Abstract
We generalize and improve the results of A. Guven, D. Israfilov, Xh. Z. Krasniqi and T. N. Shakh-Emirov. We consider the general methods of summability of Fourier series of functions from \(L_{2\pi }^{p(x)}\) with \( p\left( x\right) \ge 1\). For estimate of the error of approximation of functions by the matrix means we use a modulus of continuity constructed by the Steklov functions of the increments of considered functions without of absolute values.
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1 Introduction
Let \(p=p(x)\) will be a measurable \(2\pi \) - periodic function, \(p_{-}=\inf \left\{ p(x):x\in {\mathbb {R}} \right\} \), \(p^{-}\)\(=\sup \left\{ p(x):x\in {\mathbb {R}} \right\} \), 1 \(\le p_{-}\le p\le \)\(p^{-}\)\(<\infty \) and \(L_{2\pi }^{p} \) will be the space of all measurable \(2\pi \) - periodic functions f such that \(\int _{-\pi }^{\pi }\left| f(x)\right| ^{p(x)}dx<\infty \).
Putting
we turn \(L_{2\pi }^{p}\) into a Banach space (see [8]). We write \(\Pi _{2\pi }\) for the set of all \(2\pi \) - periodic variable exponents \(p=p(x)\ge 1\) satisfying the condition
For \(f\in L_{2\pi }^{1}\) we will consider the trigonometric Fourier series
with the partial sums \(S_{k}\left[ f\right] .\)
We will also need some notations on methods of summability of the series S[f] and modulus of continuity of f in the space \(L_{2\pi }^{p}.\)
If \(A:=\left( a_{n,k}\right) _{0\le n,k\,<\infty }\) be an infinite matrix of real numbers such that
or \(A_{0}:=\left( a_{n,k}\right) _{0\le k\le n<\infty },\) where
then
or
respectively.
Let \(f\in L_{2\pi }^{p}\) with \(p=p(x)\)\(\in \Pi _{2\pi }\). For \(n=0,1,2,...\) we denote the best approximation of f by
where the infimum is taken over all trigonometric polynomials \( T_{n}(x)=\sum _{k=-n}^{n}c_{k}e^{ikx}\) of the degrees at most n but the role of modulus of continuity of \(f\in L_{2\pi }^{p}\) in the case of a variable exponent \(p=p(x)\) play the function
It follows from the results of [9] that if \(p=p(x)\)\(\in \Pi _{2\pi }\) , then the function \(\Omega f(\cdot )_{p}\) is continuous on \( [0;\infty )\) and \(\lim _{\delta \rightarrow 0}\Omega f(\delta )_{p}=0.\) It also follows from the above definition that \(\Omega f(\delta )_{p}\) is a non-decreasing function of \(\delta \). We will call \(\Omega f(\cdot )_{p}\) the modulus of continuity of a function \(f\in L_{2\pi }^{p}\). With such modulus of continuity we can define the following class of functions:
where \(\omega \) is a function of modulus of continuity type on the interval \( [0,2\pi ],\) i.e. a nondecreasing continuous function having the following properties: \(\omega \left( 0\right) =0,\)\(\omega \left( \delta _{1}+\delta _{2}\right) \le \omega \left( \delta _{1}\right) +\omega \left( \delta _{2}\right) \) for any \(0\le \delta _{1}\le \delta _{2}\le \delta _{1}+\delta _{2}\le 2\pi \) and M is some positive constant.
It was proved in work [10, Theorem 6.1] that if the variable exponent \(p=p(x)\)\(\in \Pi _{2\pi }\) and \(f\in L_{2\pi }^{p}\), then the following Jackson-type inequality holds:
In this paper we generalize and improve the results of A. Guven, D. Israfilov, Xh. Z. Krasniqi and T. N. Shakh-Emirov. We will consider the spaces \(L_{2\pi }^{p(x)}\) with \(p\left( x\right) \ge 1\) and as a measure of approximation we will use the modulus of continuity constructed by the Steklov functions without the modulus of the increments of functions.
2 Main Results
At the begin we prove the general result.
Theorem 1
Let \(f\in L_{2\pi }^{p}\) with \(p=p(x)\)\(\in \Pi _{2\pi }\). If the conditions
for some \(\beta \ge 0\) and
hold, then
We observe that all of the lower triangular matrices satisfy (2). In this case we have:
Theorem 2
Let \(f\in L_{2\pi }^{p}\) with \(p=p(x)\)\(\in \Pi _{2\pi }\). If the conditions
for some \(\beta \ge 0\) and
hold, then
Next we will consider some special cases and we will approximate of \(f\in Lip_{p}\left( \omega ,M\right) \) with \(p=p(x)\)\(\in \Pi _{2\pi }\).
Theorem 3
Let \(f\in Lip_{p}\left( \omega ,M\right) \) with \(p=p(x)\)\(\in \Pi _{2\pi }\). If the conditions (1) for some \(\beta >0\) and (2) hold, then
In the similar way we obtain, by Theorem 2 the following
Theorem 4
Let \(f\in Lip_{p}\left( \omega ,M\right) \) with \(p=p(x)\)\(\in \Pi _{2\pi }\). If the conditions (3) for some \(\beta >0\) and (4) hold, then
Finally, we have some examples and remarks.
Example 1
One can easily verify that \(a_{n,k}=e^{-n}\sum _{j=k}^{\infty }\frac{n^{j}}{ \left( j+1\right) !},\) where \(n,k=0,1,2,...,\) satisfies the conditions (1) for any \(\beta \ge 0\) and (2).
Example 2
We can verify that \(a_{n,k}=\frac{(k+1)^{\beta }-k^{\beta }}{(n+1)^{\beta }}\) for \(k\le n\) and \(a_{n,k}=0\) for \(k>n\), where \(n,k=0,1,2,...,\) satisfies the conditions (3) for any \(\beta >1\) and (4).
Remark 1
If \(((k+1)^{-\beta }a_{n,k})\in HBVS\) for some \(\beta \ge 0\), where
then the condition (3) holds.
Remark 2
Let \(f\in Lip_{p}\left( \omega ,M\right) \) with \(\omega \left( \delta \right) =\delta ^{\alpha },\) where \(\alpha \in (0,1]\ \)and \(p=p(x)\)\(\in \Pi _{2\pi }\). From Theorem 4 the results of [1,2,3,4,5, 7] and [11, Theorem 5 (v)], [6, Theorem 5 (vi)] follow at once in the more general and improved forms.More precisely, in the mentioned papers [2,3,4] there is considered the following modulus of continuity
greater than \(\Omega f(\delta )_{p}\) and therefore the class \(Lip_{p}\left( \omega ,M\right) \) constructed by modulus \(\Omega f(\delta )_{p}\) is wider then such one constructed by \(\Omega _{p}f(\delta ).\) Moreover, in these papers and also in [7] there is the assumption \(p\left( x\right) \ge p_{-}>1,\) since for \(p_{-}=1\) the quantity \(\Omega _{p}f(\delta )\) may not tend to 0 when \(\delta \rightarrow 0\) for some \(f\in \)\(L_{2\pi }^{p}.\) In the papers [1, 5, 11, Theorem 5 (v)], [6, Theorem 5 (vi)] where \(p\left( x\right) =const.\) there is used yet bigger the integral modulus of continuity. We can note that in all above cited papers the simultaneous assumptions \(p_{-}=1\) and \(\alpha =1\) are impossible. Furthermore, in Theorem 4 there is considered very general class of matrices defined of the means \(T_{n,A}^{\text { }}\left[ f\right] \) which special cases occur in the above cited papers.
3 Lemmas
We start with two lemmas from the papers of I. I. Sharapudinov. The first one from [10, Lemma 2.3] will be formulated without a proof but the second one similar to that from the paper [9] but in more general form will be proved.
Lemma 1
(see [10, Lemma 2.3]). Let \(p=p(x)\) be a measurable \(2\pi \)—periodic function, such that 1 \(\le p_{-}\le p\le \)\(p^{-}\)\(<\infty \) and g be a function of two variables, \(2\pi \) - periodic and measurable on \(\left[ -\pi ,\pi \right] \times \left[ -\pi ,\pi \right] .\) Then
Lemma 2
(cf. [9, Lemma 3.1]). Let \(f\in L_{2\pi }^{p}\) with \(p=p(x)\)\(\in \Pi _{2\pi }\). Then
for every real \(\tau ,\) where \(f_{\frac{1}{2\lambda }}(\tau )=\lambda \int _{- \frac{1}{2\lambda }+\tau }^{\frac{1}{2\lambda }+\tau }f\left( t\right) dt\) with \(\lambda >1\).
Proof
Let \(h=\frac{1}{\left[ \lambda \right] },\)
whence \(p_{t}\left( x\right) =p_{0}\left( x+t\right) \) is a \(2\pi \) - periodic step function such that,
for \(m\pi h\le t\le \left( m+2\right) \pi h,\) since \(x_{k-\left( m+2\right) }\le x\le x_{k-\left( m-1\right) }\) for such t.
Let \(\left\| f\left( \cdot \right) \right\| _{p}\le 1\) and \(\tau \in {\mathbb {R}} .\) There exists an integer m such that
It is clear that
Further, similarly as in [9, Theorem 2.1],
and for \(x_{k-\left( m+1\right) }\le x\le x_{k-m}\)
whence
Next, by the Jensen inequality,
since similarly as in [9, p. 143] \(\int _{-\pi }^{\pi }\left| f\left( x\right) \right| ^{p_{t}\left( x\right) }dx=O\left( 1\right) .\) Thus our result follows. \(\square \)
Next we present some estimate of the kernel.
Lemma 3
If \(\beta \ge 0\) and \(0<t\le \pi \), then
Proof
Using the Abel transformation
Since \(\sum _{l=0}^{n}\frac{\sin \frac{(2l+1)t}{2}}{2\sin \frac{t}{2}}=\frac{ \sin ^{2}\frac{(n+1)t}{2}}{2\sin ^{2}\frac{t}{2}},\) the above expression does not exceed
whence the desired estimate follows. \(\square \)
Application of Lemma 3 gives next more general estimate.
Lemma 4
If (1) for some \(\beta \ge 0\) and (2) hold, then
Proof
Since
therefore
Using the Abel transformation, Lemma 3 and (1) we get
Thus
and by the assumption (2)
whence our estimate follows. \(\square \)
We will yet to need the following approximation result.
Lemma 5
Let \(f\in L_{2\pi }^{p}\) with \(p=p(x)\)\(\in \Pi _{2\pi }\) and let \(t_{n}\) be a trigonometric polynomial of the degree at most n, such that \( \left\| f-t_{n}\right\| _{p}=O\left( 1\right) \Omega f(\frac{1}{n+1} )_{p}\). If (1) for some \(\beta \ge 0\) and (2) hold, then
Proof
First of all, under the notation \(f_{h}\left( t\right) =\frac{1}{2h} \int _{-h}^{h}f\left( y+t\right) dy,\) we will prove that
and
Really, it is clear that \(a_{0}(f_{h})=a_{0}(f)\) and for \(\nu =1,2,3,...,\)
Hence
Consequently, denoting \(t_{n,h}\left( t\right) =\frac{1}{2h} \int _{-h}^{h}t_{n}\left( y+t\right) dy,\) we obtain
and therefore
Hence, by Lemma 1 and Lemma 2 with \(0<h<\frac{1}{2}\) and \(\left| t\right| \le \pi \)
and further, taking \(h=\frac{\pi }{8\nu }<\frac{1}{2}\) for \(v=1,2,...\), we have
Since, by Lemma 4
therefore, by the assumptions,
and our proof is ended. \(\square \)
4 Proofs of Main Results
4.1 Proof of Theorem 1
If \(t_{n}\) is a such polynomial that \(\left\| f-t_{n}\right\| _{p}=O\left( \Omega f(\frac{1}{n+1})_{p}\right) ,\) then
since
We note that by Lemma 5
Thus our result follows. \(\square \)
4.2 Proof of Theorem 2
We can note that the assumptions on entries of \(A_{0}\) yield
and
Thus the result follows by Theorem 1, since using the monotonicity of \( \Omega f(\delta )_{p}\) with respect to \(\delta >0\)
This ends the proof of Theorem 2. \(\square \)
4.3 Proof of Theorem 3
The subadditivity of \(\omega \) implies \(\omega \left( n\delta \right) \le n\omega \left( \delta \right) ,\) whence \(\omega \left( \lambda \delta \right) \le \left( \lambda +1\right) \omega \left( \delta \right) \) and therefore \(\frac{\omega \left( \delta _{2}\right) }{\delta _{2}}\le 2\frac{ \omega \left( \delta _{1}\right) }{\delta _{1}}\) since \(\omega \left( \delta _{2}\right) =\omega \left( \frac{\delta _{1}}{\delta _{1}}\delta _{2}\right) \le \left( \frac{\delta _{2}}{\delta _{1}}+1\right) \omega \left( \delta _{1}\right) =\left( \frac{\delta _{2}}{\delta _{1}}+\frac{\delta _{1}}{ \delta _{1}}\right) \omega \left( \delta _{1}\right) \le \left( \frac{ \delta _{2}}{\delta _{1}}+\frac{\delta _{2}}{\delta _{1}}\right) \omega \left( \delta _{1}\right) =2\frac{\delta _{2}}{\delta _{1}}\omega \left( \delta _{1}\right) ,\) where \(n\in {\mathbb {N}} _{0},\)\(\lambda \ge 0\) and \(0< \delta _{1}\le \delta _{2}.\) Hence, by ( 1) with \(\beta >0,\)
Thus, by Theorem 1, our result follows. \(\square \)
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Łenski, W., Szal, B. Trigonometric Approximation of Functions from \(L_{2\pi }^{p(x)}\). Results Math 75, 56 (2020). https://doi.org/10.1007/s00025-020-1170-0
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DOI: https://doi.org/10.1007/s00025-020-1170-0