Estimates of convolution operators of functions from $L_{2\pi}^{p(x)}$

TL;DR

This paper extends previous results on convolution operators acting on functions with variable exponent Lebesgue spaces, providing slight improvements and applications to the de la Vallée Poussin operator.

Contribution

It generalizes and slightly improves earlier work on convolution operators in variable exponent Lebesgue spaces, with new applications to approximation operators.

Findings

Improved bounds for convolution operators in $L_{2 ext{-}pi}^{p(x)}$ spaces.

Applications demonstrated for the de la Vallée Poussin operator.

Enhanced understanding of approximation in variable exponent spaces.

Abstract

We generalize and slight improve the result of I. I. Sharapudinov [Mat. Zametki, 1996, Volume 59, Issue 2, 291--302]. Some applications to the de la Vall\'{e}e Poussin operator will also be given.

Equations

$${\Vert f\Vert }_p=\inf \left\{\alpha >0:{\int }_{-\pi }^{\pi }{\left|\frac{f(x)}{\alpha }\right|}^{p(x)}dx\le 1\right\}$$

$$\left|p(x)-p(y)\right|\ln \frac{2\pi }{\left|x-y\right|}=O\left(1\right)\text{, }\left(x,y\in \left[-\pi ,\pi \right]\right).$$

$${\Vert K_{\lambda }[f]\Vert }_p=O\left(1\right){\Vert f\Vert }_p,$$

$$K_{\lambda }[f](x)={\int }_{-\pi }^{\pi }f(t)k_{\lambda }(t-x)dt.$$

$$K_{m,n}[f]=K_{m,n}[f](x)={\int }_{-\pi }^{\pi }f(t)k_{m,n}(t-x)dt$$

$${\Vert K_{m,n}[f]\Vert }_p=O\left({\mu }_{m,n}\right){\Vert f\Vert }_p,$$

$${\mu }_{m,n}=\mu \left(k_{m,n}\right)={\int }_{-h_m}^{h_m}\left|k_{m,n}(t)\right|dt.$$

$$x_k=k{h}_n-\pi ,k=0,\pm 1,\pm 2,...$$

$$s_k=\min \left\{p\left(x\right):x_{k-1}\le x\le x_{k+2}\right\},k=0,\pm 1,\pm 2,...$$

$$p_t\left(x\right)=s_k\left(x_{k-1}-t\le x\le x_{k+1}-t\right),k=0,\pm 1,\pm 2,...$$

$$p_t\left(x\right)\le p\left(x\right).$$

$$E_m\left(x\right)=\left(-\pi ,\pi \right)\backslash \left(x-h_m,x+h_m\right),$$

$$E_m\left(x\right)=\left(-\pi ,\pi \right)\backslash \left(-\pi ,x+h_m\right)$$

$$E_m\left(x\right)=\left(-\pi ,\pi \right)\backslash \left(x-h_m,\pi \right),$$

$${\Vert f\Vert }_p\le 1.$$

$${\left({\int }_{-\pi }^{\pi }{\left|K_{m,n}[f](x)\right|}^{p(x)}dx\right)}^{1/\overline{p}}$$

$$\le$$

$${\Vert f\Vert }_q\le \left(2\pi +1\right){\Vert f\Vert }_p$$

$$\left|{\int }_{E_m\left(x\right)}f(t)k_{m,n}(t-x)dt\right|$$

$$J_m=O\left(1\right).$$

$$\left|p(x)-s_k\right|=O\left(\frac{1}{\ln 2{\left(n+1\right)}^{\gamma }}\right),$$

$${\left|{\int }_{x-h_m}^{x+h_m}f(t)k_{m,n}(t-x)dt\right|}^{p(x)-s_k}$$

$$J_x$$

$$J_x=O\left(1\right)\sum _{k=0}^{2[n^{\gamma }]}{\left({\mu }_{m,n}\right)}^{s_k}{\int }_{x_k}^{x_{k+1}}{\left|\frac{1}{{\mu }_{m,n}}{\int }_{x-h_m}^{x+h_m}f(t)k_{m,n}(t-x)dt\right|}^{s_k}dx.$$

$$J_x$$

$${\int }_{-\pi }^{\pi }{\left|f(x)\right|}^{p_t\left(x\right)}dx\le {\left(2\pi +1\right)}^{\overline{p}}.$$

$$J_x^{1/\overline{p}}=O\left({\mu }_{m,n}\right)$$

$${\Vert K_{\lambda }[f]\Vert }_p=O\left({\mu }_{\lambda }\right){\Vert f\Vert }_p$$

$$Sf(x):=\frac{a_0(f)}{2}+\sum _{\nu =1}^{\infty }(a_{\nu }(f)\cos \nu x+b_{\nu }(f)\sin \nu x)$$

$$V_{n,m}[f]\left(x\right)=\frac{1}{m+1}\sum _{k=n-m}^n{S}_k[f]\left(x\right)=\frac{1}{\pi \left(m+1\right)}{\int }_{-\pi }^{\pi }f\left(t\right){\Phi }_{n,m}\left(t-x\right)dt$$

Full text

**Key words: **convolution operators, $L_{2\pi}^{p}(p=p(x))$ spaces, rate of approximation.

**2010 Mathematics Subject Classification: **47B38, 44A35, 41A35, 42A24.

1 Introduction

Let $p=p(x)$ will be a measurable $2\pi$ - periodic function, $p_{-}=\inf\left\{p(x):x\in\mathbb{R}\right\}$, $p^{-}$ $=\sup\left\{p(x):x\in\mathbb{R}\right\}$, $1$ $\leq p_{-}\leq$ $p^{-}$ $<\infty$ and $L_{2\pi}^{p}$ will be the space of measurable $2\pi$ - periodic functions $f$ such that $\int_{-\pi}^{\pi}\left|f(x)\right|^{p(x)}dx<\infty$.

Putting

[TABLE]

we turn $L_{2\pi}^{p}$ into the Banach space. We write $\Pi_{2\pi}$ for the set of all $2\pi$ - periodic variable exponents $p=p(x)\geq 1$ satisfying the condition

[TABLE]

In the paper [2] I. I. Sharapudinov proved the following theorem:

Theorem A. Let $k_{\lambda}=k_{\lambda}(x)$ $\left(1\leq\lambda<\infty\right)$ be a measurable $2\pi$ -periodic essentially bounded function (kernel) satisfying the conditions:

A∘) $\int_{-\pi}^{\pi}\left|k_{\lambda}(t)\right|dt\leq c_{1}^{\circ},$

B∘) $\sup_{t}\left|k_{\lambda}(t)\right|\leq c_{2}^{\circ}\lambda^{\eta}$,

C∘) $\left|k_{\lambda}(t)\right|\leq c_{3}^{\circ}$,  $\left(\lambda^{-\gamma}\leq\left|t\right|<\pi\right),$

where $\gamma,\eta,$ $c_{1}^{\circ},c_{2}^{\circ},c_{3}^{\circ}>0$ are independent of $\lambda$.

If $f\in L_{2\pi}^{p}$ with $p=p(x)$ $\in\Pi_{2\pi}$, then

[TABLE]

where

[TABLE]

We generalize and slight improve this result considering the wider family of two parameters convolution operators. Some applications to the de la Vallée Poussin operator will also be given.

2 Main result

Denote by $k_{m,n}=k_{m,n}(x),$ for every $0\leq m\leq n<\infty,$ a measurable $2\pi$ -periodic essentially bounded function (kernel). Let define the linear operator

[TABLE]

in space $L_{2\pi}^{p}.$ We will say that the kernel family $\left\{k_{m,n}(x)\right\}_{0\leq m\leq n<\infty}$ satisfies the conditions B) and C), respectively, if the following estimates hold:

B) $\sup_{\left|t\right|\leq h_{m}}\left|k_{m,n}(t)\right|\leq c_{2}\left(n+1\right)^{\eta}$,

C) $\left|k_{m,n}(t)\right|\leq c_{3}$,  $\left(h_{m}\leq\left|t\right|<\pi\right),$

where $h_{m}=\frac{\pi}{\left(m+1\right)^{\gamma}}$ and $\gamma$, $\eta,c_{2},c_{3}>0$ are independent of $m,n$.

For the operator $K_{m,n}[f]$ we will prove the following general estimate:

Theorem 1**.**

Let $k_{m,n}=k_{m,n}(x)$ $\left(0\leq m\leq n<\infty\right)$ satisfy the conditions B) and C). If $f\in L_{2\pi}^{p}$ with $p=p(x)$ $\in\Pi_{2\pi}$, then

[TABLE]

where

[TABLE]

Proof.

Let

[TABLE]

[TABLE]

[TABLE]

whence $p_{t}\left(x\right)=p_{0}\left(x+t\right)$ is a $2\pi$ - periodic step function such that

[TABLE]

Denote by

[TABLE]

when $\left(x-h_{m},x+h_{m}\right)\subset\left(-\pi,\pi\right)$, but

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or

[TABLE]

when $x-h_{m}<-\pi$ or $\pi<x+h_{m}$, respectively.

Let

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It is clear that for $\overline{p}=\max\left\{p\left(x\right):-\pi\leq x\leq\pi\right\}$

[TABLE]

[TABLE]

Since

[TABLE]

with $q\left(x\right)\leq p\left(x\right)$ for $-\pi\leq x\leq\pi$ (cf. [1]), by condition C),

[TABLE]

and therefore

[TABLE]

In case of integral $J_{x}$, using (1), (2) and (3), for $x_{k}\leq x\leq x_{k+1},$ we obtain that

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and therefore by condition B)

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Next,

[TABLE]

Thus

[TABLE]

By the Jensen inequality,

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Similar to [2, (17) p.295] we have

[TABLE]

Hence

[TABLE]

and our result follows. ∎

Corollary 2**.**

If we put $m=n=\lambda$ in the assumptions of Theorem 1, then for $f\in L_{2\pi}^{p}$ with $p=p(x)$ $\in\Pi_{2\pi}$ the following estimate

[TABLE]

holds, where $\mu_{\lambda}=\int_{-h_{\lambda}}^{h_{\lambda}}\left|k_{\lambda}(t)\right|dt$ and $k_{\lambda}=k_{\lambda,\lambda}$.

Remark 3**.**

If we additionally assume that $\mu_{\lambda}=O\left(1\right),$ then Corollary 2 gives Theorem A with the result [2] of I. I. Sharapudinov, under the slight weaker conditions.

3 De la Vallée Poussin operator

Let $f\in L_{2\pi}^{1}$ and consider the trigonometric Fourier series

[TABLE]

with the partial sums $S_{k}f.$ For $0\leq m\leq n,$ $m,n=0,1,2,...$ denote by

[TABLE]

the de la Vallée Poussin means of the series $Sf,$ where

[TABLE]

It is clear that the kernel family $\left\{\Phi_{m,n}(x)\right\}_{1\leq m\leq n<\infty}$ satisfies the conditions B) with $\eta=1$ and C) with $\gamma=\frac{1}{2}$. By the following calculation

[TABLE]

we obtain

[TABLE]

Hence, by Theorem 1, we have:

Theorem 4**.**

If $L_{2\pi}^{p}$ with $p=p(x)$ $\in\Pi_{2\pi}$, then

[TABLE]

From Theorem 4 we get the following corollary:

Corollary 5**.**

Let $f\in L_{2\pi}^{p}$ with $p=p(x)$ $\in\Pi_{2\pi}.$ If $m=O\left(n\right)$, then

[TABLE]

hold.

In the special case we can consider the following Fourier operator:

[TABLE]

where

[TABLE]

For this operator we have:

Corollary 6**.**

Let $f\in L_{2\pi}^{p}$ with $p=p(x)$ $\in\Pi_{2\pi}.$ If we put $m=0$ in Theorem 4, then

[TABLE]

Remark 7**.**

In the case $p\equiv 1$ the results of this section we can find in the monograph of A. Zygmund [5, Ch. II, p.70, Ch. III, p.90] (see e.g.[6, Ch. II, p.117-8] .