Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions
UDC 517.5 In terms of the best approximations of functions and generalized moduli of smoothness, direct and inverse approximation theorems are proved for Besicovitch almost periodic functions whose Fourier exponent sequences have a single limit point in infinity and their Orlicz norms are finite. Sp...
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| author | Chaichenko , S. O. Shidlich, A. L. Shulyk, T. V. Chaichenko , S. O. Shidlich, A. L. Shulyk, T. V. Shidlich, Andrii |
| author_facet | Chaichenko , S. O. Shidlich, A. L. Shulyk, T. V. Chaichenko , S. O. Shidlich, A. L. Shulyk, T. V. Shidlich, Andrii |
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In terms of the best approximations of functions and generalized moduli of smoothness, direct and inverse approximation theorems are proved for Besicovitch almost periodic functions whose Fourier exponent sequences have a single limit point in infinity and their Orlicz norms are finite. Special attention is paid to the study of cases when the constants in these theorems are unimprovable. |
| doi_str_mv | 10.37863/umzh.v74i5.7045 |
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DOI: 10.37863/umzh.v74i5.7045
UDC 517.5
S. O. Chaichenko (Donbas State Pedagog. Univ., Sloviansk, Donetsk region),
A. L. Shidlich (Inst. Math. Nat. Acad. Sci. Ukraine; Nat. Univ. Life and Environmental Sci. Ukraine, Kyiv),
T. V. Shulyk (Donbas State Pedagog. Univ., Sloviansk, Donetsk region)
DIRECT AND INVERSE APPROXIMATION THEOREMS
IN THE BESICOVITCH – MUSIELAK – ORLICZ SPACES
OF ALMOST PERIODIC FUNCTIONS
ПРЯМI ТА ОБЕРНЕНI ТЕОРЕМИ НАБЛИЖЕННЯ
У ПРОСТОРАХ БЕЗИКОВИЧА – МУСЄЛАКА – ОРЛИЧА
МАЙЖЕ ПЕРIОДИЧНИХ ФУНКЦIЙ
In terms of the best approximations of functions and generalized moduli of smoothness, direct and inverse approximation
theorems are proved for Besicovitch almost periodic functions whose Fourier exponent sequences have a single limit point
at infinity and their Orlicz norms are finite. Special attention is paid to the study of cases where the constants in these
theorems are unimprovable.
У термiнах найкращих наближень функцiй та узагальнених модулiв гладкостi доведено прямi та оберненi апрокси-
мацiйнi теореми для майже перiодичних за Безиковичем функцiй, послiдовностi коефiцiєнтiв Фур’є яких мають
єдину граничну точку в нескiнченностi, а їхнi норми Орлича є скiнченними. Особливу увагу придiлено вивченню
випадкiв, коли сталi у цих теоремах непокращуванi.
1. Introduction. The establishment of connections between the difference and differential properties
of the function being approximated and the value of the error of its approximation by some methods
was originated in the well-known works of Jackson (1911) and Bernstein (1912), in which the first
direct and inverse approximation theorems were obtained. Subsequently, similar studies were carried
out by many authors for various functional classes and for various approximating aggregates, and
their results constitute the classics of modern approximation theory. Moreover, the exact results
(in particular, in the sense of unimprovable constants) deserve special attention. A fairly complete
description of the results on obtaining direct and inverse approximation theorems is contained in the
monographs [14, 28, 30, 31].
In spaces of almost periodic functions, direct approximation theorems were established in the
papers [8, 12, 23, 24, 26]. In particular, Prytula [23] obtained direct approximation theorem for
Besicovitch almost periodic functions of the order 2 (B2-a.p. functions) in terms of the best approx-
imations of functions and their moduli of continuity. In [24] and [8], such theorems were obtained,
respectively, with moduli of smoothness of B2-a.p. functions of arbitrary positive integer order and
with generalized moduli of smoothness. In [26], direct and inverse approximation theorems were
obtained in the Besicovitch – Stepanets spaces B\scrS p. The main goal of this article is to obtain such
theorems in the Besicovitch – Musielak – Orlicz spaces B\scrS M. These spaces are natural generaliza-
tions of the all spaces mentioned above, and the results obtained can be viewed as an extension of
these results to the spaces B\scrS M.
2. Preliminaries. 2.1. Definition of the spaces \bfitB \bfscrS M. Let Bs, 1 \leq s < \infty , be the space
of all functions Lebesgue summable with the sth degrees in each finite interval of the real axis, in
c\bigcirc S. O. CHAICHENKO, A. L. SHIDLICH, T. V. SHULYK, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5 701
702 S. O. CHAICHENKO, A. L. SHIDLICH, T. V. SHULYK
which the distance is defined by the equality
D
Bs (f, g) =
\left( \mathrm{l}\mathrm{i}\mathrm{m}
T\rightarrow \infty
1
2T
T\int
- T
| f(x) - g(x)| sdx
\right) 1/s
.
Further, let T be the set of all trigonometric sums of the form \tau N (x) =
\sum N
k=1
ake
i\lambda kx, N \in \BbbN ,
where \lambda k and ak are arbitrary real and complex numbers (\lambda k \in \BbbR , ak \in \BbbC ).
An arbitrary function f is called a Besicovitch almost periodic function of order s (or Bs-a.p.
function) and is denoted by f \in Bs-a.p. [20] (Ch. 5, \S 10), [10] (Ch. 2, \S 7), if there exists a sequence
of trigonometric sums \tau 1, \tau 2, . . . from the set T such that
\mathrm{l}\mathrm{i}\mathrm{m}
N\rightarrow \infty
D
Bs (f, \tau N ) = 0.
If s1 \geq s2 \geq 1, then (see, for example, [12, 13]) Bs1 -a.p.\subset Bs2 -a.p.\subset B-a.p., where B-
a.p.:= B1-a.p. For any B-a.p. function f, there exists the average value
A\{ f\} := \mathrm{l}\mathrm{i}\mathrm{m}
T\rightarrow \infty
1
T
T\int
0
f(x)dx.
The value of the function A\{ f(\cdot )e - i\lambda \cdot \} , \lambda \in \BbbR , can be nonzero at most on a countable set. As a
result of numbering the values of this set in an arbitrary order, we obtain a set S(f) = \{ \lambda k\} k\in \BbbN of
Fourier exponents, which is called the spectrum of the function f. The numbers A\lambda k
= A\lambda k
(f) =
= A\{ f(\cdot )e - i\lambda k\cdot \} are called the Fourier coefficients of the function f. To each function f \in B-a.p.
with spectrum S(f) there corresponds a Fourier series of the form
\sum
k
A\lambda k
ei\lambda kx. If, in addition,
f \in B2-a.p., then the Parseval equality holds (see, for example, [10], Ch. 2, \S 9)
A\{ | f | 2\} =
\sum
k\in \BbbN
| A\lambda k
| 2.
Further, we will consider only those B-a.p. functions from the spaces B\scrS p, the sequences of
Fourier exponents of which have a single limit point at infinity. For such functions f, the Fourier
series are written in the symmetric form
S[f ](x) =
\sum
k\in \BbbZ
Ake
i\lambda kx, where Ak = Ak(f) = A\{ f(\cdot )e - i\lambda k\cdot \} , (2.1)
\lambda 0 := 0, \lambda - k = - \lambda k, | Ak| + | A - k| > 0, \lambda k+1 > \lambda k > 0 for k > 0.
Let \bfM = \{ Mk(t)\} k\in \BbbZ , t \geq 0, be a sequence of Orlicz functions. In other words, for every k \in \BbbZ ,
the function Mk(t) is a nondecreasing convex function for which Mk(0) = 0 and Mk(t) \rightarrow \infty as
t \rightarrow \infty . Let \bfM \ast = \{ M\ast
k (v)\} k\in \BbbZ be the sequence of functions defined by the relations
M\ast
k (v) := \mathrm{s}\mathrm{u}\mathrm{p}\{ uv - Mk(u) : u \geq 0\} , k \in \BbbZ .
Consider the set \Gamma = \Gamma (\bfM \ast ) of sequences of positive numbers \gamma = \{ \gamma k\} k\in \BbbZ such that\sum
k\in \BbbZ
M\ast
k (\gamma k) \leq 1. The modular space (or Musielak – Orlicz space) B\scrS M is the space of all
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
DIRECT AND INVERSE APPROXIMATION THEOREMS . . . 703
functions f (f \in B-a.p.) such that the following quantity (which is also called the Orlicz norm of
f ) is finite:
\| f\|
M
:= \| \{ Ak\} k\in \BbbZ \| lM(\BbbZ ) := \mathrm{s}\mathrm{u}\mathrm{p}
\Biggl\{ \sum
k\in \BbbZ
\gamma k| Ak(f)| : \gamma \in \Gamma (\bfM \ast )
\Biggr\}
. (2.2)
By definition, B-a.p. functions are considered identical in B\scrS M if they have the same Fourier series.
The spaces B\scrS M defined in this way are Banach spaces. Functional spaces of this type have
been studied by mathematicians since the 1940s (see, for example, the monographs [21, 22, 25]). In
particular, the subspaces \scrS M of all 2\pi -periodic functions from B\scrS M were considered in [3, 5]. If
all the functions Mk are identical (namely, Mk(t) \equiv M(t), k \in \BbbZ ), the spaces \scrS M coincide with
the ordinary Orlicz type spaces \scrS M [15]. If Mk(t) = \mu kt
pk , pk \geq 1, \mu k \geq 0, then \scrS M coincide
with the weighted spaces \scrS p, \mu with variable exponents [2].
If all functions Mk(u) = up
\Bigl(
p - 1/pq - 1/p\prime
\Bigr) p
, p > 1, 1/p + 1/p\prime = 1, then B\scrS M are the
Besicovitch – Stepanets spaces B\scrS p [26] of functions f \in B-a.p. with the norm
\| f\|
M
= \| f\|
B\scrS p = \| \{ A\lambda k
(f)\} k\in \BbbN \| lp(\BbbN ) =
\Biggl( \sum
k\in \BbbN
| A\lambda k
(f)| p
\Biggr) 1/p
. (2.3)
The subspaces of all 2\pi -periodic Lebesgue summable functions from B\scrS p coincide with the
well-known spaces \scrS p (see, for example, [28], Ch. XI). For p = 2, the sets B\scrS p = B\scrS 2 coincide
with the sets of B2-a.p. functions and the spaces \scrS p with the ordinary Lebesgue spases of 2\pi -
periodic square-summable functions, i.e., \scrS 2 = L2.
By G\lambda n we denote the set of all B-a.p. functions whose Fourier exponents belong to the interval
( - \lambda n, \lambda n) and define the value of the best approximation of f \in B\scrS M by the equality
E\lambda n(f)M = E\lambda n(f)B\scrS M
= \mathrm{i}\mathrm{n}\mathrm{f}
g\in G\lambda n
\| f - g\| M. (2.4)
2.2. Generalized moduli of smoothness. Let \Phi be the set of all continuous bounded nonnegative
pair functions \varphi (t) such that \varphi (0) = 0 and the Lebesgue measure of the set \{ t \in \BbbR : \varphi (t) = 0\} is
equal to zero. For an arbitrary fixed \varphi \in \Phi , consider the generalized modulus of smoothness of a
function f \in B\scrS M
\omega \varphi (f, \delta )M := \mathrm{s}\mathrm{u}\mathrm{p}
| h| \leq \delta
\mathrm{s}\mathrm{u}\mathrm{p}
\Biggl\{ \sum
k\in \BbbZ
\gamma k\varphi (\lambda kh)| Ak(f)| : \gamma \in \Gamma
\Biggr\}
, \delta \geq 0. (2.5)
Consider the connection between the modulus (2.5) and some well-known moduli of smoothness.
Let \Theta = \{ \theta j\} mj=0 be a nonzero collection of complex numbers such that
\sum m
j=0
\theta j = 0. We associate
the collection \Theta with the difference operator \Delta \Theta
h (f) = \Delta \Theta
h (f, t) =
\sum m
j=0
\theta jf(t - jh) and the
modulus of smoothness
\omega \Theta (f, \delta )M := \mathrm{s}\mathrm{u}\mathrm{p}
| h| \leq \delta
\| \Delta \Theta
h (f)\| M .
Note that the collection \Theta (m) =
\Bigl\{
\theta j = ( - 1)j
\bigl(
m
j
\bigr)
, j = 0, 1, . . . ,m
\Bigr\}
, m \in \BbbN , corresponds to the
classical modulus of smoothness of order m, i.e.,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
704 S. O. CHAICHENKO, A. L. SHIDLICH, T. V. SHULYK
\omega \Theta (m)(f, \delta )M = \omega m(f, \delta )
M
.
For any k \in \BbbZ , the Fourier coefficients of the function \Delta \Theta
h (f) satisfy the equality
| Ak(\Delta
\Theta
h (f))| = | Ak(f)|
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
m\sum
j=0
\theta je
- i\lambda kjh
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| .
Therefore, taking into account (2.2), we see that, for \varphi \Theta (t) =
\bigm| \bigm| \bigm| \sum m
j=0
\theta je
- ijt
\bigm| \bigm| \bigm| , \omega \varphi \Theta (f, \delta )M =
= \omega \Theta (f, \delta )M . In particular, for \varphi m(t) = 2m| \mathrm{s}\mathrm{i}\mathrm{n}(t/2)| m = 2
m
2 (1 - \mathrm{c}\mathrm{o}\mathrm{s} t)
m
2 , m \in \BbbN , we have
\omega \varphi m(f, \delta )M = \omega m(f, \delta )
M
.
Further, let
Fh(f, t) = fh(x) :=
1
2h
t+h\int
t - h
f(u)du
be the Steklov function of a function f \in B\scrS M. Define the differences as follows:\widetilde \Delta 1
h(f) :=
\widetilde \Delta 1
h(f, t) = Fh(f, t) - f(t) = (Fh - \BbbI )(f, t),
\widetilde \Delta m
h (f) := \widetilde \Delta m
h (f, t) = \widetilde \Delta 1
h(\Delta
m - 1
h (f), t) = (Fh - \BbbI )m(f, t) =
m\sum
k=0
km - k
\biggl(
m
k
\biggr)
Fh,k(f, t),
where m = 2, 3, . . . , Fh,0(f) := f, Fh,k(f) := Fh(Fh,k(f)) and \BbbI is the identity operator in B\scrS M.
Consider the following smoothness characteristics
\widetilde \omega m(f, \delta ) := \mathrm{s}\mathrm{u}\mathrm{p}
0\leq h\leq \delta
\| \widetilde \Delta m
h (f)\|
M
, \delta > 0.
It can be shown [6] that \omega \~\varphi m(f, \delta )M = \widetilde \omega m(f, \delta )
M
for for \~\varphi m(t) = (1 - \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c} t)m, m \in \BbbN , where
\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c} t = \{ \mathrm{s}\mathrm{i}\mathrm{n} t/t, when t \not = 0, and 1,when t = 0\} .
In the general case, moduli similar to (2.5) were studied in [3 – 5, 8, 11, 19, 26, 32, 34].
3. Main results*. 3.1. Jackson-type inequalities. In this subsection, direct theorems are
established for functions f \in B\scrS M in terms of the best approximations and generalized moduli of
smoothness. In particular, for functions f \in B\scrS M with the Fourier series of the form (2.1), we prove
Jackson-type inequalities of the kind as
E\lambda n(f)M \leq K(\tau )\omega \varphi
\biggl(
f,
\tau
\lambda n
\biggr)
M
, \tau > 0, n \in \BbbN .
Let V (\tau ), \tau > 0, be a set of bounded nondecreasing functions v that differ from a constant on
[0, \tau ].
Theorem 3.1. Assume that the function f \in B\scrS M has the Fourier series of the form (2.1).
Then, for any \tau > 0, n \in \BbbN and \varphi \in \Phi , the following inequality holds:
E\lambda n(f)M \leq Kn,\varphi (\tau )\omega \varphi
\biggl(
f,
\tau
\lambda n
\biggr)
M
, (3.1)
* The results of this section was supported by the project “Innovative methods in the theory of differential equations,
computational mathematics and mathematical modeling” (project number 0122U000670).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
DIRECT AND INVERSE APPROXIMATION THEOREMS . . . 705
where
Kn,\varphi (\tau ) := \mathrm{i}\mathrm{n}\mathrm{f}
v\in V (\tau )
v(\tau ) - v(0)
In,\varphi (\tau , v)
(3.2)
and
In,\varphi (\tau , v) := \mathrm{i}\mathrm{n}\mathrm{f}
k\in \BbbN , k\geq n
\tau \int
0
\varphi
\biggl(
\lambda kt
\lambda n
\biggr)
dv(t). (3.3)
Furthermore, there exists a function v\ast \in V (\tau ) that realizes the greatest lower bound in (3.2).
In the spaces L2 of 2\pi -periodic square-summable functions, for moduli of continuity \omega m(f ; \delta )
and \~\omega m(f ; \delta ), such result was obtained by Babenko [7], and Abilov and Abilova [6], respectively. In
the spaces \scrS p of functions of one and several variables, this result for classical moduli of smoothness
was obtained in [27] and [1], respectively. In the Musielak – Orlicz spaces \scrS M, similar result was
obtained for generalized moduli of smoothness in [3].
In the Besicovitch – Stepanets spaces B\scrS p, a similar theorem was proved in [26]. It was noted
above that in the case when all functions Mk(u) = up
\Bigl(
p - 1/pq - 1/p\prime
\Bigr) p
, p > 1, 1/p+ 1/p\prime = 1, we
have B\scrS M = B\scrS p and \| f\|
M
= \| f\|
B\scrS p . In the case p = 1, the similar equalities B\scrS M = B\scrS 1
and \| f\|
M
= \| f\|
B\scrS 1 obviously can be obtained if all Mk(u) = u, k \in \BbbZ , and the set \Gamma is
a set of all sequences of positive numbers \gamma = \{ \gamma k\} k\in \BbbZ such that \| \gamma \| l\infty (\BbbZ ) = \mathrm{s}\mathrm{u}\mathrm{p}k\in \BbbZ \gamma k \leq 1.
Comparing estimate (3.1) with the corresponding result of Theorem 1 from [26], we see that in the
case when B\scrS M = B\scrS 1, the inequality (3.1) is unimprovable on the set of all functions f \in B\scrS 1,
\| f - A0(f)\| M \not = 0. Furthermore, Theorem 1 [26] implies the existence of the function v\ast \in V (\tau )
that realizes the greatest lower bound in (3.2).
Proof. In the proof of Theorem 3.1, we mainly use the ideas outlined in [7, 16, 17, 26, 27],
taking into account the peculiarities of the spaces B\scrS M. From (2.2) and (2.4), it follows that for any
f \in B\scrS M with the Fourier series of the form (2.1), we have
E\lambda n(f)M = \| f - Sn(f)\| M = \mathrm{s}\mathrm{u}\mathrm{p}
\Biggl\{ \sum
| k| \geq n
\gamma k| Ak(f)| : \gamma \in \Gamma
\Biggr\}
, (3.4)
where Sn(f) :=
\sum
| k| <n
Ak(f)e
i\lambda kx.
By the definition of supremum, for arbitrary \varepsilon > 0 there exists a sequence \~\gamma \in \Gamma , \~\gamma = \~\gamma (\varepsilon ),
such that the following relations holds:
\sum
| k| \geq n
\~\gamma k| Ak(f)| + \varepsilon \geq \mathrm{s}\mathrm{u}\mathrm{p}
\left\{ \sum
| k| \geq n
\gamma k| Ak(f)| : \gamma \in \Gamma
\right\} .
For arbitrary \varphi \in \Phi and h \in \BbbR , consider the sequence of numbers \{ \varphi (\lambda kh)Ak(f)\} k\in \BbbZ . If there
exists a function \Delta \varphi
h(f) \in B-a.p. such that, for all k \in \BbbZ ,
Ak(\Delta
\varphi
h(f)) = \varphi (\lambda kh)Ak(f), (3.5)
then here and below we denote by \| \Delta \varphi
h(f)\| M the Orlicz norm (2.2) of the function \Delta \varphi
h(f). If
such a B-a.p. function \Delta \varphi
h(f) does not exist, then to simplify notation we also use the notation
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
706 S. O. CHAICHENKO, A. L. SHIDLICH, T. V. SHULYK
\| \Delta \varphi
h(f)\| M, meaning by it the lM-norm of the sequence \{ \varphi (\lambda kh)Ak(f)\} k\in \BbbZ . In view of (2.2) and
(3.5), we obtain
\| \Delta \varphi
hf\| M \geq \mathrm{s}\mathrm{u}\mathrm{p}
\left\{ \sum
| k| \geq n
\gamma k\varphi (\lambda kh)| Ak(f)| : \gamma \in \Gamma
\right\} \geq
\sum
| k| \geq n
\~\gamma k\varphi (\lambda kh)| Ak(f)| =
=
In,\varphi (\tau , v)
v(\tau ) - v(0)
\sum
| k| \geq n
\~\gamma k| Ak(f)| +
\sum
| k| \geq n
\~\gamma k| Ak(f)|
\biggl(
\varphi (\lambda kh) -
In,\varphi (\tau , v)
v(\tau ) - v(0)
\biggr)
.
For any u \in [0, \tau ], we get
\| \Delta \varphi
u
\lambda n
f\|
M
\geq In,\varphi (\tau , v)
v(\tau ) - v(0)
\sum
| k| \geq n
\~\gamma k| Ak(f)| +
+
\sum
| k| \geq n
\~\gamma k| Ak(f)|
\biggl(
\varphi
\biggl(
\lambda ku
\lambda n
\biggr)
- In,\varphi (\tau , v)
v(\tau ) - v(0)
\biggr)
. (3.6)
The both sides of inequality (3.6) are nonnegative and, in view of the boundedness of the function
\varphi , the series on its right-hand side is majorized on the entire real axis by the absolutely convergent
series \scrK (\varphi )
\sum
| k| \geq n
\~\gamma k| Ak(f)| , where \scrK (\varphi ) := \mathrm{m}\mathrm{a}\mathrm{x}u\in \BbbR \varphi (u). Then integrating this inequality with
respect to dv(u) from 0 to \tau , we get
\tau \int
0
\| \Delta \varphi
u
\lambda n
f\|
M
dv(u) \geq In,\varphi (\tau , v)
\sum
| k| \geq n
\~\gamma k| Ak(f)| +
+
\sum
| k| \geq n
\~\gamma k| Ak(f)|
\left( \tau \int
0
\varphi
\biggl(
\lambda ku
\lambda n
\biggr)
dv(u) - In,\varphi (\tau , v)
\right) .
By virtue of the definition of In,\varphi (\tau , v), we see that the second term on the right-hand side of
the last relation is nonnegative. Therefore, for any function v \in V (\tau ), we have
\tau \int
0
\| \Delta \varphi
u
\lambda n
f\|
M
dv(u) \geq In,\varphi (\tau , v)
\sum
| k| \geq n
\~\gamma k| Ak(f)| \geq
\geq In,\varphi (\tau , v)
\left( \mathrm{s}\mathrm{u}\mathrm{p}
\left\{ \sum
| k| \geq n
\gamma k| Ak(f)| : \gamma \in \Gamma
\right\} - \varepsilon
\right) ,
wherefrom due to an arbitrariness of choice of the number \varepsilon , we conclude that
\tau \int
0
\| \Delta \varphi
u
\lambda n
f\|
M
dv(u) \geq In,\varphi (\tau , v)E\lambda n(f)M .
Hence,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
DIRECT AND INVERSE APPROXIMATION THEOREMS . . . 707
E\lambda n(f)M \leq 1
In,\varphi (\tau , v)
\tau \int
0
\| \Delta \varphi
u
\lambda n
f\|
M
dv(u) \leq 1
In,\varphi (\tau , v)
\tau \int
0
\omega \varphi
\biggl(
f,
u
\lambda n
\biggr)
M
dv(u), (3.7)
whence taking into account nondecreasing of the function \omega \varphi , we immediately obtain relation (3.1).
Theorem 3.1 is proved.
Now we consider some realisations of Theorem 3.1. Setting \varphi \alpha (t) = 2
\alpha
2 (1 - \mathrm{c}\mathrm{o}\mathrm{s} t)
\alpha
2 , \alpha > 0,
\omega \varphi \alpha (f, \delta )M =: \omega \alpha (f, \delta )M , \tau = \pi , and v(u) = 1 - \mathrm{c}\mathrm{o}\mathrm{s}u, u \in [0, \pi ], we get the following assertion.
Corollary 3.1. For arbitrary numbers n \in \BbbN and \alpha > 0, and for any function f \in B\scrS M with
the Fourier series of the form (2.1), the following inequalities hold:
E\lambda n(f)M \leq 1
2
\alpha
2 In(
\alpha
2 )
\pi \int
0
\omega \alpha
\biggl(
f,
u
\lambda n
\biggr)
M
\mathrm{s}\mathrm{i}\mathrm{n}u du, (3.8)
where
In
\Bigl( \alpha
2
\Bigr)
= \mathrm{i}\mathrm{n}\mathrm{f}
k\in \BbbN ,k\geq n
\pi \int
0
\biggl(
1 - \mathrm{c}\mathrm{o}\mathrm{s}
\lambda ku
\lambda n
\biggr) \alpha
2
\mathrm{s}\mathrm{i}\mathrm{n}u du. (3.9)
If, in addition,
\alpha
2
\in \BbbN , then
In
\Bigl( \alpha
2
\Bigr)
=
2
\alpha
2
+1
\alpha
2
+ 1
, (3.10)
and the inequality (3.8) cannot be improved for any n \in \BbbN .
Proof. Estimate (3.8) follows from (3.7). In [27] (relation (52)), it was shown that for any \theta \geq 1
and s \in \BbbN the following inequality holds:
\pi \int
0
(1 - \mathrm{c}\mathrm{o}\mathrm{s} \theta t)s \mathrm{s}\mathrm{i}\mathrm{n} tdt \geq 2s+1
s+ 1
,
which turns into equality for \theta = 1. Therefore, setting s =
\alpha
2
and \theta =
\lambda \nu
\lambda n
, \nu = n, n + 1, . . . , and
the monotonicity of the sequence of Fourier exponents \{ \lambda k\} k\in \BbbZ , we see that for
\alpha
2
\in \BbbN , indeed, the
equality (3.10) holds.
Let us prove that in this case, the constant
\alpha
2
+ 1
2\alpha +1
in inequality (3.8) is unimprovable for
\alpha
2
\in \BbbN .
It suffices to verify that the function
f\ast (x) = \gamma + \beta e - \lambda nx + \delta e\lambda nx, (3.11)
where \gamma , \beta and \delta are arbitrary complex numbers, satisfies the equality
E\lambda n(f
\ast )
M
=
\alpha
2
+ 1
2\alpha +1
\pi \int
0
\omega \alpha
\biggl(
f\ast ,
t
\lambda n
\biggr)
M
\mathrm{s}\mathrm{i}\mathrm{n} t dt, \alpha > 0. (3.12)
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708 S. O. CHAICHENKO, A. L. SHIDLICH, T. V. SHULYK
We have E\lambda n(f
\ast )
M
= | \beta | + | \delta | , the function \| \Delta \varphi \alpha
u/\lambda n
f\ast \|
M
= 2
\alpha
2 (| \beta | + | \delta | )(1 - \mathrm{c}\mathrm{o}\mathrm{s}u)
\alpha
2 does not
decrease with respect to u on [0, \pi ]. Therefore, \omega \alpha
\biggl(
f\ast ,
u
\lambda n
\biggr)
M
= \| \Delta \varphi \alpha
u/\lambda n
f\ast \|
M
and
2\alpha +1
\alpha
2
+ 1
E\lambda n(f
\ast )
M
-
\pi \int
0
\omega \alpha
\biggl(
f\ast ,
t
\lambda n
\biggr)
M
\mathrm{s}\mathrm{i}\mathrm{n} t dt =
= (| \beta | + | \delta | )
\left( 2\alpha +1
\alpha
2
+ 1
- 2
\alpha
2
\pi \int
0
(1 - \mathrm{c}\mathrm{o}\mathrm{s} t)
\alpha
2 \mathrm{s}\mathrm{i}\mathrm{n} t dt
\right) = 0.
Corollary 3.1 is proved.
It was shown in [27] that In(s) \geq 2 when s \geq 1 and In(s) \geq 1 + 2s - 1 when s \in (0, 1).
Combining these two estimates and (3.8), we obtain the following statement, which establishes a
Jackson-type inequality with a constant uniformly bounded in the parameter n \in \BbbN .
Corollary 3.2. Assume that the function f \in B\scrS M has the Fourier series of the form (2.1) and
\| f - A0(f)\| M \not = 0. Then, for any n \in \BbbN and \alpha > 0,
E\lambda n(f)M < c\alpha \omega \alpha
\biggl(
f,
\pi
\lambda n
\biggr)
M
, (3.13)
where c\alpha = 2 - \alpha /2 for \alpha \geq 2 and c\alpha = 4 \cdot 2 - \alpha /2/3 for 0 < \alpha < 2. Furthermore, in the case where
\alpha = m \in \BbbN , the following more accurate estimate holds:
E\lambda n(f)M <
4 - 2
\surd
2
2m/2
\omega m
\biggl(
f,
\pi
\lambda n
\biggr)
M
. (3.14)
Proof. Relation (3.14) follows from the estimate In
\Bigl( \alpha
2
\Bigr)
\geq 1 +
1\surd
2
, which is a consequence of
the above estimates for the value of In(s) in the case \alpha = m \in \BbbN [27].
If the weight function v2(t) = t, then we obtain the following assertion.
Corollary 3.3. Assume that the function f \in B\scrS M has the Fourier series of the form (2.1) and
\alpha \geq 1. Then, for any 0 < \tau \leq 3\pi
4
and n \in \BbbN ,
E\lambda n(f)M \leq 1
2\alpha
\int \tau
0
\mathrm{s}\mathrm{i}\mathrm{n}\alpha
t
2
dt
\tau \int
0
\omega \alpha
\biggl(
f,
t
\lambda n
\biggr)
M
dt. (3.15)
Relation (3.15) becomes equality for the function f\ast of the form (3.11).
Inequalities (3.8) and (3.15) can be considered as an extension of the corresponding results of
Serdyuk and Shidlich [26] to the Besicovitch – Musielak spaces B\scrS M, and they coincide with them
in the case B\scrS M = B\scrS 1. In the spaces \scrS p of functions of one and several variables, analogues of
Theorem 3.1 and Corollaries 3.1 and 3.3 were proved in [27] and [1], respectively. The inequalities
of this type were also investigated in [8, 17, 27, 32, 34].
Proof. From inequality (3.7), it follows that
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DIRECT AND INVERSE APPROXIMATION THEOREMS . . . 709
E\lambda n(f)M \leq 1
2
\alpha
2 I\ast n
\Bigl( \alpha
2
\Bigr) \tau \int
0
\omega \alpha
\biggl(
f,
t
\lambda n
\biggr)
dt,
where
I\ast n
\Bigl( \alpha
2
\Bigr)
:= \mathrm{i}\mathrm{n}\mathrm{f}
k\in \BbbN , k\geq n
\tau \int
0
\biggl(
1 - \mathrm{c}\mathrm{o}\mathrm{s}
\lambda kt
\lambda n
\biggr) \alpha
2
dt, \alpha > 0, n \in \BbbN .
In [35], it is shown that for the function F\alpha (x) :=
1
x
\int x
0
| \mathrm{s}\mathrm{i}\mathrm{n} t| \alpha dt, any h \in
\biggl(
0,
3\pi
4
\biggr)
and
\alpha \geq 1, the following relation is true:
\mathrm{i}\mathrm{n}\mathrm{f}
x\geq h/2
F\alpha (x) = F\alpha (h/2). (3.16)
Since for h =
\lambda k
\lambda n
\geq 1 (k \geq n)
\tau \int
0
\biggl(
1 - \mathrm{c}\mathrm{o}\mathrm{s}
\lambda kt
\lambda n
\biggr) \alpha
2
dt = 2
\alpha
2
\tau \int
0
\bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n} \lambda kt
2\lambda n
\bigm| \bigm| \bigm| \alpha dt = 2
\alpha
2 \tau F\alpha
\biggl(
\lambda k\tau
2\lambda n
\biggr)
,
from (3.16)
\Biggl(
with \tau \in
\biggl(
0,
3\pi
4
\biggr]
and \alpha \geq 1
\Biggr)
we obtain
I\ast n
\Bigl( \alpha
2
\Bigr)
= \mathrm{i}\mathrm{n}\mathrm{f}
k\in \BbbN :k\geq n
\tau \int
0
\biggl(
1 - \mathrm{c}\mathrm{o}\mathrm{s}
\lambda kt
\lambda n
\biggr) \alpha
2
dt = \mathrm{i}\mathrm{n}\mathrm{f}
k\in \BbbN :k\geq n
2
\alpha
2
\tau \int
0
\bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n} \lambda kt
2\lambda n
\bigm| \bigm| \bigm| \alpha dt =2
\alpha
2
\tau \int
0
\mathrm{s}\mathrm{i}\mathrm{n}\alpha
t
2
dt.
For the functions f\ast of the form (3.11), the equality
E\lambda n(f
\ast )
M
=
1
2\alpha
\int \tau
0
\mathrm{s}\mathrm{i}\mathrm{n}\alpha
t
2
dt
\tau \int
0
\omega \alpha
\biggl(
f\ast ,
t
\lambda n
\biggr)
M
dt
is verified similarly to the proof of equality (3.12).
Corollary 3.2 is proved.
In the case \varphi (t) = \~\varphi m(t) = (1 - \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c} t)m, m \in \BbbN , where, by definition, \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c} t = \{ \mathrm{s}\mathrm{i}\mathrm{n} t/t, if
t \not = 0, and 1, if t = 0\} , for \tau = \pi and v(u) = 1 - \mathrm{c}\mathrm{o}\mathrm{s}u, u \in [0;\pi ], from relation (3.7) we get
E\lambda n(f)M \leq 1
\~In(m)
\pi \int
0
\~\omega m
\biggl(
f,
u
\lambda n
\biggr)
M
\mathrm{s}\mathrm{i}\mathrm{n}u du,
where
\~In(m) = \mathrm{i}\mathrm{n}\mathrm{f}
k\in \BbbN ,k\geq n
\pi \int
0
\biggl(
1 - \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}
\lambda ku
\lambda n
\biggr) m
\mathrm{s}\mathrm{i}\mathrm{n}u du.
Taking into account the estimation [33]
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710 S. O. CHAICHENKO, A. L. SHIDLICH, T. V. SHULYK
1 - \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}
\biggl(
\lambda ku
\lambda n
\biggr)
\geq 1 - \mathrm{s}\mathrm{i}\mathrm{n}u
u
\geq
\Bigl( u
\pi
\Bigr) 2
, k \geq n, u \in [0;\pi ],
we have
\~In(m) \geq
\pi \int
0
(1 - \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c} u)m \mathrm{s}\mathrm{i}\mathrm{n}u du \geq 1
\pi 2m
\pi \int
0
u2m \mathrm{s}\mathrm{i}\mathrm{n}u du =
=
2m!
\pi 2m
\left( m\sum
j=0
( - 1)j
\pi 2m - 2j
(2m - 2j)!
+
\pi 2m
2m!
( - 1)m
\right) :=
2m!
\pi 2m
K(m).
Thereby, the following corollary follows from Theorem 3.1.
Corollary 3.4. For arbitrary numbers n \in \BbbN and m > 0, and for any function f \in B\scrS M, with
the Fourier series of the form (2.1) the following inequalities hold:
E\lambda n(f)M \leq \pi 2m
2m! \cdot K(m)
\pi \int
0
\~\omega m
\biggl(
f,
u
\lambda n
\biggr)
M
\mathrm{s}\mathrm{i}\mathrm{n}u du,
where
K(m) =
m\sum
j=0
( - 1)j
\pi 2m - 2j
(2m - 2j)!
+
\pi 2m
2m!
( - 1)m.
In the case m = 1, we have 2K(1) = \pi 2 - 4 and
E\lambda n(f)M \leq \pi 2
\pi 2 - 4
\pi \int
0
\~\omega 1
\biggl(
f,
u
\lambda n
\biggr)
M
\mathrm{s}\mathrm{i}\mathrm{n}u du \leq \pi 2\lambda n
\pi 2 - 4
\pi
\lambda n\int
0
\~\omega 1 (f, u)
M
\mathrm{s}\mathrm{i}\mathrm{n}\lambda nu du.
If the weight function v2(t) = um+1, then we obtain the following assertion.
Corollary 3.5. Assume that the function f \in B\scrS M has the Fourier series of the form (2.1) and
m \geq 1. Then, for any 0 < \tau \leq \pi and n \in \BbbN ,
E\lambda n(f)M \leq \pi m - 1
\biggl(
2\lambda n
\pi 2 - 4
\biggr) m
\lambda n
\tau /\lambda n\int
0
\~\omega m(f, t)
M
tm dt. (3.17)
Ideed, applying Holder’s inequality, we find
\pi \int
0
\biggl(
1 - \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}
\lambda ku
\lambda n
\biggr) m
dum+1 \geq (m+ 1)
\pi \int
0
\biggl(
1 - \mathrm{s}\mathrm{i}\mathrm{n}u
u
\biggr) m
um du =
= (m+ 1)
\pi \int
0
(u - \mathrm{s}\mathrm{i}\mathrm{n}u)m du \geq m+ 1
\pi m - 1
\left( \pi \int
0
(u - \mathrm{s}\mathrm{i}\mathrm{n}u) du
\right) m
=
m+ 1
\pi m - 1
\biggl(
\pi 2 - 4
2
\biggr) m
.
In the spaces L2 of 2\pi -periodic square-summable functions, for moduli of smoothness \~\omega m(f ; \delta ),
the results of this kind were obtained by Abilov and Abilova [6], and Vakarchuk [32]. Note that in
the case f \in B\scrS M = L2 the inequality (3.17) follows from the result of [6] (see Theorem 1). For
m = 1 and f \in L2, the statements of Corollary 3.5 and Theorem 1 from [6] are identical, and the
constant in the right-hand side of (3.17) cannot be reduced for every fixed n.
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DIRECT AND INVERSE APPROXIMATION THEOREMS . . . 711
4. Inverse approximation theorem.
Theorem 4.1. Assume that f \in B\scrS M has the Fourier series of the form (2.1), the function
\varphi \in \Phi is nondecreasing on the interval [0, \tau ], \tau > 0, and \varphi (\tau ) = \mathrm{m}\mathrm{a}\mathrm{x}\{ \varphi (t) : t \in \BbbR \} . Then, for any
n \in \BbbN , the following inequality holds:
\omega \varphi
\biggl(
f,
\tau
\lambda n
\biggr)
M
\leq
n\sum
\nu =1
\biggl(
\varphi
\biggl(
\tau \lambda \nu
\lambda n
\biggr)
- \varphi
\biggl(
\tau \lambda \nu - 1
\lambda n
\biggr) \biggr)
E\lambda \nu (f)M . (4.1)
Proof. Let us use the proof scheme from [27] and [3], modifying it taking into account the
peculiarities of the spaces B\scrS M and the definition of the modulus \omega \varphi .
Let f \in B\scrS M. For any \varepsilon > 0 there exists a number N0 = N0(\varepsilon ) \in \BbbN , N0 > n, such that, for
any N > N0, we have
E\lambda N
(f)
M
= \| f - SN - 1(f)\| M < \varepsilon /\varphi (\tau ).
Let us set f0 := SN0(f). Then in view of (3.5), we see that
\| \Delta \varphi
h(f)\| M \leq \| \Delta \varphi
h(f0)\| M + \| \Delta \varphi
h(f - f0)\| M \leq
\leq \| \Delta \varphi
h(f0)\| M + \varphi (\tau )E\lambda N0+1
(f)
M
< \| \Delta \varphi
h(f0)\| M + \varepsilon . (4.2)
Further, let Sn - 1 := Sn - 1(f0) be the Fourier sum of f0. Then by virtue of (3.5), for | h| \leq \tau /\lambda n,
we have
\| \Delta \varphi
h(f0)\| M = \| \Delta \varphi
h(f0 - Sn - 1) + \Delta \varphi
hSn - 1\| M \leq
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \varphi (\tau )(f0 - Sn - 1) +
+
\sum
| k| \leq n - 1
\varphi (\lambda kh)| Ak(f)|
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
M
\leq
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \varphi (\tau )
N0\sum
\nu =n
H\nu +
n - 1\sum
\nu =1
\varphi
\biggl(
\tau \lambda \nu
\lambda n
\biggr)
H\nu
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
M
, (4.3)
where H\nu (x) := | A\nu (f)| + | A - \nu (f)| , \nu = 1, 2, . . . .
Now we use the following assertion from [27].
Lemma 4.1 [27]. Let \{ c\nu \} \infty \nu =1 and \{ a\nu \} \infty \nu =1 be arbitrary numerical sequences. Then the fol-
lowing equality holds for all natural N1, N2 and N N1 \leq N2 < N :
N2\sum
\nu =N1
a\nu c\nu = aN1
N\sum
\nu =N1
c\nu +
N2\sum
\nu =N1+1
(a\nu - a\nu - 1)
N\sum
i=\nu
ci - aN2
N\sum
\nu =N2+1
c\nu . (4.4)
Setting a\nu = \varphi
\biggl(
\tau \lambda \nu
\lambda n
\biggr)
, c\nu = H\nu (x), N1 = 1, N2 = n - 1 and N = N0 in (4.4), we get
n - 1\sum
\nu =1
\varphi
\biggl(
\tau \lambda \nu
\lambda n
\biggr)
H\nu (x) = \varphi
\biggl(
\tau \lambda 1
\lambda n
\biggr) N0\sum
\nu =1
H\nu (x) +
+
n - 1\sum
\nu =2
\biggl(
\varphi
\biggl(
\tau \lambda \nu
\lambda n
\biggr)
- \varphi
\biggl(
\tau \lambda \nu - 1
\lambda n
\biggr) \biggr) N0\sum
i=\nu
Hi(x) - \varphi
\biggl(
\tau \lambda \nu - 1
\lambda n
\biggr) N0\sum
\nu =n
H\nu (x).
Therefore,
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712 S. O. CHAICHENKO, A. L. SHIDLICH, T. V. SHULYK\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \varphi (\tau )
N0\sum
\nu =n
H\nu +
n - 1\sum
\nu =1
\varphi
\biggl(
\tau \lambda \nu
\lambda n
\biggr)
H\nu
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
M
\leq
\leq
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \varphi (\tau )
N0\sum
\nu =n
H\nu +
n - 1\sum
\nu =1
\biggl(
\varphi
\biggl(
\tau \lambda \nu
\lambda n
\biggr)
- \varphi
\biggl(
\tau \lambda \nu - 1
\lambda n
\biggr) \biggr) N0\sum
i=\nu
Hi - \varphi
\biggl(
\tau \lambda \nu - 1
\lambda n
\biggr) N0\sum
\nu =n
H\nu
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
M
\leq
\leq
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
n\sum
\nu =1
\biggl(
\varphi
\biggl(
\tau \lambda \nu
\lambda n
\biggr)
- \varphi
\biggl(
\tau \lambda \nu - 1
\lambda n
\biggr) \biggr) N0\sum
i=\nu
Hi
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
M
\leq
\leq
n\sum
\nu =1
\biggl(
\varphi
\biggl(
\tau \lambda \nu
\lambda n
\biggr)
- \varphi
\biggl(
\tau \lambda \nu - 1
\lambda n
\biggr) \biggr)
E\lambda \nu (f0)M . (4.5)
Combining relations (4.2), (4.3) and (4.5) and taking into account the definition of the function
f0, we see that, for | h| \leq \tau /\lambda n, the following inequality holds:
\| \Delta \varphi
h(f)\| M \leq
n\sum
\nu =1
\biggl(
\varphi
\biggl(
\tau \lambda \nu
\lambda n
\biggr)
- \varphi
\biggl(
\tau \lambda \nu - 1
\lambda n
\biggr) \biggr)
E\lambda \nu (f)M + \varepsilon
which, in view of arbitrariness of \varepsilon , gives us (4.1).
Theorem 4.1 is proved.
Consider an important special case when \varphi (t) = \varphi \alpha (t) = 2
\alpha
2 (1 - \mathrm{c}\mathrm{o}\mathrm{s} t)
\alpha
2 = 2\alpha | \mathrm{s}\mathrm{i}\mathrm{n}(t/2)| \alpha ,
\alpha > 0. In this case, the function \varphi satisfies the conditions of Theorem 4.1 with \tau = \pi . Then for
\alpha \geq 1 using the inequality x\alpha - y\alpha \leq \alpha x\alpha - 1(x - y), x > 0, y > 0 (see, for example, [18], Ch. 1),
and the usual trigonometric formulas, for \nu = 1, 2, . . . , n, we have
\varphi
\biggl(
\tau \lambda \nu
\lambda n
\biggr)
- \varphi
\biggl(
\tau \lambda \nu - 1
\lambda n
\biggr)
= 2\alpha
\biggl( \bigm| \bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n} \pi \lambda \nu
\lambda n
\bigm| \bigm| \bigm| \bigm| \alpha -
\bigm| \bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n} \pi \lambda \nu - 1
\lambda n
\bigm| \bigm| \bigm| \bigm| \alpha \biggr) \leq
\leq 2\alpha \alpha
\bigm| \bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n} \pi \lambda \nu
\lambda n
\bigm| \bigm| \bigm| \bigm| \alpha - 1 \bigm| \bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n} \pi \lambda \nu
\lambda n
- \mathrm{s}\mathrm{i}\mathrm{n}
\pi \lambda \nu - 1
\lambda n
\bigm| \bigm| \bigm| \bigm| \leq \alpha
\biggl(
2\pi
\lambda n
\biggr) \alpha
\lambda \alpha - 1
\nu (\lambda \nu - \lambda \nu - 1).
If 0 < \alpha < 1, then the similar estimate can be obtained using the inequality x\alpha - y\alpha \leq \alpha y\alpha - 1(x - y),
which holds for any x > 0, y > 0 [18] (Ch. 1). Hence, for any f \in B\scrS M, we get the following
estimate:
\omega \alpha
\biggl(
f,
\pi
\lambda n
\biggr)
M
\leq \alpha
\biggl(
2\pi
\lambda n
\biggr) \alpha n\sum
\nu =1
\lambda \alpha - 1
\nu (\lambda \nu - \lambda \nu - 1)E\lambda \nu (f)M \alpha > 0. (4.6)
It should be noted that the constant in this estimate can be improved as follows.
Theorem 4.2. Assume that f \in B\scrS M has the Fourier series of the form (2.1). Then, for any
n \in \BbbN and \alpha > 0,
\omega \alpha
\biggl(
f,
\tau
\lambda n
\biggr)
M
\leq
\biggl(
\pi
\lambda n
\biggr) \alpha n\sum
\nu =1
(\lambda \alpha
\nu - \lambda \alpha
\nu - 1)E\lambda \nu (f)M . (4.7)
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DIRECT AND INVERSE APPROXIMATION THEOREMS . . . 713
Proof. We prove this theorem similarly to the proof of Theorem 4.1. For any \varepsilon > 0, denote by
N0 = N0(\varepsilon ) \in \BbbN , N0 > n, a number such that, for any N > N0,
E\lambda N
(f)
M
= \| f - SN - 1(f)\| M < \varepsilon .
Let us set f0 := SN0(f), Sn - 1 := Sn - 1(f0) and \| \Delta \alpha
h(f)\| M := \| \Delta \varphi \alpha
h (f)\|
M
, and use relations
(4.2) and (4.3). We obtain
\| \Delta \alpha
h(f)\| M < \| \Delta \alpha
h(f0)\| M + \varepsilon (4.8)
and
\| \Delta \alpha
h(f0)\| M \leq
\bigm\| \bigm\| \bigm\| \bigm\| 2\alpha N0\sum
\nu =n
H\nu + 2\alpha
n - 1\sum
\nu =1
\bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n} \pi \lambda \nu
2\lambda n
\bigm| \bigm| \bigm| \alpha H\nu
\bigm\| \bigm\| \bigm\| \bigm\|
M
\leq
\leq
\biggl(
\pi
\lambda n
\biggr) \alpha \bigm\| \bigm\| \bigm\| \bigm\| \lambda \alpha
n
N0\sum
\nu =n
H\nu +
n - 1\sum
\nu =1
\lambda \alpha
\nu H\nu
\bigm\| \bigm\| \bigm\| \bigm\|
M
, (4.9)
where | h| \leq \pi /\lambda n and H\nu (x) = | A\nu (f)| + | A - \nu (f)| , \nu = 1, 2, . . . .
By virtue of (4.4), for a\nu = \lambda \alpha
\nu , c\nu = H\nu (x), N1 = 1, N2 = n - 1 and N = N0,
n - 1\sum
\nu =1
\lambda \alpha
\nu H\nu (x) = \lambda \alpha
1
N0\sum
\nu =1
H\nu (x) +
n - 1\sum
\nu =2
\bigl(
\lambda \alpha
\nu - \lambda \alpha
\nu - 1
\bigr) N0\sum
i=\nu
Hi(x) - \lambda \alpha
\nu - 1
N0\sum
\nu =n
H\nu (x).
Therefore, \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \lambda \alpha
n
N0\sum
\nu =n
H\nu +
n - 1\sum
\nu =1
\lambda \alpha
\nu H\nu
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
M
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
n\sum
\nu =1
\bigl(
\lambda \alpha
\nu - \lambda \alpha
\nu - 1
\bigr) N0\sum
i=\nu
Hi
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
M
\leq
\leq
n\sum
\nu =1
\bigl(
\lambda \alpha
\nu - \lambda \alpha
\nu - 1
\bigr)
E\lambda \nu (f0)M . (4.10)
Combining relations (4.8), (4.9) and (4.10) and taking into account the definition of the function
f0, we see that, for | h| \leq \tau /\lambda n, the following inequality holds:
\| \Delta \alpha
h(f)\| M \leq
\biggl(
\pi
\lambda n
\biggr) \alpha n\sum
\nu =1
\bigl(
\lambda \alpha
\nu - \lambda \alpha
\nu - 1
\bigr)
E\lambda \nu (f)M + \varepsilon
which, in view of arbitrariness of \varepsilon , gives us (4.7).
Theorem 4.2 is proved.
In (4.1), the constant \pi \alpha is exact in the sense that for any \varepsilon > 0, there exists a function
f\ast \in B\scrS M such that, for all n greater than a certain number n0, we have
\omega \alpha
\biggl(
f\ast ,
\pi
\lambda n
\biggr)
M
>
\pi \alpha - \varepsilon
\lambda \alpha
n
n\sum
\nu =1
\bigl(
\lambda \alpha
\nu - \lambda \alpha
\nu - 1
\bigr)
E\lambda \nu (f
\ast )
M
. (4.11)
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714 S. O. CHAICHENKO, A. L. SHIDLICH, T. V. SHULYK
Consider the function f\ast (x) = ei\lambda k0
x, where k0 is an arbitrary positive integer. Then E\lambda \nu (f
\ast )
M
= 1
for \nu = 1, 2, . . . , k0, E\lambda \nu (f
\ast )
M
= 0 for \nu > k0 and
\omega \alpha
\biggl(
f\ast ,
\pi
\lambda n
\biggr)
M
\geq \| \Delta \alpha
\pi
\lambda n
f\ast \|
M
\geq 2\alpha
\bigm| \bigm| \bigm| \mathrm{s}\mathrm{i}\mathrm{n} \lambda k0\pi
2\lambda n
\bigm| \bigm| \bigm| \alpha .
Since \mathrm{s}\mathrm{i}\mathrm{n} t/t tends to 1 as t \rightarrow 0, then, for all n greater than a certain number n0, the inequality
2\alpha | \mathrm{s}\mathrm{i}\mathrm{n}\lambda k0\pi /(2\lambda n)| \alpha > (\pi \alpha - \varepsilon )\lambda \alpha
k0
/\lambda n
\alpha holds, which yields (4.11).
Corollary 4.1. Suppose that f \in B\scrS M has the Fourier series of the form (2.1). Then, for any
n \in \BbbN and \alpha > 0,
\omega \alpha
\biggl(
f,
\pi
\lambda n
\biggr)
M
\leq \alpha
\biggl(
\pi
\lambda n
\biggr) \alpha n\sum
\nu =1
\lambda \alpha - 1
\nu (\lambda \nu - \lambda \nu - 1)E\lambda \nu (f)M . (4.12)
If, in addition, the Fourier exponents \lambda \nu , \nu \in \BbbN , satisfy the condition
\lambda \nu +1 - \lambda \nu \leq C, \nu = 1, 2, . . . , (4.13)
with an absolute constant C > 0, then
\omega \alpha
\biggl(
f,
\pi
\lambda n
\biggr)
M
\leq C\alpha
\biggl(
\pi
\lambda n
\biggr) \alpha n\sum
\nu =1
\lambda \alpha - 1
\nu E\lambda \nu (f)M . (4.14)
5. Constructive characteristics of the classes of functions defined by the generalized moduli
of smoothness. Let \omega be the function (majorant) given on [0, 1]. For a fixed \alpha > 0, we set
B\scrS MH\omega
\alpha =
\bigl\{
f \in B\scrS M : \omega \alpha (f, \delta )M = \scrO (\omega (\delta )), \delta \rightarrow 0+
\bigr\}
. (5.1)
Further, we consider the majorants \omega (\delta ), \delta \in [0, 1], which satisfy the following conditions: 1) \omega (\delta )
is continuous on [0, 1]; 2) \omega (\delta ) \uparrow ; 3) \omega (\delta ) \not = 0 for \delta \in (0, 1]; 4) \omega (\delta ) \rightarrow 0 for \delta \rightarrow 0; as well as the
condition
n\sum
v=1
\lambda s - 1
v \omega
\biggl(
1
\lambda v
\biggr)
= \scrO
\Bigl[
\lambda s
n\omega
\biggl(
1
\lambda n
\biggr) \Bigr]
, (5.2)
where s > 0, and \lambda \nu , \nu \in \BbbN , is a increasing sequence of positive numbers. In the case where
\lambda \nu = \nu , the condition (5.2) is the known Bari condition (Bs) (see, e.g., [9]).
Theorem 5.1. Assume that the function f \in B\scrS M has the Fourier series of the form (2.1),
\alpha > 0 and the majorant \omega satisfies the conditions 1 – 4.
1. If f \in B\scrS MH\omega
\alpha , then the following relation is true:
E\lambda n(f)M = \scrO
\biggl[
\omega
\biggl(
1
\lambda n
\biggr) \biggr]
. (5.3)
2. If the numbers \lambda \nu , \nu \in \BbbN satisfy condition (4.13) and the function \omega satisfies condition (5.2)
with s = \alpha , then relation (5.3) yields the inclusion f \in B\scrS MH\omega
\alpha .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
DIRECT AND INVERSE APPROXIMATION THEOREMS . . . 715
Proof. Let f \in B\scrS MH\omega
\alpha . Then relation (5.3) follows from (5.1) and (3.13).
On the other hand, if f \in B\scrS M, the numbers \lambda \nu , \nu \in \BbbN satisfy condition (4.13) and the function
\omega satisfies condition (5.2) with s = \alpha , and relation (5.3) holds, then, by (4.14), we get
\omega \alpha
\biggl(
f,
1
\lambda n
\biggr)
M
\leq C1
\lambda \alpha
n
n\sum
\nu =1
\lambda \alpha - 1
\nu E\lambda \nu (f) \leq
C1
\lambda \alpha
n
n\sum
\nu =1
\lambda \alpha - 1
\nu \omega
\biggl(
1
\lambda \nu
\biggr)
= \scrO
\Bigl[
\omega
\biggl(
1
\lambda n
\biggr) \Bigr]
,
where C1 = \alpha (2\pi )\alpha \cdot C. Hence, the function f belongs to the set B\scrS MH\omega
\alpha .
Theorem 5.1 is proved.
The function tr, 0 < r \leq \alpha , satisfies condition (5.2) with s = \alpha . Hence, denoting by B\scrS MHr
\alpha
the class B\scrS MH\omega
\alpha for \omega (t) = tr we establish the following statement.
Corollary 5.1. Let f \in B\scrS M have the Fourier series of the form (2.1), \alpha > 0, 0 < r \leq \alpha and
condition (4.13) holds. The function f belongs to the set B\scrS MHr
\alpha , iff the following relation is true:
E\lambda n(f)M = \scrO (\lambda - r
n ).
In the spaces \scrS p, for classical moduli of smoothness \omega m, Theorems 4.1 and 5.1 were proved in
[27] and [1]. In the spaces \scrS p, inequalities of the form (4.14) were also obtained in [29]. In spaces Lp
of 2\pi -periodic Lebesgue summable with the pth degree functions, inequalities of the kind as (4.14)
were obtained by M. Timan (see, for example, [30], Ch. 6, [31], Ch. 2). In the Musielak – Orlicz type
spaces, inequalities of the kind as (4.1) were proved in [3].
References
1. F. G. Abdullayev, P. Özkartepe, V. V. Savchuk, A. L. Shidlich, Exact constants in direct and inverse approximation
theorems for functions of several variables in the spaces \scrS p , Filomat, 33, № 5, 1471 – 1484 (2019).
2. F. Abdullayev, S. Chaichenko, M. Imash kyzy, A. Shidlich, Direct and inverse approximation theorems in the weighted
Orlicz-type spaces with a variable exponent, Turkish J. Math., 44, 284 – 299 (2020).
3. F. Abdullayev, S. Chaichenko, A. Shidlich, Direct and inverse approximation theorems of functions in the Musielak –
Orlicz type spaces, Math. Inequal. Appl., 24, № 2, 323 – 336 (2021).
4. F. Abdullayev, A. Serdyuk, A. Shidlich, Widths of functional classes defined by majorants of generalized moduli of
smoothness in the spaces \scrS p , Ukr. Math. J., 73, № 6, 841 – 858 (2021).
5. F. Abdullayev, S. Chaichenko, M. Imashkyzy, A. Shidlich, Jackson-type inequalities and widths of functional classes
in the Musielak – Orlicz type spaces, Rocky Mountain J. Math., 51, № 4, 1143 – 1155 (2021).
6. V. A. Abilov, F. V. Abilova, Problems in the approximation of 2\pi -periodic functions by Fourier sums in the space
L2(2\pi ), Math. Notes, 76, № 6, 749 – 757 (2004).
7. A. G. Babenko, On exact constant in the Jackson inequality in L2 , Math. Notes, 39, № 5, 355 – 363 (1986).
8. V. F. Babenko, S. V. Savela, Jackson – Stechkin-type inequalities for almost periodic functions, Visn. Dnipropetrovsk
Univ., 20, № 6/1, 60 – 66 (2012).
9. N. K. Bari, S. B. Stechkin, Best approximations and differential properties of two conjugate functions, Trudy Mosk.
Mat. Obshch., 5, 483 – 522 (1956) (in Russian).
10. A. S. Besicovitch, Almost periodic functions, Dover Publ., Inc., New York (1955).
11. J. Boman, Equivalence of generalized moduli of continuity, Ark. Mat., 18, 73 – 100 (1980).
12. E. A. Bredikhina, Absolute convergence of Fourier series of almost periodic functions, Dokl. Akad. Nauk SSSR,
179, № 5, 1023 – 1026 (1968) (in Russian).
13. E. A. Bredikhina, Almost periodic functions, Encyclopedia Math., 4, 543 – 545 (1984) (in Russian).
14. P. Butzer, R. Nessel, Fourier analysis and approximation, One-Dimensional Theory, Birkhäuser, Basel (1971).
15. S. Chaichenko, A. Shidlich, F. Abdullayev, Direct and inverse approximation theorems of functions in the Orlicz type
spaces \scrS M , Math. Slovaca, 69, № 6, 1367 – 1380 (2019).
16. N. I. Chernykh, On the Jackson inequality in L2 , Proc. Steklov Inst. Math., 88, 75 – 78 (1967) (in Russian).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
716 S. O. CHAICHENKO, A. L. SHIDLICH, T. V. SHULYK
17. N. I. Chernykh, On the best approximation of periodic functions by trigonometric polynomials in L2 , Mat. Zametki,
20, № 3, 513 – 522 (1967) (in Russian).
18. G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge Univ. Press (1934).
19. A. I. Kozko, A. V. Rozhdestvenskii, On Jackson’s inequality for a generalized modulus of continuity in L2 , Sb.
Math., 195, № 8, 1073 – 1115 (2004).
20. B. M. Levitan, Almost periodic functions, Gosudarstv., Izdat. Techn.-Teor. Lit., Moscow (1953) (in Russian).
21. J. Lindenstrauss, L. Tzafriri, Classical Banach spaces I: Sequence spaces, Berlin (1977).
22. J. Musielak, Orlicz spaces and modular spaces, Springer, Berlin (1983).
23. Ya. G. Pritula, Jackson’s inequality for B2 -almost periodic functions, Izv. Vyssh. Uchebn. Zaved. Mat., 8, 90 – 93
(1972) (in Russian).
24. Ya. G. Pritula, M. M. Yatsymirskyi, Estimates of approximations of B2 -almost periodic functions, Visn. L’viv. Univ.
Ser. Mekh.-Mat., 21, 3 – 7 (1983) (in Ukrainian).
25. M. M. Rao, Z. D. Ren, Applications of Orlicz spaces, Marcel Dekker Inc., New York, Basel (2002).
26. A. S. Serdyuk, A. L. Shidlich, Direct and inverse theorems on the approximation of almost periodic functions in
Besicovitch – Stepanets spaces, Carpathian Math. Publ., 13, № 3, 687 – 700 (2021) (see also arXiv preprint, arXiv:
2105.06796).
27. A. I. Stepanets, A. S. Serdyuk, Direct and inverse theorems in the theory of the approximation of functions in the
space \scrS p , Ukr. Math. J., 54, № 1, 126 – 148 (2002).
28. A. I. Stepanets, Methods of approximation theory, VSP, Leiden, Boston (2005).
29. M. D. Sterlin, Exact constants in inverse theorems of approximation theory, Sov. Math. Dokl., 13, 160 – 163 (1972).
30. A. F. Timan, Theory of approximation of functions of a real variable, Fizmatgiz, Moscow (1960) (in Russian).
31. M. F. Timan, Approximation and properties of periodic functions, Nauk. dumka, Kiev (2009) (in Russian).
32. S. B. Vakarchuk, Jackson-type inequalities with generalized modulus of continuity and exact values of the n-widths
for the classes of (\psi , \beta )-differentiable functions in L2. I, Ukr. Math. J., 68, № 6, 823 – 848 (2016).
33. S. B. Vakarchuk, V. I. Zabutnaya, Inequalities between best polynomial approximations and some smoothness
characteristics in the space L2 and widths of classes of functions, Math. Notes, 99, № 2, 222 – 242 (2016)
34. S. N. Vasil’ev, The Jackson – Stechkin inequality in L2[ - \pi , \pi ], Proc. Steklov Inst. Math., Suppl., 1, S243-S253
(2001) (in Russian).
35. V. R. Voitsekhivs’kyj, Jackson type inequalities in approximation of functions from the space \scrS p , Approx. Theory
and Relat. Top., Proc. Inst. Math. Nat. Acad. Sci. Ukraine, 35, 33 – 46 (2002) (in Ukrainian).
Received 14.12.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 5
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| id | umjimathkievua-article-7045 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:31:13Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b8/213cbef1b828f1fbb58b98fd491c61b8.pdf |
| spelling | umjimathkievua-article-70452022-07-13T07:53:28Z Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions Chaichenko , S. O. Shidlich, A. L. Shulyk, T. V. Chaichenko , S. O. Shidlich, A. L. Shulyk, T. V. Shidlich, Andrii пряма апроксимаційна теорема, обернена апроксимаційна теорема, нерівність типу Джексона, узагальнений модуль гладкості. direct approximation theorem, inverse approximation theorem, Jackson type inequality, generalized module of smoothness UDC 517.5 In terms of the best approximations of functions and generalized moduli of smoothness, direct and inverse approximation theorems are proved for Besicovitch almost periodic functions whose Fourier exponent sequences have a single limit point in infinity and their Orlicz norms are finite. Special attention is paid to the study of cases when the constants in these theorems are unimprovable. УДК 517.5Прямі та обернені теореми наближення в просторах Безиковича-Мусєлака-Орлича майже періодичних функційУ термiнах найкращих наближень функцiй та узагальнених модулiв гладкостi доведено прямi та оберненi апроксимацiйнi теореми для майже перiодичних за Безиковичем функцiй, послiдовностi коефiцiєнтiв Фур’є яких мають єдину граничну точку в нескiнченностi, а їхнi норми Орлича є скiнченними. Особливу увагу придiлено вивченню випадкiв, коли сталi у цих теоремах непокращуванi. Institute of Mathematics, NAS of Ukraine 2022-06-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7045 10.37863/umzh.v74i5.7045 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 5 (2022); 701 - 716 Український математичний журнал; Том 74 № 5 (2022); 701 - 716 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7045/9242 Copyright (c) 2022 Андрій Любомирович Шидліч |
| spellingShingle | Chaichenko , S. O. Shidlich, A. L. Shulyk, T. V. Chaichenko , S. O. Shidlich, A. L. Shulyk, T. V. Shidlich, Andrii Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions |
| title | Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions |
| title_alt | Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions |
| title_full | Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions |
| title_fullStr | Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions |
| title_full_unstemmed | Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions |
| title_short | Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions |
| title_sort | direct and inverse approximation theorems in the besicovitch – musielak – orlicz spaces of almost periodic functions |
| topic_facet | пряма апроксимаційна теорема обернена апроксимаційна теорема нерівність типу Джексона узагальнений модуль гладкості. direct approximation theorem inverse approximation theorem Jackson type inequality generalized module of smoothness |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7045 |
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