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.

Introduction

The study of various summability means of double Fourier series have been done by several authors, for example, Chow [2], Sharma [11], Łenski [6], and Ustina [15]. Dealing with the first arithmetic means of double Fourier series, Hasegawa [4] 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 [12] 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 [8] 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 [5]. 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 [9]. 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 [1] and [7].

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 [15].

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 $\chi (1,1)-\chi (1,0)-\chi (0,1)+\chi (0,0) = 1$ [10].

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 [16]).

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

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

• 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 $(C,1,1)$ 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 $d_{m,n}$ [3].

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 [17]

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

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 $Lip(\alpha ,\beta )$ [4] 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

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

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)

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,

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

 □

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 1–3, 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.

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

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 1–4 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 [9], 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.