Inverse approximation and GBS of bivariate Kantorovich type sampling series

TL;DR

This paper establishes inverse approximation results for bivariate Kantorovich sampling series, analyzes their approximation rates in specific function spaces, and provides practical kernel examples for application.

Contribution

It introduces inverse theorems and approximation rate analysis for bivariate Kantorovich sampling series, extending the theoretical understanding of these operators.

Findings

Inverse approximation results for bivariate Kantorovich series

Rate of approximation in Bogel space for GBS operators

Examples of kernels applicable to the theory

Abstract

In this paper, we derive an inverse result for bivariate Kantorovich type sampling series for the space of all continuous functions with upto second order partial derivatives are continuous and bounded on $R^2.$ Further, we prove the rate of approximation in the Bogel space of continuous functions for the GBS (Generalized Boolean Sum) of these operators. Finally, we give some examples for the kernel to which the theory can be applied

Equations

$$(G_w^{\chi }f)(x,y)$$

$$m_{(p_1,p_2)}^{\eta }(\chi ,u,v):=\sum _{k=-\infty }^{\infty }\sum _{j=-\infty }^{\infty }\chi (u-k,v-j)(u-k{)}^{p_1}(v-j{)}^{p_2}$$

$$M_{(p_1,p_2)}^{\eta }(\chi ,u,v):=\underset{(x,y)\in {\mathbb{R}}^2}{\sup }\sum _{k=-\infty }^{\infty }\sum _{j=-\infty }^{\infty }|\chi (u-k,v-j)||u-k{|}^{p_1}|y-j{|}^{p_2}$$

$$M_{\eta }(\chi ):=\underset{|p|=\eta }{\max }{M}_{(p_1,p_2)}^{\eta }(\chi ).$$

$$(S_{w}f)(x,y)=\sum_{k\in Z}\sum_{j\in Z}\chi(wx-k,wy-j)\Bigg{[}w^{2}\int_{\frac{k}{w}}^{\frac{k+1}{w}}\int_{\frac{j}{w}}^{\frac{j+1}{w}}f(u,v)\ du\ dv\Bigg{]},\ \ \ \ \ \forall(x,y)\in\mathbb{R}^{2},$$

$$m_{(p_1,p_2)}^{\eta }(\chi ,u,v)=\{\begin{array}{ll}\text{0,}& \text{if,}\eta =1,2,\mathrm{3...},r-1\\ \text{c,}& \text{if,}\eta =r\end{array}$$

$$\Vert G_{w}f-f{\Vert }_{\infty }\le \frac{c}{r!}\frac{M_r(\chi )}{w^r}.H$$

$$M_r(\chi )=\underset{\eta =r}{\max }{M}_{(p_1,p_2)}^r(\chi ).$$

$$A_{r}=\Big{\|}\frac{\delta^{r}f}{\delta x^{r}}\Big{\|}_{\infty}\ and\ B_{r}=\Big{\|}\frac{\delta^{r}f}{\delta y^{r}}\Big{\|}_{\infty}.$$

$$|(G_{w}f)(x,y)-f(x,y)|$$

$$f(u,v)$$

$$=$$

$$\Vert G_{w}f-f{\Vert }_{\infty }\le \frac{c}{r!}\frac{M_r(\chi )}{w^r}.H$$

$$M_r(\chi )=\underset{\eta =r}{\max }{M}_{(p_1,p_2)}^r(\chi ).$$

$$(G_w{P}_{r-1})(x,y)=(P_{r-1})(x,y)$$

$$(S_{w}f)(x,y)=(G_{w}f)(x,y)+\frac{1}{2w}(G_w{f}_x)(x,y)+\frac{1}{2w}(G_w{f}_y)(x,y)+{\hat{R}}_2^w(x,y),$$

$$\displaystyle{\hat{R}_{2}^{w}(x,y)=\frac{1}{2!}\sum_{k\in\mathbb{Z}}\sum_{j\in\mathbb{Z}}\chi(wx-k,wy-j)\Bigg{[}w^{2}\int_{\frac{k}{w}}^{\frac{k+1}{w}}\int_{\frac{j}{w}}^{\frac{j+1}{w}}\Big{\{}f_{xx}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}}\bigg{(}u-\frac{k}{w}\bigg{)}^{2}$$

$$\displaystyle+2f_{xy}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}\bigg{(}u-\frac{k}{w}\bigg{)}\bigg{(}v-\frac{j}{w}\bigg{)}+f_{yy}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}\bigg{(}v-\frac{j}{w}\bigg{)}^{2}\Big{\}}dudv\Bigg{]},$$

$$f(u,v)$$

$$=$$

$$(S_{w}f)(x,y)$$

$$\big{|}\hat{R_{2}^{w}}(x,y)\big{|}\leq\sum_{k\in\mathbb{Z}}\sum_{j\in\mathbb{Z}}\big{|}\chi(wx-k,wy-j)\big{|}\Big{[}\frac{1}{6w^{2}}\mid f_{xx}\mid+\frac{1}{4w^{2}}\mid f_{xy}\mid+\frac{1}{6w^{2}}\mid f_{yy}\mid\Big{]}.$$

$$\mid \widehat{R_2^w}(x,y)\mid \le \frac{7M}{12w^2}{M}_{(0,0)}^0(\chi )<+\infty ,$$

$$\Vert S_{w}f-f{\Vert }_{\infty }=o(w^{-1})$$

$$(S_{w}f)(x,y)=(G_{w}f)(x,y)+\frac{1}{2w}(G_w{f}_x)(x,y)+\frac{1}{2w}(G_w{f}_y)(x,y)+{\hat{R}}_2^w(x,y)$$

$$\displaystyle w\Big{[}S_{w}f)(x,y)-f(x,y)\Big{]}$$

$$I_1$$

$$\lim_{w\rightarrow\infty}\mid I_{1}\mid\ \ \leq\lim_{w\rightarrow\infty}\Bigg{|}\frac{1}{2}(f_{x}+f_{y})+w\hat{R}_{2}^{w}\Bigg{|}.$$

$$f_x+f_y=0.$$

Full text

1 Introduction

The theory of generalized sampling series was first initiated by P. L. Butzer and his school Stens1 and Stens2 . Let $\chi:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ be a suitable kernel function. Then, the two dimensional generalized sampling series of a function $f:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ is defined by

[TABLE]

where $w>0$ and $(x,y)\in{\mathbb{R}}^{2}.$ These operators have great importance in the development of models for signal recovery. These type of operators have been studied by many authors (cf. bv1 ; BM* ; PLB2 ; PLB1 etc.).

The Kantorovich type generalizations of approximation operators is an important subject in approximation theory and they are the method to approximate Lebesgue integrable functions. In the last few decades, the Kantorovich modifications of several operators were constructed and their approximation behaviour have been studied. We mention some of the work in this direction e.g., GV3 ; PNAI ; gupta1 ; BM1 ; Tuncer ; gupta2 etc.

Bardaro et.al.BVBS , studied the rate of convergence of sampling Kantorovich operators in the general settings of Orlicz spaces. Danilo and Vinti COS , Bardaro BM extended their study in the multivariate setting and obtained the rate of convergence for functions in Orlicz spaces and also obtained the rate of approximation for the family of sampling Kantorovich operators in the uniform norm, for bounded and uniformly continuous, functions belonging to Lipschitz classes and for functions in Orlicz spaces in Dani . The non-linear version of sampling Kantorovich series has been studied in Dan , Zam and DV . Butzer and Stens studied the saturation results for the univariate generalised sampling operators in (Stens2 ). Recently, the inverse results for the Kantorovich type sampling operators have been derived for the spaces $C^{2}(\mathbb{R})$ and $C^{1}(\mathbb{R})$ in (GV3** ), (BCG ) respectively.

Let $C^{r}(\mathbb{R}^{2})$ denotes the space of all continuous functions with upto $r$-th order partial derivatives are continuous and bounded on $\mathbb{R}^{2}.$ Assume that $\chi\in C^{0}(\mathbb{R}^{2})$ be fixed. For any $\eta\in\mathbb{N}_{0}=\mathbb{N}\cup\{0\},$ $p=(p_{1},p_{2})\in\mathbb{N}_{0}^{2}$ with $|p|=p_{1}+p_{2}=\eta,$ we define the algebraic moments as,

[TABLE]

and the absolute moments by,

[TABLE]

and,

[TABLE]

We can easily see that, for $\xi,\eta\in\mathbb{N}_{0}$ with $\xi<\eta,$ $M_{\eta}(\chi)<+\infty$ implies that $M_{\xi}(\chi)<+\infty.$ Also note that, when $\chi$ is compactly supported then we have, $M_{\eta}(\chi)<\infty$, for every $\eta\in\mathbb{N}_{0}.$ (see bv1 )

The bivariate Kantorovich version of the generalized sampling series is defined as,

[TABLE]

where $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a locally integrable function, such that the above series is convergent for every $(x,y)\in\mathbb{R}^{2}$ and $\chi:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a kernel function.

We assume that the following conditions hold;

$(\textbf{K1})\ \ \chi$ is integrable and bounded at the origin.

(\textbf{K2})\ The series $\displaystyle{\sum_{k\in\mathbb{Z}}\sum_{j\in\mathbb{Z}}}\chi(u-k,v-j)=1,$ for every $(u,v)\in\mathbb{R}^{2},$

$(\textbf{K3})\ \ M_{(p_{1},p_{2})}^{\eta}(\chi)<+\infty$ , for some $(p_{1},p_{2})\in\mathbb{N}^{2}.$

Theorem 1.1

Let $\chi$ be the kernel satisfying the following moment condition,

[TABLE]

for every $(u,v)\in{\mathbb{R}}^{2}$ and some $r\in\mathbb{N}^{+}.$ Then, for every $f\in C^{r}(\mathbb{R}^{2})$, we have,

[TABLE]

where,

[TABLE]

Here, $\ H=\binom{r}{0}A_{r}+\binom{r}{1}A_{r-1}B_{1}+\binom{r}{2}A_{r-2}B_{2}+...+\binom{r}{r}B_{r},$ with

[TABLE]

Proof

. By using (1), we have

[TABLE]

Let $f\in C^{r}(\mathbb{R}^{2}).$ Now, by the Taylor’s formula in two variables with Lagrange’s form of remainder, we have

[TABLE]

$\mbox{where},(x,y),(u,v)\in\mathbb{R}^{2}\ \ and\ \theta_{u,x}\in(x,u)\ \mbox{and}\ \theta_{v,y}\in(y,v).$ Now, if we substitute $x=\frac{k}{w}\ \mbox{and}\ y=\frac{j}{w},\ \ \ k,j\in\mathbb{Z},\ w>0,$ for every u\in\Big{(}\frac{k}{w},\frac{k+1}{w}\Big{)},\\ v\in\Big{(}\frac{j}{w},\frac{j+1}{w}\Big{)} in the above formula, then,

[TABLE]

Now, we have

$|(G_{w}f)(x,y)-f(x,y)|$

[TABLE]

Now, using (3) and then taking the supremum norm on both sides, we obtain,

[TABLE]

where,

[TABLE]

This completes the proof.

Theorem 1.2

Let $\chi$ be kernel, which satisfies the moment condition (3) then,

[TABLE]

for every $w>0$, i.e., $(G_{w}f)$ maps a homogeneous polynomial of degree $(r-1)$ into a homogeneous polynomial of same degree, where $(P_{r-1})(x,y)$ denotes any homogeneous algebraic polynomial of degree $r-1.$

Proof

. Proceeding in a manner similar to the proof of Remark 2 in PLB2 , we can easily obtain the proof of this theorem.

Now, we derive the representation formula which relates the bivariate Kantorovich sampling series with the bivariate generalized sampling operators.

Theorem 1.3

Let $f\in C^{(2)}(\mathbb{R}^{2}).$ Then,

[TABLE]

where, the remainder $\hat{R}_{2}^{w}(x,y)$ of order $2$, is the absolutely convergent series for every $(x,y)\in\mathbb{R}^{2}$ as;

[TABLE]

where, $\theta_{k,w}(u)$ and $\theta_{j,w}(v)$ are measurable functions, such that $\frac{k}{w}<\theta_{k,w}(u)<\frac{k+1}{w}$ and $\frac{j}{w}<\theta_{j,w}(v)<\frac{j+1}{w},\ \ k,j\in\mathbb{Z},$ and for every u\in\Big{(}\frac{k}{w},\frac{k+1}{w}\Big{)},\\ v\in\Big{(}\frac{j}{w},\frac{j+1}{w}\Big{)},\ w>0.

Proof

. By the Taylor’s formula with Lagrange’s form of remainder for $r=2$, we have,

[TABLE]

where $(x,y)\ and\ (u,v)\in\mathbb{R}^{2}\ \ and\ \theta_{u,x}\in(x,u)\ \mbox{and}\ \theta_{v,y}\in(y,v).$ Now, if we substitute $x=\frac{k}{w}\ \mbox{and}\ y=\frac{j}{w},\ \ \ k,j\in\mathbb{Z},\ w>0,$ for every u\in\Big{(}\frac{k}{w},\frac{k+1}{w}\Big{)},v\in\Big{(}\frac{j}{w},\frac{j+1}{w}\Big{)} in the above formula, then $\frac{k}{w}<\theta_{u,\frac{k}{w}}:=\theta_{k,w}(u)<\frac{k+1}{w}$ and $\frac{j}{w}<\theta_{v,\frac{j}{w}}:=\theta_{j,w}(v)<\frac{j+1}{w}.$ So, we have

[TABLE]

$\mbox{Consider},\displaystyle w^{2}\int_{\frac{k}{w}}^{\frac{k+1}{w}}\int_{\frac{j}{w}}^{\frac{j+1}{w}}f(u,v)\ du\ dv$

[TABLE]

where, \displaystyle R_{2}^{w}=w^{2}\int_{\frac{k}{w}}^{\frac{k+1}{w}}\int_{\frac{j}{w}}^{\frac{j+1}{w}}\frac{1}{2!}\Bigg{[}f_{xx}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}\bigg{(}u-\frac{k}{w}\bigg{)}^{2}+2f_{xy}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}\\ \bigg{(}u-\frac{k}{w}\bigg{)}\bigg{(}v-\frac{j}{w}\bigg{)}+f_{yy}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}\bigg{(}v-\frac{j}{w}\bigg{)}^{2}\Bigg{]}dudv.

Now, by using (4) in the defintion of $(S_{w}f)(x,y)$, we have,

[TABLE]

where, \displaystyle\hat{R}_{2}^{w}(x,y)=\frac{1}{2!}\sum_{k\in\mathbb{Z}}\sum_{j\in\mathbb{Z}}\chi(wx-k,wy-j)\ \ \Bigg{[}w^{2}\int_{\frac{k}{w}}^{\frac{k+1}{w}}\int_{\frac{j}{w}}^{\frac{j+1}{w}}\Big{\{}f_{xx}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}\\ \bigg{(}u-\frac{k}{w}\bigg{)}^{2}+2f_{xy}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}\bigg{(}u-\frac{k}{w}\bigg{)}\bigg{(}v-\frac{j}{w}\bigg{)}\\ +f_{yy}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}\bigg{(}v-\frac{j}{w}\bigg{)}^{2}\Big{\}}dudv\Bigg{]}.

Indeed, the above series $\hat{R}_{2}^{w}(x,y)$ is absolutely convergent for every

$(x,y)\in\mathbb{R}^{2}$ and for every $w>0$ as,

\displaystyle\big{|}\hat{R}_{2}^{w}(x,y)\big{|}\leq\frac{1}{2!}\sum_{k\in\mathbb{Z}}\sum_{j\in\mathbb{Z}}|\chi(wx-k,wy-j)|\ \ \Big{|}w^{2}\int_{\frac{k}{w}}^{\frac{k+1}{w}}\int_{\frac{j}{w}}^{\frac{j+1}{w}}\Big{\{}f_{xx}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}\\ \bigg{(}u-\frac{k}{w}\bigg{)}^{2}+2f_{xy}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}\bigg{(}u-\frac{k}{w}\bigg{)}\bigg{(}v-\frac{j}{w}\bigg{)}\\ +f_{yy}\big{(}\theta_{k,w}(u),\theta_{j,w}(v)\big{)}\bigg{(}v-\frac{j}{w}\bigg{)}^{2}\Big{\}}dudv\Big{|}.

Thus, we have

[TABLE]

Since, all the second order partial derivatives are bounded, i.e., $\mid f_{xx}\mid\leq M_{1},$ $\mid f_{xy}\mid\leq M_{2},\ \mbox{and}\mid f_{yy}\mid\leq M_{3},$ we have,

[TABLE]

where, $M=\mbox{max}\{M_{1},M_{2},M_{3}\}.$

This is the desired resut.

Now, we prove the inverse result by using the representation formula, derived in Theorem $3.$

Theorem 1.4

Let $\chi$ be the kernel function satisfying the condition in (3) for every $(u,v)\in\mathbb{R}^{2}$ with $r=2.$ Let $f\in C^{2}(\mathbb{R}^{2})$ and suppose that;

[TABLE]

Then, $f=g(y-x)$ for some arbitrary function $g.$

Proof

. By the representation formula of Theorem $3$, we have

[TABLE]

Now,

[TABLE]

Now, we estimate $I_{1}$,

[TABLE]

Using the convergence results of $(G_{w}f_{x})$ and $(G_{w}f_{y})$, we obtain,

[TABLE]

Since, \displaystyle{\lim_{w\rightarrow\infty}}\big{[}(S_{w}f)(x,y)-f(x,y)\big{]}=0\ \ ,\ \displaystyle{\lim_{w\rightarrow\infty}}\big{[}(G_{w}f)(x,y)-f(x,y)\big{]}=0 and using (5), we have, $\displaystyle\lim_{w\longrightarrow\infty}|w\ \hat{R}_{2}^{w}(x,y)|=0,$ then the equation (6) gives,

[TABLE]

The solution of the above first order partial differential equation is given by,

[TABLE]

for some arbitrary function $g.$

Remark 1

If, in addition, $f$ satisfies the initial condition $f(0,y)=f_{0}(y)=y,$ then,

[TABLE]

i.e., if $f$ satisfies the initial condition $f(0,y)=f_{0}(y)=y,$ then $f$ turns out to be linear.

Theorem 1.5

Let $\chi$ be the kernel satisfying the moment condition (3) with $r\in N$ then, $S_{w}$ maps algebraic polynomials of degree atmost $r-1$ into algebraic polynomials of the same degree.

Proof

. By using Theorem $3$ and the condition (3), we can easily obtain the desired result.

2 Approximation in the Space of B$\ddot{o}$gel Continuous Functions

In recent years, the study of GBS operators of certain linear positive operators is an interesting topic in approximation theory and function theory. In this section, we shall give a generalization of the GBS operator for the $B$-continuous functions. For this, we shall introduce a GBS operator associated with the bivariate Kantorovich type sampling operators and study some of its smoothness properties. Karl B$\ddot{o}$gel BOG1 ; BOG2 introduced the concepts of $B$-continuous and $B$-differentiable functions. In approximation theory, the well-known Korovkin theorem is developed for B-continuous functions by Badea et al. in BAD1 ; BAD2 . In DOB , Dobrescu and Matei proved that any $B$-continuous function on a bounded interval can be approximated uniformly by boolean sum of bivariate Bernstein polynomials. The approximation properties of Bernstein-Stancu polynomials associated with GBS operators was considered in MIC . Agrawal et al. PNA6 studied the degree of approximation for bivariate Lupaş-Durrmeyer type operators based on Pólya distribution with associated GBS operators. Recently, many researchers have made significant contributions on this topic. We refer the reader to some of the related papers (PNA3 ; PNA2 ; BAR ; FAR ; KAJ ; POP ).

Let $X$ and $Y$ be compact intervals. A function $f:X\times X\rightarrow\mathbb{R}$ is called $B$-continuous (B$\ddot{o}$gel Continuous) at a point $(x_{0},y_{0})\in X\times Y$ if

[TABLE]

for any $(x_{0},y_{0})\in X\times Y,$ where $\Delta_{(x,y)}f[x_{0},y_{o};x,y]=f(x,y)-f(x,y_{0})-f(x_{0},y)+f(x_{0},y_{0}).$ The function $f:X\times X\rightarrow\mathbb{R}$ is $B$-bounded if there exists $\Omega>0$ such that $\mid\Delta_{(x,y)}f[u,v;x,y]\mid\ \leq\Omega,$ for every $(x,y),(u,v)\in X\times Y.$ Throughout this paper, we denote $B_{b}(X\times Y)$ and $C_{b}(X\times Y)$ be the space of all $B$-bounded and $B$-continuous functions on $X\times Y$ respectively. Let $C_{b}(\mathbb{R}^{2})$ be the space of all $B$-continuous functions defined on $\mathbb{R}^{2}.$

Motivated by the above work, we construct the bivariate Kantorovich type sampling operator $(S_{w}),$ associated with GBS operators, for any $f\in C_{b}(\mathbb{R}{{}^{2}})$ as,

[TABLE]

for all $(x,y)\in\mathbb{R}^{2}.$ The GBS operator of bivariate Kantorovich type is defined as,

$(K_{w}^{\chi})(f;x,y)$

[TABLE]

where, $w\in\mathbb{R}^{+}$ and $(x,y)\in\mathbb{R}^{2}.$ We shall estimate the rate of convergence of the sequences of these operators to $f\in C_{b}(\mathbb{R}{{}^{2}})$ in terms of the modulus of continuity in B$\ddot{o}$gel sense.

Let $f\in C_{b}(\mathbb{R}{{}^{2}}).$ Then, the mixed modulus of smoothness of $f$ is given by,

[TABLE]

for all $(x,y),(u,v)\in\mathbb{R}^{2}$ and for any $(\delta_{1},\delta_{2})\in(0,\infty)\times(0,\infty)$ with $\omega_{B}:(0,\infty)\times(0,\infty)\rightarrow\mathbb{R}.$ The properties of $\omega_{B}$ can be found in (Badea C ).

Theorem 2.1

Let $f\in C_{b}(\mathbb{R}^{2}).$ Then, the operator $(K_{w}^{\chi})$ satisfies the following estimate,

[TABLE]

where, A=\frac{1}{2w}\big{[}M_{0,0}^{0}+2M_{1,0}^{1}\big{]}, B=\frac{1}{2w}\big{[}M_{0,0}^{0}+2M_{0,1}^{1}\big{]} and, C=\frac{1}{4w^{2}}\big{[}M_{0,0}^{0}+2M_{1,0}^{1}+2M_{0,1}^{1}+4M_{1,1}^{2}\big{]}.

Proof

. Since, $w_{B}(f;a\ \delta_{1},b\ \delta_{2})\leq(1+a)(1+b)\ \omega_{mixed}(f;\delta_{1},\delta_{2}),$ for $a,b\ >0,$ we have,

[TABLE]

for every $(x,y),(u,v)\in\mathbb{R}^{2}$ and for any $\delta_{1},\delta_{2}>0.$ Now, applying the operator $(K_{w}^{\chi})$ on the definition of $\Delta_{(x,y)}f[u,v;x,y],$ we obtain,

[TABLE]

In veiw of (7), we have

[TABLE]

Using the definition of $(S_{w}),$ we obtain

[TABLE]

Similarly, we get

[TABLE]

and S_{w}\big{(}|u-x|.|v-y|\big{)}=\frac{1}{4w^{2}}\big{[}M_{0,0}^{0}+2M_{1,0}^{1}+2M_{0,1}^{1}+4M_{1,1}^{2}\big{]}:=C\ . On substituting the values of $A,B$ and $C$ in (8), we get the desired result.

Now, we define a B-differentiable (B$\ddot{o}$gel differentiable) function. A function $f:X\times Y(\subset\mathbb{R}^{2})\rightarrow\mathbb{R}$ is said to be B-differentiable at $(x_{0},y_{0}),$ if the following limit exists finitely,

[TABLE]

The B-differential of $f$ at any point $(x_{0},y_{0})$ is denoted by $D_{B}(f;x_{0},y_{0}).$ We denote the space of all B-differentiable functions defined on $X\times Y$ as $D_{b}(X\times Y).$

Theorem 2.2

Let $f\in D_{b}(\mathbb{R}^{2})$ and $D_{B}f\in B(\mathbb{R}^{2}).$ Then, for each $(x,y)\in\mathbb{R}^{2},$ we have

[TABLE]

where, D=\frac{1}{4w^{2}}\big{[}M_{0,0}^{0}+2M_{1,0}^{1}+2M_{0,1}+4M_{1,1}\big{]}, E=\frac{1}{6w^{3}}\big{[}M_{0,0}^{1}+3M_{2,0}^{2}+3M_{1,0}^{1}+2M_{0,1}^{1}+6M_{2,1}^{3}+6M_{1,1}^{2}\big{]}, F=\frac{1}{6w^{3}}\big{[}M_{0,0}^{0}+3M_{0,2}^{2}+3M_{0,1}^{1}+2M_{1,0}^{1}+6M_{1,2}^{3}+6M_{1,1}^{2}\big{]} and, G=\frac{1}{9w^{4}}\big{[}M_{0,0}^{0}+3M_{2,0}^{2}+3M_{0,2}^{2}+3M_{0,1}^{1}+3M_{1,0}^{1}+9M_{2,2}^{4}+9M_{1,2}^{3}+9M_{2,1}^{3}+9M_{1,1}^{2}\big{]}.

Proof

. If $f\in D_{b}(\mathbb{R}^{2})$, then we have

[TABLE]

where, $p\in(x,u)$ and $q\in(y,v).$ Using the definition of $\Delta_{(x,y)}f[u,v;x,y],$ we have

[TABLE]

Since, $D_{B}f\in B(\mathbb{R}^{2}),$ using the above equality, we have

\big{|}S_{w}(\Delta_{(x,y)}f[u,v;x,y];x,y)\big{|}

[TABLE]

Now, Using the monotonocity of mixed modulus of smoothness $\omega_{B}$, we can write

\Big{|}(K_{w}^{\chi})(f;x,y)-f(x,y)\Big{|}

[TABLE]

By the definition of $(S_{w}),$ we get the following estimates,

[TABLE]

Using these estimates in (LABEL:9), we can easily get the result.

The mixed K- functional (Badea ; Cottin ) can be defined as,

[TABLE]

where, $g_{1}\in C_{B}^{2,0},$ $g_{2}\in C_{B}^{0,2}$ and $h\in C_{B}^{2,2}.$ Here, $C_{B}^{p,q}$ denotes the space of the functions $f\in C_{b}(\mathbb{R}^{2})$ with continuous mixed partial derivatives $D_{B}^{\xi,\eta}f,$ $0\leq\xi\leq p,$ $0\leq\eta\leq q.$

The mixed partial derivatives are defined as,

[TABLE]

and,

[TABLE]

where, \Delta_{x}f\big{(}[x,x_{0}];y_{0}\big{)}=f(x,y_{0})-f(x_{0},y_{0}) and, \Delta_{y}f\big{(}x_{0};[y_{0},y]\big{)}=f(x_{0},y)-f(x_{0},y_{0}). Similarly, we can define the higher order mixed partial derivative as in case of ordinary derivatives.

Now, we estimate the order of convergence in terms of mixed K- functional.

Theorem 2.3

Let $(K_{w}^{\chi})$ be GBS operator of $(S_{w})$. Then, for every $f\in C_{b}(\mathbb{R}^{2}),$

[TABLE]

Here, H=\frac{1}{3w^{2}}\big{[}M_{0,0}+3M_{2,0}+3M_{1,0}\big{]}, J=\frac{1}{3w^{2}}\big{[}M_{0,0}+3M_{0,2}+3M_{0,1}\big{]} and L=\frac{1}{9w^{4}}\big{[}M_{0,0}+3M_{2,0}+3M_{0,2}+3M_{0,1}+3M_{1,0}+9M_{2,2}+9M_{1,2}+9M_{2,1}+9M_{1,1}\big{]}.

Proof

. Since $g_{1}\in C_{B}^{2,0},$ the Taylor’s formula for $g_{1}$ is given by

[TABLE]

On applying the operator $(K_{w}^{\chi})$ both the sides, we obtain

[TABLE]

Now, \big{|}K_{w}^{\chi}(g_{1};x,y)-g_{1}(x,y)\big{|}

[TABLE]

Since, S_{w}\big{(}(u-x)^{2}\big{)}=\frac{1}{3w^{2}}\big{[}M_{0,0}^{0}+3M_{2,0}^{2}+3M_{1,0}^{1}\big{]}:=H, we have

[TABLE]

Similarly, we obatin

[TABLE]

where, J:=S_{w}\big{(}(v-y)^{2};x,y\big{)}=\frac{1}{3w^{2}}\big{[}M_{0,0}^{0}+3M_{0,2}^{2}+3M_{0,1}^{1}\big{]}.

For, $h\in C_{B}^{2,2}(\mathbb{R}^{2}),$ we have

[TABLE]

Using the definition of $(K_{w}^{\chi})$, we get

[TABLE]

where, L:=S_{w}\big{(}(u-x)^{2}(v-x)^{2};x,y\big{)}=\frac{1}{9w^{4}}\big{[}M_{0,0}^{0}+3M_{2,0}^{2}+3M_{0,2}^{2}+3M_{0,1}^{1}+3M_{1,0}^{1}+9M_{2,2}^{4}+9M_{1,2}^{3}+9M_{2,1}^{3}+9M_{1,1}^{2}\big{]}.

Thus, for $f\in C_{b}(\mathbb{R}^{2}),$ we have

\big{|}K_{w}^{\chi}(f;x,y)-f(x,y)\big{|}

[TABLE]

Taking the infimum over all $g_{1}\in C_{B}^{2,0},g_{2}\in C_{B}^{0,2}$ and $h\in C_{B}^{2,2},$ we obtain the required result.

3 Examples of the kernel

Now, we give the kernel which will satisfy the assumptions (K1) to (K3) and condition (3) of section $2$. There are many examples of the univariate kernels satisfying the conditions (K1) to (K3) in one variable, given in PLB1 ; COS , but some of the them fail to satisfy the moment condition (3), for eg., Fejer’s kernel, see Dan . To find the kernel satisfying the moment condition (3) also, Butzer and Stens have given a result using well known B-spline kernels in Stens2 .

The central B-splines (univariate) of order $n\in\mathbb{N}$ has the form ,

[TABLE]

where $t\in\mathbb{R}$ and $x_{+}^{r}=max\{x^{r},0\}.$

The Fourier transform of $M_{n}$ is given as,

[TABLE]

where $n\in\mathbb{N},v\in\mathbb{R}.$

Theorem 3.1

.Stens2 . For $r\in\mathbb{N}$, $r\geq 2,$ let $\epsilon_{0}<\epsilon_{1}<...<\epsilon_{r-1}$ be any given real numbers, and let $a_{\mu_{r}}$ , $\mu=0,1,2,...,r-1,$ be the unique solutions of the linear system of equations;

[TABLE]

*for every $j=0,1,2,...,r-1,$, where i denotes the imaginary unit. Then

[TABLE]

*is a polynomial spline of order $r$, satisfying the moment condition $(3)$ and having compact support contained in $[\epsilon_{0}-r/2,\epsilon_{r-1}+r/2].$

Here, for $r=3$ and $\epsilon_{0}=2,$ we get,

[TABLE]

where,

[TABLE]

for $x\in\mathbb{R}.$

With the help of (10) and (11), we define a bivariate kernel as,

[TABLE]

As $\chi_{3}(x)$ satisfies all the required conditions (see bv1 , Stens2 ), so the kernel $\bar{\chi}(x,y)$ will also satisfy all the assumptions of section $2.$

The following result (see PLB2 ) provides us another tool to construct some more examples of kernel satisfying the moment condition (3).

Theorem 3.2

*.bv1 Let $\chi:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ be a continuous and bounded function such that $M_{r}(\chi)<+\infty$ for some $r\in\mathbb{N}_{0},$ $p=(p_{1},p_{2})\in\mathbb{N}_{0}^{2}$ with $|p|=p_{1}+p_{2}\leq r,$ the double series in the definition of $M_{r}(\chi)$ is unifomly convergent on each compact subset of $\mathbb{R}^{2}$ for each $p$ such that $|p|=r,$ then, the following two statements are equivalent for $c\in\mathbb{R},$

(a) for every $(x,y)\in\mathbb{R}^{2},$*

[TABLE]

(b)

[TABLE]

Using the above result, we can see that the Bochner-Riesz kernel, defined as,

[TABLE]

for $\gamma>\frac{1}{2}$, where, $J_{\lambda}$ is the Bessel function of order $\lambda,$ also satisfies all the required conditions for $\gamma>\frac{5}{2}$ (see bv1 ).