On Strong Approximation in Generalized Hölder and Zygmund Spaces ^†

Abstract

The strong approximation of a function is a useful tool to analyze the convergence of its Fourier series. It is based on the summability techniques. However, unlike matrix summability methods, it uses non-linear methods to derive an auxiliary sequence using approximation errors generated by the series under analysis. In this paper, we give some direct results on the strong means of Fourier series of functions in generalized Hölder and Zygmund spaces. To elaborate its use, we deduce some corollaries.

1. Introduction

The strong approximation, rather that providing approximations, is a tool of analysis. It is based on the summability techniques. However, unlike matrix summability methods, it uses non-linear methods to derive an auxiliary sequence using approximation errors generated by the series under analysis. This auxiliary sequence is further used to analyze the convergence properties of the series. To know more about the development of strong approximation methods, one can see the articles by Hyslop [1] and, Mittal and Kumar [2]. They give simple settings for strong approximation along with some comparison results.

2. Preliminaries

The classical $L^p[0,2\pi ]$ spaces define the foundations of Fourier analysis. An $L^p[0,2\pi ],$ $1\le p\le \infty$ space contains $2\pi$-periodic, Lebesgue integrable functions, which have finite norms denoted by ${\parallel ·\parallel }_p$ and defined by

$$\begin{array}{c}\hfill {\parallel f\parallel }_p=\left\{\begin{array}{cc}{\displaystyle {\left({\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }{\left|f\left(x\right)\right|}^{p}dx\right)}^{1/p},}\hfill & 1\le p<\infty ,\hfill \\ \underset{0\le x<2\pi }{\mathrm{ess}\sup }\left|f\left(x\right)\right|,\hfill & p=\infty .\hfill \end{array}\right.\end{array}$$

To measure the smoothness of the functions, we use the moduli of smoothness. For $f\in L^p[0,2\pi ],$ the r^th-order modulus of smoothness ${\omega }_r{(f;t)}_p$ is defined by

$$\begin{array}{c}\hfill {\omega }_r{(f;t)}_p=\underset{0<h\le t}{sup}{\parallel {\Delta }_h^{r}f(·)\parallel }_p,\end{array}$$

where ${\Delta }_h^{r}f\left(x\right)={\sum }_{k=0}^r{(-1)}^k\left(\genfrac{}{}{0pt}{}{r}{k}\right)f(x+(r-k)h)$ denotes the rth-order forward difference of f at x with step-size $h.$ The most basic spaces, which encode the smoothness properties of the functions, are the Hölder spaces. For periodic functions, Hölder spaces were first introduced for the space of $2\pi$-periodic continuous functions [3] (p. 51). Later, Das et al. [4] extended them to $L^p[0,2\pi ]$ spaces. They defined the Hölder spaces $H_{\alpha }\left(L^p\right),0<\alpha \le 1$ to contain functions $f\in L^p[0,2\pi ]$ such that ${\omega }_1{(f;t)}_p=O\left(t^{\alpha }\right).$ The norm for $f\in H_{\alpha }\left(L^p\right)$ is given by ${\parallel f\parallel }_{H_{\alpha }\left(L^p\right)}={\parallel f\parallel }_p+{\left|f\right|}_{H_{\alpha }\left(L^p\right)},$ where

$$\begin{array}{c}\hfill {\left|f\right|}_{H_{\alpha }\left(L^p\right)}=\underset{t>0}{sup}{\displaystyle \frac{{\omega }_1{(f;t)}_p}{t^{\alpha }}}.\end{array}$$

As a further generalization, Das et al. [5] generalized Hölder spaces, $H_{\alpha }\left(L^p\right)$ to $H_{\omega }\left(L^p\right)$ spaces where $\omega :[0,\infty )\to [0,\infty )$ is a non-decreasing function with ${lim}_{t\to 0^{+}}\omega \left(t\right)=0.$ The $H_{\omega }\left(L^p\right)$ spaces contain $f\in L^p[0,2\pi ]$ such that ${\omega }_1{(f;t)}_p=O\left(\omega \left(t\right)\right).$ For $f\in H_{\omega }\left(L^p\right),$ the norm is defined by ${\parallel f\parallel }_{H_{\omega }\left(L^p\right)}:={\parallel f\parallel }_p+{\left|f\right|}_{H_{\omega }\left(L^p\right)},$ where

$$\begin{array}{c}\hfill {\left|f\right|}_{H_{\omega }\left(L^p\right)}=\underset{t>0}{sup}{\displaystyle \frac{{\omega }_1{(f;t)}_p}{\omega \left(t\right)}}.\end{array}$$

The Zygmund spaces are defined using the second-order modulus of smoothness. The Zygmund spaces $Z_{\alpha }\left(L^p\right),0<\alpha \le 2$ are defined as containing the functions $f\in L^p[0,2\pi ]$ such that ${\omega }_2{(f;t)}_p=O\left(t^{\alpha }\right).$ The norm for $f\in Z_{\alpha }\left(L^p\right)$ is defined by ${\parallel f\parallel }_{Z_{\alpha }\left(L^p\right)}={\parallel f\parallel }_p+{\left|f\right|}_{Z_{\alpha }\left(L^p\right)},$ where

$$\begin{array}{c}\hfill {\left|f\right|}_{Z_{\alpha }\left(L^p\right)}=\underset{t>0}{sup}{\displaystyle \frac{{\omega }_2{(f;t)}_p}{t^{\alpha }}}.\end{array}$$

The Zygmund spaces $Z_{\alpha }\left(L^p\right)$ can be generalized in the same way as Hölder spaces $H_{\alpha }\left(L^p\right)$. Let $\omega :[0,\infty )\to [0,\infty )$ be a non-decreasing function with ${lim}_{t\to 0^{+}}\omega \left(t\right)=0.$ Then, $Z_{\omega }\left(L^p\right)$ generalizes $Z_{\alpha }\left(L^p\right)$ spaces via the requirement ${\omega }_2{(f;t)}_p=O\left(\omega \left(t\right)\right)$ for $f\in L^p[0,2\pi ].$ The norm for $f\in Z_{\omega }\left(L^p\right)$ is defined by ${\parallel f\parallel }_{Z_{\omega }\left(L^p\right)}={\parallel f\parallel }_p+{\left|f\right|}_{Z_{\omega }\left(L^p\right)},$ where

$$\begin{array}{c}\hfill {\left|f\right|}_{Z_{\omega }\left(L^p\right)}=\underset{t>0}{sup}{\displaystyle \frac{{\omega }_2{(f;t)}_p}{\omega \left(t\right)}}.\end{array}$$

For $\omega \left(t\right)=t^{\alpha },0<\alpha \le 1,$ the generalized Hölder space $H_{\omega }\left(L^p\right)$ becomes Hölder space $H_{\alpha }\left(L^p\right).$ Similarly, for $\omega \left(t\right)=t^{\alpha },0<\alpha \le 2,$ the generalized Zygmund space $Z_{\omega }\left(L^p\right)$ coincides with Zygmund space $Z_{\alpha }\left(L^p\right).$ Because of the fact that ${\omega }_2{(f;t)}_p\le 2{\omega }_1{(f;t)}_p,$ the Hölder spaces are the subsets of corresponding Zygmund spaces. However, as the Zygmund norm is not same as Hölder norm on Hölder spaces, the Zygmund spaces do not generalize the Hölder spaces.

Let f be a $2\pi$-periodic Lebesgue integrable function. Then, the partial sums $S_n(f;x),n=1,2,\dots$ of the trigonometric Fourier series of f can be written as

$$\begin{array}{c}\hfill S_n(f;x)={\displaystyle \frac{1}{2\pi }}{\int }_0^{\pi }\left(f(x+t)+f(x-t)\right)D_n\left(t\right)dt,n=0,1,2,\dots ,\end{array}$$

where $D_n\left(t\right)$ denotes the Dirichlet kernel of order n given by

$$\begin{array}{c}\hfill D_n\left(t\right)=1+2\sum _{k=1}^{n}coskt=\left\{\begin{array}{cc}2n+1,\hfill & t=2m\pi ,m\in \mathbb{Z},\hfill \\ {\displaystyle \frac{sin\left({\displaystyle \frac{(2n+1)t}{2}}\right)}{sin\left({\displaystyle \frac{t}{2}}\right)}},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

The properties of the Dirichlet kernels can be found in [6] (p. 235). $R_n(f;x),$ the nth residual in the approximation of f by partial sums $S_n(f;x)$ can be written as follows:

$$\begin{array}{c}\hfill R_n(f;x):=S_n(f;x)-f\left(x\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^{\pi }{\Delta }_t^{2}f(x-t)D_n\left(t\right)dt,n=0,1,2,\dots .\end{array}$$

The summability techniques are an extension of the notion of convergence of a series in which we try to attach a limit to a non-convergent sequence. The summability methods may also increase the rate of convergence of an already convergent series. A summability matrix $T=\left(a_{n,k}\right),n,k=0,1,2,\dots$ is called a regular summability matrix if it satisfies:

(i)

${lim}_{n\to \infty }{a}_{n,k}=0$,

(ii)

${\sum }_{k=0}^{\infty }\left|a_{n,k}\right|\le M\ge 0,\forall n,$

(iii)

${\sum }_{k=0}^{\infty }{a}_{n,k}=1,\forall n.$

Let $T=\left(a_{n,k}\right),n,k=0,1,2,\dots$ be an infinite-dimensional, lower triangular summability matrix. Then, the sequence

$$\begin{array}{c}\hfill T_n(f;x)=\sum _{k=0}^n{a}_{n,k}{S}_k(f;x),n=0,1,2,\dots ,\end{array}$$

defines the T-means of the trigonometric Fourier series of $f.$ The difference ${\rho }_n(f;x):=T_n(f;x)-f\left(x\right)$ denotes the nth residual in the approximation of f by T-means of the Fourier series. If

$$\begin{array}{c}\hfill \sum _{k=0}^n{a}_{n,k}=1,n=0,1,2,\dots ,\end{array}$$

(1)

then we can write

$$\begin{array}{c}\hfill {\rho }_n(f;x)=\sum _{k=0}^n{a}_{n,k}{R}_k(f;x)={\displaystyle \frac{1}{2\pi }}{\int }_0^{\pi }{\Delta }_t^{2}f(x-t)K_n\left(t\right)dt,\end{array}$$

where $K_n\left(t\right)$ is the kernel generated by T and defined by $K_n\left(t\right):={\sum }_{k=0}^n{a}_{n,k}{D}_k\left(t\right),n=0,1,2,\dots .$ If $T=\left(a_{n,k}\right)$ is such that $a_{n,k}\ge 0,n,k=0,1,2,\dots ,$ then for $\lambda >0,$ the T-strong means of the Fourier series are defined by

$$\begin{array}{c}\hfill U_n(f,\lambda ,x)={\left(\sum _{k=0}^n{a}_{n,k}{\left|R_k(f;x)\right|}^{\lambda }\right)}^{1/\lambda }.\end{array}$$

In this paper, we present estimates of $U_n(f,\lambda ,x)$ in $H_{\omega }\left(L^p\right)$ and $Z_{\omega }\left(L^p\right)$ spaces. These estimates help in gaining insights of the approximation error in Fourier approximation.

3. Results

First, we give some auxiliary results as lemmas to make the proof of the main results concise.

Lemma 1.

Let $D_k\left(t\right)$ be the kth Dirichlet kernel. Then,

(i)

$|D_k\left(t\right)|=O(k+1),t\in [0,\pi ].$

(ii)

$|D_k\left(t\right)|=O(1/t),t\in (0,\pi ].$

The lemma can be proved easily.

Lemma 2.

Let ${\left\{f_n\right\}}_{n=0}^{\infty }$ be a sequence of $2\pi$-periodic Lebesgue measurable functions. Then, for any $p\ge 1$ and $0<\lambda \le p$

$$\begin{array}{c}\hfill {∥{\left(\sum _{k=0}^{\infty }{\left|f_k\left(x\right)\right|}^{\lambda }\right)}^{1/\lambda }∥}_p\le {\left(\sum _{k=0}^{\infty }{∥f_k∥}_p^{\lambda }\right)}^{1/\lambda }.\end{array}$$

Using generalized Minkowski inequality, the Lemma 2 can proved easily. Now, we present the main results.

Theorem 1.

Let $T=\left(a_{n,k}\right)$ be a summability matrix such that $a_{n,k}\ge 0,n,k=0,1,2,\dots .$ Then, for $f\in H_{\omega }\left(L^p\right)$ and $0<\lambda \le p$,

$$\begin{array}{c}\hfill \parallel U_n{(f,\lambda ,x)\parallel }_{H_{\omega }\left(L^p\right)}=O{\left(\sum _{k=0}^n{a}_{n,k}{\left(\omega \left({\displaystyle \frac{\pi }{k+1}}\right)+{\int }_{\frac{\pi }{k+1}}^{\pi }{\displaystyle \frac{\omega \left(t\right)}{t}}dt\right)}^{\lambda }\right)}^{1/\lambda },\end{array}$$

provided f satisfies the following conditions:

(i)

${\Delta }_h^1{U}_n(f,\lambda ,x)=O\left(U_n({\Delta }_h^{1}f,\lambda ,x)\right).$

(ii)

$\parallel {\Delta }_h^1{\Delta }_t^2{f\left(x\right)\parallel }_p=O\left({\omega }_1(f;t){\parallel {\Delta }_h^{1}f\left(x\right)\parallel }_p\right).$

Proof. 

By the definition of ${\parallel ·\parallel }_{H_{\omega }\left(L^p\right)},$ we have

$$\begin{array}{c}\hfill \parallel U_n{(f,\lambda ,x)\parallel }_{H_{\omega }\left(L^p\right)}={\parallel U_n(f,\lambda ,x)\parallel }_p+{\left|U_n(f,\lambda ,x)\right|}_{H_{\omega }\left(L^p\right)}.\end{array}$$

We first estimate $\parallel U_n{(f,\lambda ,x)\parallel }_p.$ Using Lemma 2,

$$\begin{array}{cc}\hfill \parallel U_n{(f,\lambda ,x)\parallel }_p& ={∥{\left(\sum _{k=0}^n{a}_{n,k}{\left|R_k(f,x)\right|}^{\lambda }\right)}^{1/\lambda }∥}_p\le {\left(\sum _{k=0}^n{a}_{n,k}{∥R_k(f,x)∥}_p^{\lambda }\right)}^{1/\lambda }.\hfill \end{array}$$

(2)

Using the generalized Minkowski inequality, Lemma 1 and the definition of $R_k(f,x)$, we have

$$\begin{array}{cc}\hfill {∥R_k(f,x)∥}_p& ={∥{\displaystyle \frac{1}{2\pi }}{\int }_0^{\pi }{\Delta }_t^{2}f(x-t)D_k\left(t\right)dt∥}_p\hfill \\ \hfill & =O\left({\int }_0^{\pi }{\omega }_2{(f;t)}_p\left|D_k\left(t\right)\right|dt\right)\hfill \\ \hfill & =O\left({\int }_0^{\pi }{\omega }_1{(f;t)}_p\left|D_k\left(t\right)\right|dt\right)\hfill \\ \hfill & =O\left({\int }_0^{\frac{\pi }{k+1}}{\omega }_1{(f;t)}_p\left|D_k\left(t\right)\right|dt+{\int }_{\frac{\pi }{k+1}}^{\pi }{\omega }_1{(f;t)}_p\left|D_k\left(t\right)\right|dt\right)\hfill \\ \hfill & =O\left(\omega \left({\displaystyle \frac{\pi }{k+1}}\right)+{\int }_{\frac{\pi }{k+1}}^{\pi }{\displaystyle \frac{\omega \left(t\right)}{t}}dt\right).\hfill \end{array}$$

(3)

From (2) and (3), we have

$$\begin{array}{cc}\hfill \parallel U_n{(f,\lambda ,x)\parallel }_p& =O{\left(\sum _{k=0}^n{a}_{n,k}{\left(\omega \left({\displaystyle \frac{\pi }{k+1}}\right)+{\int }_{\frac{\pi }{k+1}}^{\pi }{\displaystyle \frac{\omega \left(t\right)}{t}}dt\right)}^{\lambda }\right)}^{1/\lambda }.\hfill \end{array}$$

(4)

Now, we estimate ${\left|U_n(f,\lambda ,x)\right|}_{H_{\omega }\left(L^p\right)}.$ Using condition 1, we have

$$\begin{array}{cc}\hfill \parallel {\Delta }_h^1{U}_n{(f,\lambda ,x)\parallel }_p& =O\left(\parallel U_n({\Delta }_h^{1}f,\lambda ,x){\parallel }_p\right)\hfill \\ \hfill & =O\left({∥{\left(\sum _{k=0}^n{a}_{n,k}{\left|R_k({\Delta }_h^{1}f,x)\right|}^{\lambda }\right)}^{1/\lambda }∥}_p\right).\hfill \end{array}$$

Using Lemma 2, the definition of $R_k(f,x)$ and condition 1, we have

$$\begin{array}{cc}\hfill \parallel {\Delta }_h^1{U}_n{(f,\lambda ,x)\parallel }_p& =O{\left(\sum _{k=0}^n{a}_{n,k}{∥R_k({\Delta }_h^{1}f,x)∥}_p^{\lambda }\right)}^{1/\lambda }\hfill \\ \hfill & =O\left(\parallel {\Delta }_h^1{f\left(x\right)\parallel }_p{\left(\sum _{k=0}^n{a}_{n,k}{\left({\int }_0^{\pi }{\omega }_1(f;t)\left|D_k\left(t\right)\right|dt\right)}^{\lambda }\right)}^{1/\lambda }\right).\hfill \end{array}$$

Following the calculation in (3), we have

$$\begin{array}{cc}\hfill \parallel {\Delta }_h^1{U}_n{(f,\lambda ,x)\parallel }_p& =O\left(\parallel {\Delta }_h^1{f\left(x\right)\parallel }_p{\left(\sum _{k=0}^n{a}_{n,k}{\left(\omega \left({\displaystyle \frac{\pi }{k+1}}\right)+{\int }_{\frac{\pi }{k+1}}^{\pi }{\displaystyle \frac{\omega \left(t\right)}{t}}dt\right)}^{\lambda }\right)}^{1/\lambda }\right).\hfill \end{array}$$

Taking the supremum for $0<h\le u$ on both sides

$$\begin{array}{c}\hfill {\omega }_1\left(U_n(f,\lambda ,x);u\right)=O\left({\omega }_1(f;u){\left(\sum _{k=0}^n{a}_{n,k}{\left(\omega \left({\displaystyle \frac{\pi }{k+1}}\right)+{\int }_{\frac{\pi }{k+1}}^{\pi }{\displaystyle \frac{\omega \left(t\right)}{t}}dt\right)}^{\lambda }\right)}^{1/\lambda }\right).\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill |U_n{(f,\lambda ,x)|}_{H^{\omega }\left(L^p\right)}& =\underset{u>0}{sup}{\displaystyle \frac{{\omega }_1{\left(U_n(f,\lambda ,x);u\right)}_p}{\omega \left(u\right)}}\hfill \\ \hfill & =O\left(\underset{u>0}{sup}{\displaystyle \frac{{\omega }_2{(f;u)}_p}{\omega \left(u\right)}}{\left(\sum _{k=0}^n{a}_{n,k}{\left(\omega \left({\displaystyle \frac{\pi }{k+1}}\right)+{\int }_{\frac{\pi }{k+1}}^{\pi }{\displaystyle \frac{\omega \left(t\right)}{t}}dt\right)}^{\lambda }\right)}^{1/\lambda }\right)\hfill \\ \hfill & =O{\left(\sum _{k=0}^n{a}_{n,k}{\left(\omega \left({\displaystyle \frac{\pi }{k+1}}\right)+{\int }_{\frac{\pi }{k+1}}^{\pi }{\displaystyle \frac{\omega \left(t\right)}{t}}dt\right)}^{\lambda }\right)}^{1/\lambda }.\hfill \end{array}$$

(5)

Combining (4) and (5), we have

$$\begin{array}{c}\hfill \parallel U_n{(f,\lambda ,x)\parallel }_{H_{\omega }\left(L^p\right)}=O\left({\left(\sum _{k=0}^n{a}_{n,k}{\left(\omega \left({\displaystyle \frac{\pi }{k+1}}\right)+{\int }_{\frac{\pi }{k+1}}^{\pi }{\displaystyle \frac{\omega \left(t\right)}{t}}dt\right)}^{\lambda }\right)}^{1/\lambda }\right),\end{array}$$

which completes the proof of the theorem. □

Depending on the value of $\lambda ,$ condition 1 in Theorem 1 can be relaxed. More precisely, the following holds.

Corollary 1.

Let $T=\left(a_{n,k}\right)$ be a summability matrix such that $a_{n,k}\ge 0,n,k=0,1,2,\dots .$ Then, for $f\in H_{\omega }\left(L^p\right)$ and $1\le \lambda \le p$

$$\begin{array}{c}\hfill \parallel U_n{(f,\lambda ,x)\parallel }_{H_{\omega }\left(L^p\right)}=O{\left(\sum _{k=0}^n{a}_{n,k}{\left(\omega \left({\displaystyle \frac{\pi }{k+1}}\right)+{\int }_{\frac{\pi }{k+1}}^{\pi }{\displaystyle \frac{\omega \left(t\right)}{t}}dt\right)}^{\lambda }\right)}^{1/\lambda }\end{array}$$

provided f satisfies $\parallel {\Delta }_h^1{\Delta }_t^2{f\left(x\right)\parallel }_p=O\left({\omega }_1(f;t){\parallel {\Delta }_h^{1}f\left(x\right)\parallel }_p\right).$

Proof. 

In the light of the Minkowski inequality for sequence spaces, the condition 1 in Theorem 1 holds for $\lambda \ge 1.$ Then, the corollary follows from Theorem 1. □

Since, for $\omega \left(t\right)=t^{\alpha },0<\alpha \le 1,$$H_{\omega }\left(L^p\right)$ space reduces to $H_{\alpha }\left(L^p\right)$ space, we have the following corollary for $f\in H_{\alpha }\left(L^p\right)$.

Corollary 2.

Let $T=\left(a_{n,k}\right)$ be a summability matrix such that $a_{n,k}\ge 0,n,k=0,1,2,\dots$ and satisfies (1). Then, for $f\in H_{\alpha }\left(L^p\right)$ and $0<\lambda \le p$

$$\begin{array}{c}\hfill \parallel U_n{(f,\lambda ,x)\parallel }_{H_{\alpha }\left(L^p\right)}=O\left(1\right),\end{array}$$

provided f satisfies the following conditions:

(i)

${\Delta }_h^1{U}_n(f,\lambda ,x)=O\left(U_n({\Delta }_h^{1}f,\lambda ,x)\right).$

(ii)

$\parallel {\Delta }_h^1{\Delta }_t^2{f\left(x\right)\parallel }_p=O\left({\omega }_1(f;t){\parallel {\Delta }_h^{1}f\left(x\right)\parallel }_p\right).$