On estimates of deviation of conjugate functions from matrix operators of their Fourier series by some expressions with $r$-differences of the entries

TL;DR

This paper extends previous results on the deviation of conjugate functions from matrix operators of Fourier series, providing new estimates involving $r$-differences of matrix entries.

Contribution

It introduces novel bounds for conjugate Fourier series deviations using $r$-differences, expanding prior work to conjugate series cases.

Findings

Extended deviation estimates to conjugate Fourier series.

Derived bounds involving $r$-differences of matrix entries.

Generalized previous results to broader classes of Fourier series.

Abstract

We extend the results of the authors from [Abstract and Applied Analysis, Volume 2016, Article ID 9712878] to the case conjugate Fourier series.

Equations

$$\Vert f{\Vert }_{{}_{L^p}}$$

$$\Vert f{\Vert }_{{}_C}$$

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

$$\tilde{S}f(x):=\sum _{\nu =1}^{\infty }(a_{\nu }(f)\sin \nu x-b_{\nu }(f)\cos \nu x)$$

$$\tilde{f}\left(x\right):=-\frac{1}{\pi }{\int }_0^{\pi }{\psi }_x\left(t\right)\frac{1}{2}\cot \frac{t}{2}dt=\underset{ϵ\to 0^{+}}{\lim }\tilde{f}\left(x,ϵ\right),$$

$$\tilde{f}\left(x,ϵ\right):=-\frac{1}{\pi }{\int }_{ϵ}^{\pi }{\psi }_x\left(t\right)\frac{1}{2}\cot \frac{t}{2}dt,$$

$$a_{n,k}\ge 0\text{ when }k,n=0,1,2,...\text{, }\underset{n\to \infty }{\lim }{a}_{n,k}=0\text{ and }\sum _{k=0}^{\infty }{a}_{n,k}=1.$$

$$\left(\begin{array}[]{c}T_{n,A}^{\text{ }}f\left(x\right)\\ \widetilde{T}_{n,A}^{\text{ }}f\left(x\right)\end{array}\right):=\sum_{k=0}^{\infty}a_{n,k}\left(\begin{array}[]{c}S_{k}f\left(x\right)\\ \widetilde{S}_{k}f\left(x\right)\end{array}\right)\text{ \ \ \ }\left(n=0,1,2,...\right).$$

$${\tilde{\omega }}_2{\left(f,\delta \right)}_X=\underset{0<t\le \delta }{\sup }\Vert {\psi }_{\cdot }\left(t\right){\Vert }_X.$$

$${\Vert T_{n,A}^{}f-f\Vert }_X=O\left(H\left(\frac{\pi }{n+1}\right)\left(\frac{\pi }{n+1}+A_{n,r}\right)\right),$$

$${\int }_u^{\pi }{t}^{-2}\omega \left(t\right)dt=O\left(H\left(u\right)\right)\text{ when }u\in \left[0,\pi \right],$$

$${\int }_0^{u}H\left(t\right)dt=O\left(uH\left(u\right)\right)\text{ when }u\in \left[0,\pi \right].$$

$${\left[\sum _{l=0}^n\sum _{k=l}^{r+l-1}{a}_{n,k}\right]}^{-1}=O\left(1\right),$$

$${\Vert T_{n,A}^{}f-f\Vert }_X=O\left(H\left(\frac{\pi }{n+1}\right)A_{n,r}\right)$$

$$\sum _{k=0}^{\infty }\left(k+1\right)a_{n,k}=O\left(n+1\right),$$

$${\Vert T_{n,A}^{}f-f\Vert }_X=O\left(\omega \left(\frac{\pi }{n+1}\right)+H\left(\frac{\pi }{n+1}\right)A_{n,r}\right).$$

$${\Vert T_{n,A}^{}f-f\Vert }_X$$

$$\left.\begin{array}[]{c}\left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot\right)\right\|_{X}\\ \left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot,\frac{\pi}{r\left(n+1\right)}\right)\right\|_{X}\end{array}\right\}=O\left(H\left(\frac{\pi}{n+1}\right)\left(\frac{\pi}{n+1}+A_{n,r}\right)\right).$$

$$\left.\begin{array}[]{c}\left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot\right)\right\|_{X}\\ \left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot,\frac{\pi}{r\left(n+1\right)}\right)\right\|_{X}\end{array}\right\}=O\left(H\left(\frac{\pi}{n+1}\right)A_{n,r}\right).$$

$$\left.\begin{array}[]{c}\left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot\right)\right\|_{X}\\ \left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot,\frac{1}{r}A_{n,r}\right)\right\|_{X}\end{array}\right\}=O\left(H\left(A_{n,r}\right)A_{n,r}\right).$$

$$\left.\begin{array}[]{c}\left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot\right)\right\|_{X}\\ \left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot,\frac{\pi}{r\left(n+1\right)}\right)\right\|_{X}\end{array}\right\}=O\left(\omega\left(\frac{\pi}{n+1}\right)+H\left(\frac{\pi}{n+1}\right)A_{n,r}\right),$$

$${\int }_0^u{t}^{-1}\omega \left(t\right)dt=O\left(\omega \left(u\right)\right)\text{ when }u\in \left[0,2\pi \right],$$

$$\displaystyle\left.\begin{array}[]{c}\left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot\right)\right\|_{X}\\ \left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot,\frac{\pi}{r\left(n+1\right)}\right)\right\|_{X}\end{array}\right\}$$

$$\sum _{k=m}^{\infty }\left|a_{n,k}-a_{n,k+r}\right|\le K\sum _{k=m/c}^{\infty }\frac{a_{n,k}}{k},$$

$$\left.\begin{array}[]{c}\left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot\right)\right\|_{X}\\ \left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot,\frac{\pi}{r\left(n+1\right)}\right)\right\|_{X}\end{array}\right\}=O\left(\frac{H\left(\frac{\pi}{n+1}\right)}{n+1}+\sum_{k=1}^{n}a_{n,k}\frac{H\left(\frac{\pi}{k+1}\right)}{k+1}\right),$$

$$D_{k,r}^{\circ }\left(t\right)=\frac{\sin \frac{\left(2k+r\right)t}{2}}{2\sin \frac{rt}{2}}\text{, }{\widetilde{D^{\circ }}}_{k,r}\left(t\right)=\frac{\cos \frac{\left(2k+r\right)t}{2}}{2\sin \frac{rt}{2}}\text{ and }{\tilde{D}}_{k,r}\left(t\right)=\frac{\cos \frac{t}{2}-\cos \frac{\left(2k+r\right)t}{2}}{2\sin \frac{rt}{2}}.$$

$${\tilde{T}}_{n,A}^{}f\left(x\right)-\tilde{f}\left(x\right)=\frac{1}{\pi }{\int }_0^{\pi }{\psi }_x\left(t\right)\sum _{k=0}^{\infty }{a}_{n,k}{\widetilde{D^{\circ }}}_{k,1}\left(t\right)dt$$

$${\tilde{T}}_{n,A}^{}f\left(x\right)-\tilde{f}\left(x,\frac{\pi }{r\left(n+1\right)}\right)$$

$$\left|{\widetilde{D^{\circ }}}_{k,1}\left(t\right)\right|\le \frac{\pi }{2\left|t\right|},\left|{\tilde{D}}_{k,1}\left(t\right)\right|\le \frac{\pi }{\left|t\right|}$$

$$\left|{\tilde{D}}_{k,1}\left(t\right)\right|\le \frac{1}{2}k\left(k+1\right)\left|t\right|\text{ and }\left|{\tilde{D}}_{k,1}\left(t\right)\right|\le k+1.$$

$$\sum _{k=n}^m{a}_k\sin kx$$

Full text

**Key words: **Rate of approximation, summability of Fourier series

**2000 Mathematics Subject Classification: **42A24

1 Introduction

Let $X=L^{p}$ or $X=C$, where $L^{p}\ (1\leq p\leq\infty)\;\left[C\right]$ be the class of all $2\pi$–periodic real–valued functions, integrable in the Lebesgue sense with $p$–th power when $p\geq 1$ and essentially bounded when $p=\infty$ $\left[\text{continuous}\right]$ over $Q=$ $[-\pi,\pi]$ with the norm

[TABLE]

and consider the trigonometric Fourier series

[TABLE]

with the partial sums $S_{k}f$ and the conjugate one

[TABLE]

with the partial sums $\widetilde{S}_{k}f$. We know that if $f\in L,$ then

[TABLE]

where

[TABLE]

with $\psi_{x}\left(t\right):=f\left(x+t\right)-f\left(x-t\right),$ exists for almost all  $x$ [6, Th.(3.1)IV].

Let $A:=\left(a_{n,k}\right)$ be an infinite matrix of real numbers such that

[TABLE]

We will use the notation $A_{n,r}=\sum_{k=0}^{\infty}\left|a_{n,k}-a_{n,k+r}\right|,$ for $r\in\mathbb{N}.$

The $A-$transformation  of  $S_{k}f$ and of $\widetilde{S}f$ be given, by a matrix convention, as follows

[TABLE]

In this paper, we study the upper bounds of $\left\|\widetilde{T}_{n,A}^{\text{ }}f-\widetilde{f}\right\|_{X}$ and $\left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot,\epsilon\right)\right\|_{X}$ by the second conjugate modulus of continuity of $f$ in the space $X$ defined by the formula

[TABLE]

We will also used the second classical modulus of continuity of $f$ in the space $X$ defined by the formula $\omega_{2}\left(f,\delta\right)_{X}:=\sup_{0\leq\left|t\right|\leq\delta}\left\|\varphi_{.}\left(t\right)\right\|_{X},$ where $\varphi_{x}\left(t\right):=f\left(x+t\right)+f\left(x-t\right)-2f\left(x\right).$

We will consider a function $\omega$ of modulus of continuity type on the interval $[0,2\pi],$ i.e. a nondecreasing continuous function having the following properties: $\omega\left(0\right)=0,$ $\omega\left(\delta_{1}+\delta_{2}\right)\leq\omega\left(\delta_{1}\right)+\omega\left(\delta_{2}\right)$ for any $0\leq\delta_{1}\leq\delta_{2}\leq\delta_{1}+\delta_{2}\leq 2\pi$.

The deviation $T_{n,A}^{\text{ }}f-f$ was estimated in the paper [2] ( see also [1, Theorems 3.4, p. 290] and [5]) as follows:

Theorem A. Let $f\in X_{\omega}=\left\{f\in X:\omega_{2}\left(f,\delta\right)_{X}=O\left(\omega\left(\delta\right)\right)\text{ {when\ }}\delta\in\left[0,2\pi\right]\right\}$ and $r\in\mathbb{N}.$ Then,

[TABLE]

where a function of modulus of continuity type $\omega$ satisfies the condition

[TABLE]

with $H(u)\geq 0$, such that

[TABLE]

Additionally, if

[TABLE]

then

[TABLE]

but if

[TABLE]

then

[TABLE]

Theorem B. If $f\in X$ and a matrix $A$ is such that $\left(\ref{114}\right)$ holds, then for $r\in\mathbb{N}$

[TABLE]

From our theorems we also derived a corollary for a matrix $A$ satisfying the following condition $\sum_{k=m}^{\infty}\left|a_{n,k}-a_{n,k+r}\right|\leq K\sum_{k=n/c}^{\infty}\frac{a_{n,k}}{k}.$

2 Statement of the results

Let $X_{\omega}=\left\{f\in X:\widetilde{\omega}_{2}\left(f,\delta\right)_{X}=O\left(\omega\left(\delta\right)\right)\text{ when\ }\delta\in\left[0,2\pi\right]\right\}.$ We present the estimates of the quantities $\left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot\right)\right\|_{X}$ and $\left\|\widetilde{T}_{n,A}^{\text{ }}f\left(\cdot\right)-\widetilde{f}\left(\cdot,\epsilon\right)\right\|_{X}$ simultaneously. Finally, we give a corollary and a remark.

Theorem 1

If $f\in X_{\omega},$ where $\omega$ satisfies the condition $\left(\ref{111}\right)$ such that $\left(\ref{112}\right)$ holds and $r\in\mathbb{N}$, then

[TABLE]

Additionally, if a matrix $A$ is such that $\left(\ref{113}\right)$ is true, then

[TABLE]

Theorem 2

If $f\in X_{\omega},$ where $\omega$ satisfies the condition $\left(\ref{111}\right)$ such that $\left(\ref{112}\right)$ holds, $r\in\mathbb{N}$ and a matrix $A$ is such that $\left(\ref{113}\right)$ is true, then

[TABLE]

Theorem 3

If $f\in X_{\omega},$ where $\omega$ satisfies the condition $\left(\ref{111}\right)$ such that $\left(\ref{112}\right)$ holds and $r\in\mathbb{N}$, then

[TABLE]

were in the case of the first estimate $\omega$ satisfies the extra condition

[TABLE]

but in the case of the second estimate a matrix $A$ is such that $\ \left(\ref{114}\right)$ is true.

Theorem 4

If $f\in X$ and $r\in\mathbb{N}$, then

[TABLE]

were in the case of the first estimate $\widetilde{\omega}_{2}$ instead of $\omega$ satisfies the extra condition $\left(\ref{202}\right)$, but in the case of the second estimate a matrix $A$ is such that $\left(\ref{114}\right)$ is true.

Corollary 1

From Theorem 4 it follows that if $f\in X_{\omega},$ where $\omega$ satisfies the condition $\left(\ref{111}\right)$ such that $\left(\ref{112}\right)$ is true and

[TABLE]

with some $c>1$ and $r\in\mathbb{N},$ holds, then

[TABLE]

were in the case of the first estimate $\widetilde{\omega}_{2}$ instead of $\omega$ satisfies the extra condition $\left(\ref{202}\right)$, but in the case of the second estimate a matrix $A$ is such that $\left(\ref{114}\right)$ is true.

Remark 1

We note that our extra conditions $\left(\ref{113}\right)$ and $\left(\ref{114}\right)$ for a lower triangular infinite matrix $A$ always hold.

3 Auxiliary results

We begin this section by some notations from [4] and [6, Section 5 of Chapter II]. Let for $r=1,2,...$

[TABLE]

It is clear by [6] that $\widetilde{S}_{k}f\left(x\right)=-\frac{1}{\pi}\int_{-\pi}^{\pi}f\left(x+t\right)\widetilde{D}_{k,1}\left(t\right)dt,$ whence

[TABLE]

and

[TABLE]

Next, we present the known estimates and relations.

Lemma 1

[6]** If  $0<\left|t\right|\leq\pi$ then

[TABLE]

and, for any real $t,$ we have

[TABLE]

Lemma 2

[4]** Let $r\in N,$ $l\in Z$ and $a:=(a_{n})\subset\mathbb{C}$. If $x$ $\neq$ $\frac{2l\pi}{r}$ , then for every $m\geq n$

[TABLE]

We additionally need two estimates with a function of modulus of continuity type $\omega$.

Lemma 3

[2]** If $\left(\ref{111}\right)$ and $\left(\ref{112}\right)$ hold, then, for $c\geq 1$ and $\beta>\alpha>0,$

[TABLE]

Lemma 4

[2]** If $\left(\ref{111}\right)$ and $\left(\ref{112}\right)$ hold, then, for $b\geq 1,$

[TABLE]

Finally, we present very useful property of such function $\omega$.

Lemma 5

[6]** A function $\omega$ of modulus of continuity type on the interval $[0,2\pi]$ satisfies the following condition $\delta_{2}^{-1}\omega\left(\delta_{2}\right)\leq 2\delta_{1}^{-1}\omega\left(\delta_{1}\right)$ for$\ \delta_{2}\geq\delta_{1}>0.$

4 Proofs of the results

4.1 Proof of Theorem 1

It is clear that for an odd $r$

[TABLE]

and for an even $r$

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Then,

[TABLE]

By Lemma 1

[TABLE]

Since, by Lemma 2,

[TABLE]

[TABLE]

whence

[TABLE]

Hence and by Lemma 1,

[TABLE]

and therefore

[TABLE]

Using Lemmas 3, 4, with $c=b=r$, and the estimates $\left|\sin\frac{t}{2}\right|\geq\frac{\left|t\right|}{\pi}$ for $t\in\left[0,\pi\right],$ $\left|\sin\frac{rt}{2}\right|\geq\frac{rt}{\pi}-2m$ for $t\in\left[\frac{2m\pi}{r},\frac{2m\pi}{r}+\frac{\pi}{r}\right],$ where $m\in\left\{0,...,\left[\ r/2\right]\right\},$ we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Analogously

[TABLE]

Similarly, by Lemma 1, Lemmas 3, 4, with $c=b=r$ and the estimates $\left|\sin\frac{t}{2}\right|\geq\frac{\left|t\right|}{\pi}$ for $t\in\left[0,\pi\right],$ $\left|\sin\frac{rt}{2}\right|\geq 2\left(m+1\right)-\frac{rt}{\pi}$ for $t\in\left[\frac{2\left(m+1\right)\pi}{r}-\frac{\pi}{r},\frac{2\left(m+1\right)\pi}{r}-\frac{\pi}{r\left(n+1\right)}\right],$ where $m\in\left\{0,...,\left[r/2\right]-1\right\},$ we get

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus

[TABLE]

Collecting these estimates we obtain the first result.

Applying condition $\left(\ref{113}\right)$ we have

[TABLE]

[TABLE]

and the second result also follows. $\blacksquare$

4.2 Proof of Theorem 2

Analogously, as in the proof of Theorem 1, we consider an odd $r$ and an even $r.$ Then,

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

[TABLE]

respectively. Since $A_{n,r}\leq 2,$ therefore we can estimate our terms analogously as in the proof of Theorem 1 with $A_{n,r}$ instead of $\frac{\pi}{n+1}$ and thus we obtain the desired estimate. $\blacksquare$

4.3 Proof of Theorem 3

Similarly, as in the proof of Theorem 1

[TABLE]

By Lemma 1 and $\left(\ref{114}\right)$

[TABLE]

and by Lemma 1 and $\left(\ref{202}\right)$

[TABLE]

Further, by the same lemmas and conditions as in the above proofs and Lemma 5, we obtain with \kappa=\left\{\begin{array}[]{c}1\text{ when }r\text{ is even,}\\ 0\text{ when }r\text{ is odd,}\end{array}\right. that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus our proof is complete. $\blacksquare$

4.4 Proof of Theorem 4

Let as above

[TABLE]

[TABLE]

and

[TABLE]

Further, taking $\tau_{m}^{1}=\left[\frac{\pi}{rt-2m\pi}\right]$ and $\tau=\left[\frac{\pi}{rt}\right]$, using Lemma 5, we obtain with \kappa=\left\{\begin{array}[]{c}1\text{ when }r\text{ is even,}\\ 0\text{ when }r\text{ is odd,}\end{array}\right. that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Next, taking $\tau_{m}^{2}=\left[\frac{\pi}{-rt+2\left(m+1\right)\pi}\right]$ , we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus the result follows. $\blacksquare$

4.5 Proof of Corollary 1

Theorem 4 implies that

[TABLE]

Since $\left(\ref{200}\right)$

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[TABLE]

[TABLE]

one has

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[TABLE]

[TABLE]

[TABLE]

If $\left(\ref{111}\right)$ and $\left(\ref{112}\right)$ hold, then

[TABLE]

[TABLE]

and therefore

[TABLE]

Since

[TABLE]

the result follows. $\blacksquare$