∥∥∥∥∥
2ν+1−1∑
µ=2ν
(1−rµ)Aµ(x, f)
∥∥∥∥∥∥
L
p(.)
ω (T)
=
=
∥∥∥∥∥
2ν+1−1∑
µ=2ν
{Sµ(f, x)− S2ν−1(f, x)}(rµ+1− rµ)+
+{S2ν+1−1(f, x)− S2ν−1(f, x)}(1− r2ν+1
)
∥∥∥∥∥
L
p(.)
ω (T)
≤
≤ c162ν+1(1− r)E2ν−1(f)
L
p(.)
ω (T)
. (2.11)
Using monotonicity of the best approximation sequence En(f)
L
p(.)
ω (T )
from (2.11) we have∥∥∥∥∥∥
2ν+1−1∑
µ=2ν
(1−rµ)Aν(x, f)
∥∥∥∥∥∥
γ
L
p(.)
ω (T)
≤ c172(ν+1)γ(1− r)γEγ2ν−1(f)
L
p(.)
ω (T)
≤
≤ c18(1− r)γ
2ν−1∑
µ=2ν−1
Eγ2ν−1(f)
L
p(.)
ω (T)
. (2.12)
Consideration of (2.10 ) and (2.12 ) gives us we
I1 ≤ c19
{
(1− r)γ‖A1(x, f)‖γ
L
p(.)
ω (T)
+
+
m∑
ν=1
(1− r)γ
2ν−1∑
µ=2ν−1
µγ−1Eγµ(f)
L
p(.)
ω (T)
}1/γ
≤
≤ c20(1− r)
n∑
µ=0
(µ+ 1)γ−1Eγµ(f)
L
p(.)
ω (T)
1/γ
. (2.13)
Applying Theorem 1.4 for the sequence {λk}∞0 satisfying the condition (1.5 ) and (2.1), we obtain
I2 ≤ c21
∥∥∥∥∥∥
∞∑
ν=2m+1
Aν(x, f)
∥∥∥∥∥∥
L
p(.)
ω (T)
≤ c22En(f)
L
p(.)
ω (T)
. (2.14)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
LINEAR METHODS OF SUMMING FOURIER SERIES AND APPROXIMATION . . . 1355
Therefore according to (2.8 ), ( 2.13) and (2.14 ), we have
Rr(f, λ)
L
p(.)
ω (T)
≤
∥∥∥∥∥∥
2m+1−1∑
ν=0
(1−rν)Aν(x, f)
∥∥∥∥∥∥
L
p(.)
ω (T)
+
+
∥∥∥∥∥
∞∑
ν=2m+1
(1−rν)Aν(x, f)
∥∥∥∥∥
L
p(.)
ω (T)
≤
≤ c20(1− r)
n∑
µ=0
(µ+ 1)γ−1Eγµ(f)
L
p(.)
ω (T)
1/γ
+ c22En(f)
L
p(.)
ω (T)
≤
≤ c23(1− r)
{
n∑
ν=0
(ν + 1)γ−1Eγν (f)
L
p(.)
ω (T)
}1/γ
≤
≤ c24(1− r)
∞∑
ν=0
rν(ν + 1)γ−1Eγν (f)
L
p(.)
ω (T)
1/γ
.
Theorem 1.2 is proved.
1. Akgün R., Kokilashvili V. On converse theorems of trigonometric approximation in weighted variable exponent
Lebesgue spaces // Banach J. Math. Anal. – 2011. – 5, № 1. – P. 70 – 82.
2. Akgün R. Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent // Ukr.
Math. J. – 2011. – 63, № 1. – P. 1 – 26.
3. Diening L., Höstö P., Nekvinda A. Open problems in variable exponent and Sobolev spaces // Function Spaces,
Different. Operators and Nonlinear Anal.: Proc. Conf. held in Milovy (Bohemian-Moravian Uplands, May 29 – June
2, 2004). – Praha, 2005. – P. 38 – 58.
4. Gavriljuk V. G. Linear summability methods for Fourier series and best approximation // Ukr. Math. Zh. –1963. – 15,
№ 4. – P. 412 – 418 (in Russian).
5. Guven A. Trigonometric, approximation of functions in weighted Lp spaces // Sarajevo J. Math. – 2009. – 5, № 17. –
P. 99 – 108.
6. Guven A., Israfilov D. M. Approximation by means of Fourier trigonometric series in weighted Orlicz spaces // Adv.
Stud. Contemp. Math. (Kyundshang). – 2009. – 19, № 2. – P. 283 – 295.
7. Ibragimov I. I., Mamedkhanov D. I. A constructive characterization of a certain class of functions // Dokl. Akad.
Nauk SSSR. – 1975. – 223, № 1. – P. 35 – 37 (in Russian).
8. Israfilov D. M. Approximation by p-Faber polynomials in the weighted Smirnov class Ep(G,w) and the Bieberbach
polynomials // Constr. Approxim. – 2001. – 17. – P. 335 – 351.
9. Israfilov D. M., Akgün R. Approximation in weighted Smirnov – Orlicz classes // J. Math. Kyoto Univ. – 2006. – 46. –
P. 755 – 770.
10. Israfilov D. M., Guven Ali. Approximation by trigonometric polynomials in weighted Orlicz spaces // Stud. Math. –
2006. – 174, № 2. – P. 147 – 168.
11. Israfilov D . M., Kokilashvili V. , Samko S. G. Approximation in weighted Lebesgue spaces and Smirnov spaces with
variable exponent // Proc. A. Razmadze Math. Inst. – 2007. – 143. – P. 25 – 35.
12. Jafarov S. Z. Approximation by rational functions in Smirnov – Orlicz classes // J. Math. Anal. and Appl. – 2011. –
379. – P. 870 – 877.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
1356 S. Z. JAFAROV
13. Jafarov S. Z. Approximation by polynomials and rational functions in Orlicz spaces // J. Comput. Anal. and Appl. –
2011. – 13, № 5. – P. 953 – 962.
14. Jafarov S. Z. The inverse theorem of approximation of the function in Smirnov – Orlicz classes // Math. Inequalit.
and Appl. – 2012. – 12, № 4. – P. 835 – 844.
15. Jafarov S. Z. On approximation in weighted Smirnov – Orlicz classes // Complex Variables and Elliptic Equat. – 2012. –
57, № 5. – P. 567 – 577.
16. Kokilashvili V., Samko S. G. Singular integrals weighted Lebesgue spaces with variable exponent // Georg. Math. J. –
2003. – 10, № 1. – P. 145 – 156.
17. Kokilashvili V., Samko S. Singular integrals and potentials in some Banach function spaces with variable exponent //
J. Funct. Spaces Appl. – 2003. – 1, № 1. – P. 45 – 59.
18. Kokilashvili V., SamkoS. G. Operators of harmonic analysis in weighted spaces with non-standard growth // J. Math.
Anal. and Appl. – 2009. – 352. – P. 15 – 34.
19. Kokilashvili V., Samko S. G. A refined inverse inequality of approximation in weighted variable exponent Lebesgue
spaces // Proc. A. Razmadze Math. Inst. – 2009. – 151. – P. 134 – 138.
20. Kokilashvili V., Tsanava Ts. On the norm estimate of deviation by linear summability means and an extension of the
Bernstein inequality // Proc. A. Razmadze Math. Inst. – 2010. – 154. – P. 144 – 146.
21. Kokilashvili V. On a progress in the theory of integral operators in weighted Banach function spaces // Function
Spaces, Different. Operators and Nonlinear Anal.: Proc. Conf. held in Milovy (Bohemian-Moravian Uplands, May
29 – June 2, 2004). – Praha, 2005. – P. 152 – 174.
22. Kováčik O., Rákosnik J. On spaces Lp(x) and W k,p(x) // Czechoslovak Math. J. – 1991. – 41, № 116. – P. 592 – 618.
23. Mamedkhanov D. M. Approximation in complex plane and singular operators with a Cauchy kernel: Dissertation
doct. phys.-math. nauk. – Tbilisi, 1984 (in Russian).
24. Samko S. G. Diferentiation and integration of variable order an the spaces Lp(x) // Operator Theory for Complex and
Hypercomplex Analysis (Mexico City, 1994). Contemp. Math. – 1998. – 212. – P. 203 – 219.
25. Samko S. G. On a progres in the theory of Lebesgue spaces with variable exponent: maximal and singular operators
// Integral Transforms Spec. Funct. – 2005. – 16, № 5-6. – P. 461 – 482.
26. Stechkin S. B. The approximation of periodic functions by Fejér sums // Trudy Math. Inst. Steklov. – 1961. – 2. –
P. 48 – 60 (in Russian).
27. Sharapudinov I. I. The topology of the space Lp(t)([0, 1]) // Mat. Zametki. – 1979. – 26, № 4. – P. 613 – 632 (in
Russian).
28. Sharapudinov I. I. Approximation of functions in the metric of the space Lp(t)([a, b]) and quadrature (in Russian) //
Constructive Function Theory 81 (Varna, 1981). – Sofia: Publ. House Bulgar. Acad. Sci., 1983. – P. 189 – 193.
29. Timan M. F. Best approximation of a function and linear methods of summing Fourier series // Izv. Akad. Nauk SSSR
Ser. Math. – 1965. – 29. – P. 587 – 604 (in Russian).
30. Timan M. F. The approximation of continuous periodic functions by linear operators which are constructed on the
basis of their Fourier series // Dokl. Akad. Nauk SSSR. – 1968. – 181. – S. 1339 – 1342 (in Russian).
31. Timan M. F. Some linear summation processes for Fourier series and best approximation // Dokl. Akad. Nauk SSSR . –
1962. – 145 . – S. 741 – 743 (in Russian).
32. Timan M. F. Deviation of harmonic functions from their boundary values and best approximation // Dokl. Akad.
Nauk SSSR. – 1962. – 145 . – S. 1008 – 1009.
Received 17.07.13,
after revision — 14.05.14
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 10
|