Theory and Modern Applications

# Trigonometric approximation of functions $$f(x,y)$$ of generalized Lipschitz class by double Hausdorff matrix summability method

## Abstract

In this paper, we establish a new estimate for the degree of approximation of functions $$f(x,y)$$ belonging to the generalized Lipschitz class $$Lip ((\xi _{1}, \xi _{2} );r )$$, $$r \geq 1$$, by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from $$Lip ((\alpha ,\beta );r )$$ and $$Lip(\alpha ,\beta )$$ in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and $$(C, \gamma , \delta )$$ means.

## 1 Introduction

The study of various summability means of double Fourier series have been done by several authors, for example, Chow , Sharma , Łenski , and Ustina . Dealing with the first arithmetic means of double Fourier series, Hasegawa  obtained the following:

### Theorem A

If a continuous function $$f(x, y)$$ of period 2π with respect to both x and y belongs to $$Lip (\alpha , \beta )$$, where $$0<\alpha <l$$ and $$0<\beta <1$$, then

$$\bigl\vert \sigma _{m,n}(x,y) -f(x,y) \bigr\vert = O \bigl( m^{-\alpha }+n^{- \beta } \bigr)$$

uniformly in $$(x, y)$$ as m and n independently tend to infinity.

If $$\alpha =\beta =1$$, then

$$\bigl\vert \sigma _{m,n}(x,y) -f(x,y) \bigr\vert = O \bigl( m^{-1} \log m +n^{-1} \log n \bigr)$$

uniformly in $$(x, y)$$ as m and n independently tend to infinity.

Siddiqui and Mohammadzadeh  investigated the approximation by Cesàro and B means of double Fourier series. Stepanets [13, 14] has established estimates of approximation for certain classes of periodic functions and differentiable periodic functions of two variables by linear methods of summation of their Fourier sums. Móricz and Shi  proved the following result for the approximation to continuous functions by Cesàro means of double Fourier series.

### Theorem B

If $$f \in E(\alpha , \beta )$$, $$0 < \alpha$$, $$\beta \leq 1$$, $$\gamma , \delta \geq 0$$, then

\begin{aligned} \bigl\| \sigma _{mn}^{\gamma \delta } (f,x,y) - f(x,y\bigr\| =& O \biggl( \frac{1}{(m+1)^{\alpha }}+ \frac{1}{(n+1)^{\beta }} \biggr) \quad \textit{if } 0 < \alpha , \beta \leq 1, \\ =& O \biggl( \frac{1}{(m+1)^{\alpha }}+ \frac{\log (n+2)}{(n+1)} \biggr) \quad \textit{if } 0 < \alpha < \beta = 1, \\ =& O \biggl( \frac{\log (m+2)}{(m+1)}+ \frac{\log (n+2)}{(n+1)} \biggr) \quad \textit{if } \alpha = \beta = 1. \end{aligned}

The degree of approximation using Gauss–Weierstrass integrals was also investigated by Khan and Ram . Recently, error and bounds of certain bivariate functions by almost Euler means of double Fourier series for the functions of Lipschitz and Zygmund classes was estimated by Rathor and Singh . To find the approximation of functions of two-dimensional torus, in this paper, we obtain a new estimate for trigonometric approximation of functions $$f(x,y)$$ of generalized Lipschitz class by double Hausdorff matrix summability method of double Fourier series. For other summability methods of approximation, see  and .

## 2 Definitions and preliminaries

Let $$\sum_{m=0}^{\infty }\sum_{n=0}^{\infty } g_{m,n}$$ be double series with the sequence of $$(m,n)$$th partial sums

$$s_{m,n}=\sum_{j=0}^{m}\sum _{k=0}^{n} g_{j,k}.$$

A double Hausdorff matrix has the entries

$$h_{m,n}^{j,k}= \binom{m }{j} \binom{n }{k} \Delta ^{m-j}_{1} \Delta ^{n-k}_{2} \mu _{j,k},$$

where $$\{ \mu _{j,k} \}$$ is any real or complex sequence, and

$$\Delta ^{m-j}_{1} \Delta ^{n-k}_{2} \mu _{j,k} = \sum_{w=0}^{m-j} \sum _{z=0}^{n-k} (-1)^{j+k} \binom{m-j }{w} \binom{n-k }{z} \mu _{j+w,k+z} .$$

If $$t_{m,n}^{H} = \sum_{j=0}^{m}\sum_{k=0}^{n} h_{m,n}^{j,k} s_{j,k} \rightarrow g$$ as $$m \rightarrow \infty$$ and $$n \rightarrow \infty$$, then $$\sum_{m=0}^{\infty }\sum_{n=0}^{\infty } g_{m,n}$$ is said to be summable to the sum g by the double Hausdorff matrix summability method .

A necessary and sufficient condition for double Hausdorff matrix summability method to be regular is there exists a function $$\chi (s,t) \in BV[0,1]\times [0,1]$$ such that

$$\int _{0}^{1} \int _{0}^{1} \bigl\vert d \chi (s,t) \bigr\vert < \infty$$

and

$$\mu _{m,n} = \int _{0}^{1} \int _{0}^{1} s^{m} t^{n} \,d \chi (s,t),$$

where $$\chi (s,0)=\chi (s,0^{+})=\chi (0^{+},t)=\chi (0,t) = 0$$, $$0\leq s$$, $$t \leq 1$$, and $$\chi (1,1)-\chi (1,0)-\chi (0,1)+\chi (0,0) = 1$$ .

It is easy to see that the absolute value of the measure $$d \chi (s,t)$$ can me majorized by $$K_{1} K_{2} \,ds \,dt$$ for some constants $$K_{1}$$ and $$K_{2}$$ (see ).

The important particular cases of double Hausdorff matrix summability means are as follows:

1. 1

Almost Euler summability means ($$(E,q_{1},q_{2})$$ means) if $$\mu _{m,n} = \frac{1}{(1+q_{1})^{m}}\frac{1}{(1+q_{2})^{n}}$$.

2. 2

$$(E,1,1)$$ means if $$q_{1}=1$$ and $$q_{2}=1$$ in $$(E,q_{1},q_{2})$$ means.

3. 3

$$(C, \gamma , \delta )$$ means if $$\mu _{m,n} = \frac{1}{A^{\gamma }_{m}}\frac{1}{A^{\delta }_{n}}$$, where $$\gamma , \delta \geq -1$$ and $$A^{\gamma }_{m} = \binom{{\gamma +m} }{m }$$, $$A^{\delta }_{n} = \binom{{\delta +n} }{n }$$.

4. 4

$$(C,1,1)$$ means if $$\gamma =\delta =1$$ in $$(C, \gamma , \delta )$$ means.

Let $$f(x,y)$$ be a Lebesgue-integrable function of period 2π with respect to both variables x and y and summable in the fundamental square $$Q:(-\pi ,\pi ) \times (-\pi ,\pi )$$. The double Fourier series of $$f(x,y)$$ is given by

\begin{aligned} f(x,y)&=\sum_{m=0}^{\infty } \sum_{n=0}^{\infty } \lambda _{m,n} [ a_{m,n} \cos mx \cos ny +b_{m,n} \sin mx \cos ny \\ &\quad{} + c_{m,n} \cos mx \sin ny + d_{m,n} \sin mx \cos ny ] \end{aligned}
(1)

with $$(m,n)$$th partial sums $$s_{m,n}(f;(x,y))$$, where

\begin{aligned}& \lambda _{m,n}= \textstyle\begin{cases} 1/4 & \text{for } m=n=0, \\ 1/2 & \text{for } m>0, n=0 \mbox{ and } m=0, n>0, \\ 1 & \text{for } m>0, n>0 , \end{cases}\displaystyle \\& a_{m,n}=\pi ^{-2} \iint _{Q} f(x,y) \cos mx \cos ny \,dx \,dy, \end{aligned}

and similar expressions for $$b_{m,n}$$, $$c_{m,n}$$, and $$d_{m,n}$$ .

We define the $$L^{r}$$ norm by

$$\Vert f \Vert _{r}= \textstyle\begin{cases} \{ \frac{1}{4\pi } \int _{0}^{2\pi } \int _{0}^{2\pi } \vert f(x,y) \vert ^{r} \,dx \,dy \} ^{1/r}, & r\geq 1, \\ \operatorname*{ess\,sup}_{0\leq x,y \leq 2\pi } \vert f(x,y) \vert , & r=\infty . \end{cases}$$

The degree of approximation of a function $$f :\mathbb{R}^{2} \rightarrow \mathbb{R}$$ by a trigonometric polynomial 

\begin{aligned} t_{m,n}(x,y)&=\sum _{j=0}^{m}\sum_{k=0}^{n} \lambda _{m,n} [ a_{j,k} \cos mx \cos ny +b_{j,k} \sin mx \cos ny \\ &\quad {}+ c_{j,k} \cos mx \sin ny + d_{j,k} \sin mx \cos ny ] \end{aligned}

of order $$(m+n)$$ is defined by

$$E_{m,n}\bigl(f,L^{r}\bigr) =\min_{0\leq x,y \leq2\pi} \Vert t_{m,n}-f \Vert _{r} .$$

A function $$f :\mathbb{R}^{2} \rightarrow \mathbb{R}$$ of two variables x and y is said to belong to the class $$Lip(\alpha ,\beta )$$  if

$$\bigl\vert f(x+u,y+v)-f(x,y) \bigr\vert =O\bigl( \vert u \vert ^{\alpha } + \vert v \vert ^{\beta }\bigr), \quad 0< \alpha \leq 1, 0< \beta \leq 1,$$

to the class $$Lip ((\alpha ,\beta );r )$$ if

$$\biggl\{ \frac{1}{4\pi } \int _{0}^{2\pi } \int _{0}^{2\pi } \bigl\vert f(x+u,y+v)-f(x,y) \bigr\vert ^{r} \,dx \,dy \biggr\} ^{1/r}= O \bigl( \vert u \vert ^{\alpha } + \vert v \vert ^{\beta } \bigr),\quad r\geq 1,$$

and to the class $$Lip ((\xi _{1},\xi _{2});r )$$ if

$$\biggl\{ \frac{1}{4\pi } \int _{0}^{2\pi } \int _{0}^{2\pi } \bigl\vert f(x+u,y+v)-f(x,y) \bigr\vert ^{r} \,dx \,dy \biggr\} ^{1/r}= O \bigl(\xi _{1}(u) + \xi _{2}(v) \bigr),\quad r\geq 1,$$

where $$\xi _{1}$$ and $$\xi _{2}$$ are moduli of continuity, that is, nonnegative nondecreasing continuous functions such that $$\xi _{1}(0) =\xi _{2}(0) = 0$$, $$\xi _{1}(u_{1} + u_{2}) \le \xi _{1}(u_{1}) + \xi _{1}(u_{2})$$, and $$\xi _{2}(v_{1} + v_{2}) \le \xi _{2}(v_{1}) + \xi _{2}(v_{2})$$.

If $$\xi _{1}(u)=u^{\alpha }$$ and $$\xi _{2}(v)=v^{\beta }$$, $$0<\alpha \leq 1$$, $$0 < \beta \leq 1$$, then the class $$Lip ((\xi _{1},\xi _{2});r )$$ coincides with $$Lip ((\alpha ,\beta );r )$$. As $$r \rightarrow \infty$$, $$Lip ((\alpha ,\beta );r )$$ reduces to $$Lip(\alpha ,\beta )$$. Clearly, $$Lip(\alpha ,\beta ) \subseteq Lip ((\alpha ,\beta );r ) \subseteq Lip ((\xi _{1},\xi _{2});r )$$.

We define the forward difference operator Δ as $$\Delta \mu _{k} = \mu _{k} - \mu _{k+1}$$; also, $$\Delta ^{n+1}\mu _{k}=\Delta (\Delta ^{n} \mu _{k} )$$, $$k\geq 0$$. We denote

\begin{aligned}& \begin{aligned} \phi (u,v)&=(1/4) \bigl[f(x+u,y+v)+f(x+u,y-v) +f(x-u,y+v)+ f(x-u,y-v) \\ &\quad{} -4f(x,y) \bigr], \end{aligned} \\& M_{m}^{H}(u)= \frac{K_{1}}{2\pi }\sum _{j=0}^{m} \int _{0}^{1} \binom{m }{j} s^{j} (1-s)^{m-j} \,ds \frac{\sin (j+\frac{1}{2} )u}{\sin \frac{u}{2}}, \\& K_{n}^{H}(v) = \frac{K_{2}}{2\pi }\sum _{k=0}^{n} \int _{0}^{1} \binom{N }{K} t^{k} (1-t)^{n-k} \,dt \frac{\sin (k+\frac{1}{2} )v}{\sin \frac{v}{2}}. \end{aligned}

## 3 Result

The object of this paper is obtaining the degree of approximation of functions $$f(x,y)$$ of generalized Lipschitz class by double Hausdorff matrix summability means of its double Fourier series:

### Theorem 1

If $$f(x,y)$$ is a 2π periodic function with respect to both variables x and y, Lebesgue integrable in $$(-\pi ,\pi )\times (-\pi ,\pi )$$ and belonging to the class $$Lip ((\xi _{1}, \xi _{2});r )$$ ($$r \geq 1$$), then the degree of approximation of $$f(x,y)$$ by double Hausdorff matrix summability means

$$t_{m,n}^{H}= \sum_{j=0}^{m} \sum_{k=0}^{n} \int _{0}^{1} \int _{0}^{1} \binom{m }{j} \binom{n }{k} s^{j}(1-s)^{m-j} t^{k}(1-t)^{n-k} \,d \chi (s,t) s_{j,k}$$

of double Fourier series (1) satisfies

$$\begin{gathered} \bigl\Vert t_{m,n}^{H} - f \bigr\Vert _{r} = O \biggl(\frac{1}{(m+1)} \int _{\frac{1}{m+1}}^{\pi } \frac{ \xi _{1}(u)}{u^{2}} \,du + \frac{1}{(n+1)} \int ^{\pi }_{\frac{1}{n+1}} \frac{\xi _{2}(v)}{v^{2}} \,dv \biggr) \\ \quad \textit{for } m,n=0,1,2,\dots . \end{gathered}$$
(2)

## 4 Lemmas

For the proof of our theorems, we need the following lemmas.

### Lemma 1

$$\vert M_{m}^{H}(u) \vert = O (m+1 )$$ for $$0< u \leq \frac{1}{m+1}$$, and $$\vert K_{n}^{H}(v) \vert = O (n+1 )$$ for $$0< v \leq \frac{1}{n+1}$$.

### Proof

Since $$\vert \sin mu \vert \leq mu$$ for $$0< u\leq \frac{1}{m+1}$$ and $$\sin (u/2)\geq (u/\pi )$$, we have

\begin{aligned} \bigl\vert M_{m}^{H}(u) \bigr\vert =& \Biggl\vert \frac{K_{1}}{2\pi }\sum_{j=0}^{m} \int _{0}^{1} \binom{m }{j} s^{j} (1-s)^{m-j} \,ds \frac{\sin (j+\frac{1}{2} )u}{\sin \frac{u}{2}} \Biggr\vert \\ =& \frac{K_{1}}{2\pi }\sum_{j=0}^{m} \int _{0}^{1} \binom{m }{j} s^{j} (1-s)^{m-j} \,ds \frac{ \vert \sin (j+\frac{1}{2} )u \vert }{ \vert \sin \frac{u}{2} \vert } \\ \leq & \frac{K_{1}}{2\pi }\sum_{j=0}^{m} \int _{0}^{1} \binom{m }{j} s^{j} (1-s)^{m-j} \,ds \frac{ (j+ \frac{1}{2} )u }{ \vert \frac{u}{\pi } \vert } \\ =& K_{1} \pi \biggl(m+ \frac{1}{2} \biggr) \int _{0}^{1} \sum_{j=0}^{m} \binom{m }{j} s^{j} (1-s)^{m-j} \,ds \\ =& K_{1} \pi \biggl(m+ \frac{1}{2} \biggr) \int _{0}^{1} ( s+1-s)^{m} \,ds \\ =& O ( m+1 ). \end{aligned}

Similarly, for $$0< v \leq \frac{1}{n+1}$$,

$$\big\vert K_{n}^{H}(v) \big\vert = O ( n+1 ).$$

□

### Lemma 2

$$\vert M_{m}^{H}(u) \vert = O (\frac{1}{(j+1)u^{2}} )$$ for $$\frac{1}{m+1} < u \leq \pi$$, and $$\vert K_{n}^{H}(v) \vert = O (\frac{1}{(k+1)v^{2}} )$$ for $$\frac{1}{n+1} < v \leq \pi$$.

### Proof

Since $$\sin (m+1) u \leq 1$$ for $$\frac{1}{m+1} < u \leq \pi$$ and $$\sin (u/2)\geq (u/\pi )$$, we get

\begin{aligned} \Biggl\vert \sum_{j=0}^{m} \int _{0}^{1} \binom{m }{j} s^{j} (1-s)^{m-j} e^{i (j+\frac{1}{2} )u} \,ds \Biggr\vert =& \int _{0}^{1} e^{iu/2} \sum _{j=0}^{m} \binom{m }{j} s^{j} (1-s)^{m-j} e^{iju} \,ds \\ =& \int _{0}^{1} e^{iu/2} \bigl(1-s+s e^{iu} \bigr)^{m} \,ds \\ =& O \biggl(\frac{ 1}{(m+1)} \biggr) \biggl( \frac{e^{iu/2} (e^{i(m+1)u}-1 )}{e^{iu}-1} \biggr). \end{aligned}

Equating the imaginary parts of both sides, we get

$$\Biggl\vert \sum_{j=0}^{m} \int _{0}^{1} \binom{m }{j} s^{k} (1-s)^{m-j} \sin \biggl(k+\frac{1}{2} \biggr) \,ds \Biggr\vert = O \biggl( \frac{1}{(m+1)u} \biggr).$$

Therefore

\begin{aligned} \bigl\vert M_{m}^{H}(u) \bigr\vert =& \Biggl\vert \frac{K_{1}}{2\pi } \sum_{j=0}^{m} \int _{0}^{1} \binom{ m }{j } s^{j} (1-s)^{m-j} \frac{\sin (j+\frac{1}{2} )u}{\sin \frac{u}{2}} \,ds \Biggr\vert \\ \leq & \frac{K_{1}}{2 u} \Biggl\vert \sum_{j=0}^{m} \int _{0}^{1} \binom{ m }{j } s^{j} (1-s)^{m-j} \sin \biggl(j+\frac{1}{2} \biggr)u \,ds \Biggr\vert \\ =& O \biggl(\frac{1}{(m+1) u^{2}} \biggr). \end{aligned}

Similarly, for $$\frac{1}{n+1} < v \leq \pi$$,

$$\big\vert K_{n}^{H}(v) \big\vert = O \biggl(\frac{1}{(n+1)v^{2}} \biggr).$$

□

### Lemma 3

If $$f(x,y)\in Lip ((\xi _{1},\xi _{2});r )$$ ($$r\geq 1$$), then $$\Vert \phi (u,v)) \Vert _{r} = O ( \xi _{1}(u) + \xi _{2}(v) )$$.

### Proof

Clearly,

\begin{aligned}& \begin{aligned} \bigl\vert \phi (u,v) \bigr\vert &= \frac{1}{4} \bigl\vert f(x+u,y+v)+ f(x+u,y-v) +f(x-u,y+v) +f(x-u,y-v)-4f(x,y) \bigr\vert \\ &\leq \frac{1}{4} \bigl[ \bigl\vert f(x+u,y+v)-f(x,y) \bigr\vert + \bigl\vert f(x+u,y-v)-f(x,y) \bigr\vert \\ &\quad {} + \bigl\vert f(x-u,y+v)-f(x,y) \bigr\vert + \bigl\vert f(x-u,y-v)-f(x,y) \bigr\vert \bigr], \end{aligned} \\& \begin{aligned} \bigl\Vert \phi (u,v) \bigr\Vert _{r} &\leq \frac{1}{4} \bigl[ \bigl\Vert f(x+u,y+v)-f(x,y) \bigr\Vert _{r} + \bigl\Vert f(x+u,y-v)-f(x,y) \bigr\Vert _{r} \\ &\quad {} + \bigl\Vert f(x-u,y+v)-f(x,y) \bigr\Vert _{r} + \bigl\Vert f(x-u,y-v)-f(x,y) \bigr\Vert _{r} \bigr] \\ &= O \bigl( \xi _{1}(u)+\xi _{2}(v) \bigr). \end{aligned} \end{aligned}

□

## 5 Proof of Theorem 1

The $$(m,n)$$th partial sum of the double Fourier series (1) is given by

$$s_{m,n}\bigl(f;(x,y)\bigr)-f(x,y)=\frac{1}{4\pi ^{2}} \int _{0}^{\pi } \int _{0}^{\pi } \phi (u,v) \frac{\sin (m+\frac{1}{2})u \sin (n+\frac{1}{2})v }{\sin \frac{u}{2} \sin \frac{v}{2}} \,du \,dv.$$

Denoting the double Hausdorff matrix sums of $$s_{m,n}$$ by $$t_{m,n}^{H}$$, we have

\begin{aligned}& \begin{aligned}[b] t_{m,n}^{H} (x,y) -f(x,y) &= \sum _{j=0}^{m} \sum_{k=0}^{n} h_{m,n}^{j,k} \bigl\{ s_{j,k}\bigl(f;(x,y) \bigr)-f(x,y) \bigr\} \\ &= \int _{0}^{\pi } \int _{0}^{\pi } \phi (u,v) \sum _{j=0}^{m} \sum_{k=0}^{n} h_{m,n}^{j,k} \frac{\sin (j+\frac{1}{2})u \sin (k+\frac{1}{2})v }{\sin \frac{u}{2} \sin \frac{v}{2}} \,du \,dv \\ &= \int _{0}^{\pi } \int _{0}^{\pi } \phi (u,v) M_{m}^{H}(u) K_{n}^{H}(v) \,du \,dv, \end{aligned} \end{aligned}
(3)
\begin{aligned}& \bigl\Vert t_{m,n}^{H} - f \bigr\Vert _{r} = \int _{0}^{\pi } \int _{0}^{\pi } \bigl\Vert \phi (u,v) \bigr\Vert _{r} M_{m}^{H}(u) K_{n}^{H}(v) \,du \,dv \\& \hphantom{\bigl\Vert t_{m,n}^{H} - f \bigr\Vert _{r}}= \biggl( \int _{0}^{\frac{1}{m+1}} \int _{0}^{ \frac{1}{n+1}} + \int _{0}^{\frac{1}{m+1}} \int ^{\pi }_{ \frac{1}{n+1}}+ \int ^{\pi }_{\frac{1}{m+1}} \int _{0}^{ \frac{1}{n+1}} + \int ^{\pi }_{\frac{1}{m+1}} \int ^{ \pi }_{\frac{1}{n+1}} \biggr) \\ \end{aligned}
(4)
\begin{aligned}& \hphantom{\bigl\Vert t_{m,n}^{H} - f \bigr\Vert _{r}} \quad \bigl\Vert \phi (u,v) \bigr\Vert _{r} M_{m}^{H}(u) K_{n}^{H}(v) \,du \,dv \\& \hphantom{\bigl\Vert t_{m,n}^{H} - f \bigr\Vert _{r}}= I_{1}+I_{2}+I_{3}+I_{4}, \quad \text{say}. \end{aligned}
(5)

Using Lemmas 1 and 3, we obtain

\begin{aligned} \vert I_{1} \vert =& \int _{0}^{\frac{1}{m+1}} \int _{0}^{ \frac{1}{n+1}} \bigl\Vert \phi (u,v) \bigr\Vert _{r} M_{m}^{H}(u) K_{n}^{H}(v) \,du \,dv \\ =& O \biggl( \int _{0}^{\frac{1}{m+1}} \int _{0}^{ \frac{1}{n+1}} \bigl(\xi _{1}(u)+\xi _{2}(v)\bigr) (m+1) (n+1) \,du \,dv \biggr) \\ =& O \biggl( (m+1) (n+1) \int _{0}^{\frac{1}{m+1}} \int _{0}^{\frac{1}{n+1}} \bigl(\xi _{1}(u)+\xi _{2}(v)\bigr) \,du \,dv \biggr) \\ =&O \biggl[(m+1) (n+1) \biggl( \int _{0}^{\frac{1}{m+1}} \int _{0}^{\frac{1}{n+1}} \xi _{1}(u)\,du \,dv + \int _{0}^{ \frac{1}{m+1}} \int _{0}^{\frac{1}{n+1}} \xi _{2}(v)\,du \,dv \biggr) \biggr] \\ =&O \biggl[(m+1) (n+1) \biggl( \int _{0}^{\frac{1}{m+1}} \frac{\xi _{1}(u)}{n+1} \,du + \int _{0}^{\frac{1}{m+1}} \frac{\xi _{2} (\frac{1}{(n+1)} )}{n+1} \,dv \biggr) \biggr] \\ =& O \biggl[(m+1) (n+1) \biggl( \frac{\xi _{1} (\frac{1}{(m+1)} )}{(m+1)(n+1)} + \frac{\xi _{2} (\frac{1}{(n+1)} )}{(m+1)(n+1)} \biggr) \biggr] \\ =& O \biggl( \xi _{1} \biggl(\frac{1}{m+1} \biggr) +\xi _{2} \biggl( \frac{1}{n+1} \biggr) \biggr). \end{aligned}

Again by Lemmas 13, we have

\begin{aligned} \vert I_{2} \vert =&O \biggl[ \int _{0}^{\frac{1}{m+1}} \int ^{ \pi }_{\frac{1}{n+1}} \bigl(\xi _{1}(u)+\xi _{2}(v)\bigr) \frac{(m+1)}{(n+1)v^{2}} \,du \,dv \biggr] \\ =& O \biggl[ \frac{(m+1)}{(n+1)} \biggl( \int _{0}^{ \frac{1}{m+1}} \xi _{1}(u) \,du \int ^{\pi }_{\frac{1}{n+1}} \frac{dv}{v^{2}} + \int _{0}^{\frac{1}{m+1}} \,du \int ^{ \pi }_{\frac{1}{n+1}} \frac{\xi _{2}(v)}{v^{2}} \,dv \biggr) \biggr] \\ =& O \biggl[ \frac{(m+1)}{(n+1)} \biggl(\xi _{1} \biggl( \frac{1}{m+1} \biggr)\frac{1}{(m+1)} \biggl((n+1)-\frac{1}{\pi } \biggr) + \frac{1}{(m+1)} \int ^{\pi }_{\frac{1}{n+1}} \frac{\xi _{2}(v)}{v^{2}} \,dv \biggr) \biggr] \\ =& O \biggl( \xi _{1} \biggl(\frac{1}{m+1} \biggr) + \frac{1}{(n+1)} \int ^{\pi }_{\frac{1}{n+1}} \frac{\xi _{2}(v)}{v^{2}} \,dv \biggr). \end{aligned}
(6)

Similarly,

\begin{aligned} \vert I_{3} \vert =&O \biggl[ \int _{\frac{1}{m+1}}^{\pi } \int ^{ \frac{1}{n+1}}_{0} \bigl(\xi _{1}(u)+\xi _{2}(v)\bigr) \frac{(n+1)}{(m+1)u^{2}} \,du \,dv \biggr] \\ =& O \biggl[ \frac{(n+1)}{(m+1)} \biggl( \int _{\frac{1}{m+1}}^{ \pi } \frac{ \xi _{1}(u)}{u^{2}} \,du \int ^{\frac{1}{n+1}}_{0} \,dv + \int _{\frac{1}{m+1}}^{\pi } \frac{du}{u^{2}} \int ^{ \frac{1}{n+1}}_{0} \xi _{2}(v) \,dv \biggr) \biggr] \\ =& O \biggl(\frac{1}{(m+1)} \int _{\frac{1}{m+1}}^{\pi } \frac{ \xi _{1}(u)}{u^{2}} \,du + \xi _{2} \biggl(\frac{1}{n+1} \biggr) \biggr). \end{aligned}
(7)

Also, using Lemmas 2 and 3, we get

\begin{aligned} \vert I_{4} \vert =& O \biggl[ \int _{\frac{1}{m+1}}^{\pi } \int ^{ \pi }_{\frac{1}{n+1}} \bigl(\xi _{1}(u)+\xi _{2}(v)\bigr) \frac{1}{(m+1)u^{2}} \frac{1}{(n+1)v^{2}}\,du \,dv \biggr] \\ =& O \biggl[ \frac{1}{(m+1)(n+1)} \biggl( \int ^{\pi }_{ \frac{1}{m+1}} \frac{\xi _{1}}{u^{2}} \,du \int _{\frac{1}{n+1}}^{ \pi } \frac{1}{v^{2}} \,dv+ \int ^{\pi }_{\frac{1}{m+1}} \frac{1}{u^{2}} \,du \int ^{\pi }_{\frac{1}{n+1}} \frac{\xi _{2}}{v^{2}} \,dv \biggr) \biggr] \\ =& O \biggl(\frac{1}{(m+1)} \int _{\frac{1}{m+1}}^{\pi } \frac{ \xi _{1}(u)}{u^{2}} \,du + \frac{1}{(n+1)} \int ^{\pi }_{ \frac{1}{n+1}} \frac{\xi _{2}(v)}{v^{2}} \,dv \biggr). \end{aligned}
(8)

Next,

\begin{aligned}& \begin{aligned} \frac{1}{(m+1)} \int ^{\pi }_{\frac{1}{m+1}} \frac{\xi _{1}(u)}{u^{2}} \,du &\geq \frac{1}{(m+1)} \xi _{1} \biggl(\frac{1}{m+1} \biggr) \int ^{ \pi }_{\frac{1}{m+1}} \frac{1}{u^{2}} \,dt \\ &=\frac{1}{(m+1)} \xi _{1} \biggl(\frac{1}{m+1} \biggr) \biggl\{ - \frac{1}{u} \biggr\} ^{\pi }_{\frac{1}{m+1}} \\ &=\xi _{1} \biggl(\frac{1}{m+1} \biggr) \biggl\{ 1- \frac{1}{(m+1)\pi } \biggr\} \\ &\geq \frac{1}{2} \xi _{1} \biggl(\frac{1}{m+1} \biggr), \end{aligned} \\& \text{or}\quad \xi _{1} \biggl(\frac{1}{m+1} \biggr) = O \biggl( \frac{1}{(m+1)} \int ^{\pi }_{\frac{1}{m+1}} \frac{\xi _{1}(u)}{u^{2}} \,dt \biggr) . \end{aligned}
(9)

Similarly,

$$\xi _{2} \biggl(\frac{1}{(n+1)} \biggr) = O \biggl( \frac{1}{(n+1)} \int ^{\pi }_{\frac{1}{n+1}} \frac{\xi _{2}(v)}{v^{2}} \,dt \biggr).$$
(10)

Combining equations (5)–(10), we have

$$\bigl\Vert t_{m,n}^{H} - f \bigr\Vert _{r} = O \biggl(\frac{1}{(m+1)} \int _{\frac{1}{m+1}}^{\pi } \frac{ \xi _{1}(u)}{u^{2}} \,du + \frac{1}{(n+1)} \int ^{\pi }_{\frac{1}{n+1}} \frac{\xi _{2}(v)}{v^{2}} \,dv \biggr) .$$

This completes the proof of Theorem 1.

## 6 Corollaries

From the main theorem we derive the following corollaries.

### Corollary 1

If $$f(x,y)$$ is a 2π periodic function with respect to both variables x and y, Lebesgue integrable in $$(-\pi ,\pi )\times (-\pi ,\pi )$$ and belonging to the class $$Lip ((\alpha , \beta );r )$$ ($$r \geq 1$$), then the degree of approximation of $$f(x,y)$$ by means $$t_{m,n}^{H}$$ of double Fourier series (1) satisfies

\begin{aligned}& \bigl\Vert t_{m,n}^{H}-f \bigr\Vert _{r}= \textstyle\begin{cases} O ( (m+1)^{-\alpha }+(n+1)^{-\beta } ), & 0< \alpha < 1, 0< \beta < 1, \\ O ( (m+1)^{-\alpha }+ \frac{\log (n+1)\pi }{(n+1)} ), & 0< \alpha < 1, \beta =1, \\ O ( \frac{\log (m+1)\pi }{(m+1)} +(n+1)^{-\beta } ), & \alpha =1, 0< \beta < 1, \\ O ( \frac{\log (m+1)\pi }{(m+1)} + \frac{\log (n+1)\pi }{(n+1)} ), & \alpha =\beta =1, \end{cases}\displaystyle \end{aligned}

for $$m,n=0,1,2,\dots$$.

### Corollary 2

If $$f(x,y)$$ is a 2π periodic function with respect to both variables x and y, Lebesgue integrable in $$(-\pi ,\pi )\times (-\pi ,\pi )$$ and belonging to the class $$Lip(\alpha ,\beta )$$, then the degree of approximation of $$f(x,y)$$ by double Hausdorff matrix summability means $$t_{m,n}^{H}$$ of double Fourier series (1) satisfies

\begin{aligned}& \bigl\Vert t_{m,n}^{H}-f \bigr\Vert _{\infty }= \textstyle\begin{cases} O ( (m+1)^{-\alpha }+(n+1)^{-\beta } ), & 0< \alpha < 1, 0< \beta < 1, \\ O ( (m+1)^{-\alpha }+ \frac{\log (n+1)\pi }{(n+1)} ), & 0< \alpha < 1, \beta =1, \\ O ( \frac{\log (m+1)\pi }{(m+1)} +(n+1)^{-\beta } ), & \alpha =1, 0< \beta < 1, \\ O ( \frac{\log (m+1)\pi }{(m+1)} + \frac{\log (n+1)\pi }{(n+1)} ), & \alpha =\beta =1, \end{cases}\displaystyle \end{aligned}

for $$m,n=0,1,2,\dots$$.

### Corollary 3

If $$f(x,y)$$ is a 2π periodic function with respect to both variables x and y, Lebesgue integrable in $$(-\pi ,\pi )\times (-\pi ,\pi )$$ and belonging to the class $$Lip ((\xi _{1}, \xi _{2});r )$$, then the degree of approximation of $$f(x,y)$$ by almost Euler summability means

$$t_{m,n}^{E}= \frac{1}{(1+q_{1})^{m}} \frac{1}{(1+q_{2})^{n}}\sum _{j=0}^{m}\sum_{k=0}^{n} \binom{m }{j} \binom{n }{k} q_{1}^{m-j} q_{2}^{n-k} s_{j,k}$$

of double Fourier series (1) satisfies

$$\bigl\Vert t_{m,n}^{E} - f \bigr\Vert _{r} = O \biggl(\frac{1}{(m+1)} \int _{\frac{1}{m+1}}^{\pi } \frac{ \xi _{1}(u)}{u^{2}} \,du + \frac{1}{(n+1)} \int ^{\pi }_{\frac{1}{n+1}} \frac{\xi _{2}(v)}{v^{2}} \,dv \biggr)$$

for $$m,n=0,1,2,\dots$$.

### Corollary 4

For $$\gamma , \delta \geq -1$$, the Cesàro means $$\sigma _{m,n}^{\gamma , \delta }$$ of order γ and δ, that is, $$(C, \gamma , \delta )$$ means of double Fourier series, are given by

$$\sigma _{m,n}^{\gamma ,\delta }= \frac{1}{A^{\gamma }_{m}}\frac{1}{A^{\delta }_{n}}\sum _{j=0}^{m}\sum_{k=0}^{n} A_{m-j}^{\gamma -1} A_{n-k}^{\delta -1} s_{j,k},$$

where $$A^{\gamma }_{m} = \binom{{\gamma +m} }{m }$$ and $$A^{\delta }_{n} = \binom{{\delta +n} }{n }$$.

If $$f(x,y)$$ is a 2π periodic function with respect to both variables x and y, Lebesgue integrable in $$(-\pi ,\pi )\times (-\pi ,\pi )$$ and belonging to the class $$Lip ((\xi _{1}, \xi _{2});r )$$, then the degree of approximation of $$f(x,y)$$ by $$(C, \gamma , \delta )$$ means of double Fourier series (1), satisfies

$$\bigl\Vert \sigma _{m,n}^{\gamma ,\delta } - f \bigr\Vert _{r} = O \biggl( \frac{1}{(m+1)} \int _{\frac{1}{m+1}}^{\pi } \frac{ \xi _{1}(u)}{u^{2}} \,du + \frac{1}{(n+1)} \int ^{\pi }_{ \frac{1}{n+1}} \frac{\xi _{2}(v)}{v^{2}} \,dv \biggr)$$

for $$m,n=0,1,2,\dots$$.

## 7 Conclusion

We established the degree of approximation of a function $$f(x,y)$$ belonging to the generalized Lipschitz class by double Hausdorff matrix summability means of its double Fourier series in the form of equation (2). If $$\xi _{1}=u^{\alpha }$$ and $$\xi _{2}=v^{\beta }$$, then Theorem 1 reduces to Corollary 1, and as $$r \rightarrow \infty$$, Corollary 1 reduces to Corollary 2. Independent proofs of Corollaries 14 can be developed along the same lines as that of Theorem 1. Results similar to Corollaries 3 and 4 can be derived for $$(E,1,1)$$ means and $$(C,1,1)$$ means of its double Fourier series. In this way, we can obtain some more different results by changing $$\xi _{1}$$, $$\xi _{2}$$, and $$\mu _{m,n}$$ under given conditions. For functions $$f(x,y)$$ belonging to the Zygmund classes $$Zyg(\alpha ,\beta )$$ and $$Zyg(\alpha ,\beta ;p)$$ discussed in , the degree of approximation using double Hausdorff matrix summability means and hence almost Euler means of its double Fourier series can be obtained similarly to Theorem 1.

Not applicable.

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Not applicable.

Not applicable.

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