\(^\ast \)Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India, e-mails: pnappfma@gmail.com, dharmendrak.dav@gmail.com.

\(^{\ast \ast }\)Department of Mathematics, University of Prishtina, Prishtina, Kosovo, e-mail: behar.baxhaku@uni-pr.edu, corresponding author.

1 Introduction

In 1912, Bernstein [ 9 ] gave a simple and elegant proof of the Weierstrass approximation theorem by defining a sequence of positive linear operators as follows.

For \(\zeta \in \mathcal{C}(\mathcal{I})=\{ f:\mathcal{I}\rightarrow \mathbb {R}|\; f\; is\; continuous\} \), \(\mathcal{I}=[0,1]\), the \(n\)-th Bernstein polynomial \(\mathcal{B}_n:\mathcal{C}(\mathcal{I})\rightarrow \mathcal{C}(\mathcal{I})\) is defined by

Later several authors introduced various generalizations of these polynomials and studied their approximation properties (see [ 22 ] ).

In the past two decades, the development of \(q\)-calculus has been an active area of research. In \(1987\), Lupas [ 29 ] was the first person who introduced a \(q\)-analogue of the Bernstein polynomials and established some approximation results. A decade later, Phillips [ 35 ] gave another \(q\)-analogue, \(\mathcal{L}_{n,q}:\mathcal{C}(\mathcal{I})\rightarrow \mathcal{C}(\mathcal{I})\) of the Bernstein polynomials which became very popular. He defined it as

where \(0{\lt}q{\lt}1\) and \(\mathfrak {p}_{n,k}(q;x)={n\choose k}_{q}x^{k}(1-x)^{n-k}_{q},\; x\in \mathcal{I}.\) Several generalizations have been studied for the definition of q-Bernstein polynomials given by (1.2), for instance, Muraru [ 31 ] proposed a \(q\)-analogue of the Bernstein-Schurer operators and examined convergence behaviour by virtue of the modulus of continuity. Agrawal et al. [ 2 ] defined Bernstein-Schurer-Stancu operators based on \(q\)-integers and discussed the local and global results. Dalmanog̎lu [ 33 ] gave a \(q\)-analogue of the Bernstein-Kantorovich polynomials. Gupta [ 22 ] introduced a sequence of Bernstein-Durrmeyer operators based on \(q\)-integers and established some approximation theorems.

Gupta and Radu [ 24 ] discussed statistical approximation properties of another kind of \(q\)-Baskakov-Kantorovich operators. Mahmudov [ 30 ] introduced an alternate form of the \(q\)-analogue of Bernstein-Kantorovich operators considered in [ 33 ] . Aliaga and Baxhaku [ 4 ] defined a bivariate extension of the \(q\)-Bernstein type operators involving parameter \(\lambda \) and examined their degree of approximation.

Recently, Mursaleen et al. [ 32 ] proposed a sequence of generalized Bernstein operators based on \(q\)-integers. Cai [ 12 ] introduced another generalization of Bernstein operators based on \(q\)-integers and derived some convergence theorems and shape preserving properties. Based on the Phillips \(q\)-Bernstein polynomials [ 35 ] , generalized Bézier curves and surfaces were introduced in ( [ 15 ] , [ 16 ] , [ 34 ] ). For a detailed account of the work in this direction, we refer the reader to the book [ 5 ] . Before proceeding further, let us mention some important basic definitions and notations of \(q\)-calculus. For \(q{\gt}0\), and each nonnegative integer \(l\), the q-integer \([l]_q\) and the q-factorial \([l]_q!\) are, respectively, given by

and

For the non-negative integers \(n,l\) satisfying \(l\leq n\), the q-binomial coefficients are defined by

In 2017, Chen et al. [ 13 ] introduced generalized Bernstein operators (1.1) involving a real parameter ‘\(\alpha \)’ satisfying \(0\leq \alpha \leq 1\), as

where the \(\alpha \)-Bernstein basis function \(P_{n,k,\alpha }(x),\) for \(n\geq 2,\) is given by

with \({n-2\choose -2}={n-2\choose -1}=0\). Clearly \(P_{n,k,\alpha }(x),\) verifies the following recurrence relation

The authors [ 13 ] established the uniform convergence theorem and the Voronovskaja-type asymptotic formula etc. For the special case \(\alpha =1\), the operators (1.3) reduce to (1.1).

For any \(\zeta \in \mathcal{C}(\mathcal{I}),\) Khosravian-Arab et al. [ 27 ] presented a new family of Bernstein operators as follows:

where

and

\(a_0(n)\) and \(a_1(n)\) being two unknown sequences, which may be defined in an appropriate manner. If \(a_0(n)=1,\) and \(a_1(n)=-1,\) then (1.5) reduces to (1.1).

Recently, Kajla and Acar [ 26 ] for any \(\zeta \in \mathcal{C}(\mathcal{I}),\) constructed a new family of \(\alpha \)-Bernstein operators defined by

where \(P^{M,1}_{n,k,\alpha }(x)=a(x,n)P_{n-1,k,\alpha }(x)+a(1-x,n)P_{n-1,k-1,\alpha }(x)\) and \(a(x,n)=a_1(n)x+a_0(n)\), and investigated some approximation properties such as Korovkin type theorem and a Voronovskaja type asymptotic formula. Clearly, for \(a_0(n)=1,\) and \(a_1(n)=-1,\) (1.4) includes (1.3).

In 2019, Gupta et al. [ 23 ] presented a Kantorovich variant of the operators (1.5) and determined various approximation results.

Motivated by the above research, for \(\zeta \in \mathcal{C}(\mathcal{I})\) endowed with the uniform norm \(\| .\| \), we define the \(q\)-analogue of the operators \(\eqref{je1}\) as follows:

where \(P^{M,1}_{n,k,\alpha }(q;x)=a(x,n,q)P_{n-1,k,\alpha }(q;x)+a(1-x,n,q)P_{n-1,k-1,\alpha }(q,x)\) with \(P_{n,k,\alpha }(q;x)={n-2\choose k}_{q}(1-\alpha )(qx)^{k}(1-x)_{q}^{n-k-1}+{n-2\choose k-2}_{q}(1-\alpha )q^{k-2}x^{k-1}(1-qx)_{q}^{n-k}+{n\choose k}_{q} \alpha (qx)^{k}(1-qx)_{q}^{n-k}\) and \(a(x,n,q)=a_1(n,q)x+a_0(n,q)\). For \(a_1(n,q)=-1\) and \(a_0(n,q)=1\) and \(\alpha =1\), we get \(q\)-Bernstein polynomials given by (1.2).

The purpose of the given article is to examine the convergence behavior of the operators \(\eqref{je2}\) by virtue of the Lipschitz class and the \(K\)-functional. Next, we study a bivariate extension of these operators and determine the convergence estimates by virtue of the moduli of continuity and the \(K\)-functional. Further, we study the Voronovskaja and Grüss Voronovskaja type theorems and establish a quantitative result for functions in the Lipschitz class. Lastly, we extend our study to the corresponding GBS operators.

2 Preliminaries

Throughout this paper, we assume that \(2a_0(n,q)+a_1(n,q)=1\).

In what follows, let \((q_{n})\) be a sequence in \((0,1)\) such that

3 Rate of convergence of the operators \(\mathcal{G}^{M,1}_{\lowercase {n},\lowercase {\alpha }}\left(.,\lowercase {q_n};\lowercase {x}\right)\)

First, we prove the uniform convergence theorem for the operators \((\ref{je2})\).

Next, we establish a Voronovskaja type asymptotic result for the operators \((\ref{je2})\).

where \(\theta \) lies between \(\hslash \) and \(x\). Operating by \(\mathcal{G}^{M,1}_{n,\alpha }(.,q_n;x)\) in the above equation, we obtain

Using the well known properties of modulus of continuity, for any \(\rho {\gt}0\), we have

Therefore,

Using Cauchy-Schwarz inequality, remark 2.3 and choosing \(\rho =[n]_{q_{n}}^{-\frac{1}{2}}\), we get

as \(n\to \infty \), uniformly in \(x \in \mathcal{I}\).

Hence,

as \(n\to \infty \), uniformly in \(x \in \mathcal{I}\).

Consequently,

uniformly in \(x\in \mathcal{I}\). Thus, from equation \(\eqref{de2}\) and remark 2.3, we get

uniformly in \(x \in \mathcal{I}\), which completes the proof.

Now, we obtain an approximation theorem for the operators \((\ref{je2})\) by virtue of the second order modulus of continuity by using a smoothing process, e.g. Steklov mean. For \(\zeta \in \mathcal{C}(\mathcal{I})\), the Steklov mean is defined as

It is known that for the function \(\zeta _{h}(x)\), there hold the following properties:

• \(\| \zeta _{h}-\zeta \| \leq \omega _{2}(\zeta ,h)\)

• \(\zeta ^{\prime }_{h}, \zeta ^{\prime \prime }_{h}\in \mathcal{C}(\mathcal{I})\quad and\quad \| \zeta ^{\prime }_{h}\| \leq \tfrac {5}{h}\omega (\zeta ,h), ~ \| \zeta ^{\prime \prime }_{h}\| \leq \tfrac {9}{h^{2}}\omega _{2}(\zeta ,h)\),

where \(\omega (\zeta ;h)\) and \(\omega _{2}(\zeta ;h)\) denote the first and second order modulus of continuity. In what follows, for all \(x\in \mathcal{I}\), let \(\mu ^{\alpha ,m}_{n,q_n}(x):=\mathcal{G}^{M,1}_{n,\alpha }\left(\phi ^{m}_{x}(\hslash );q_n;x\right), m\in \mathbb {N}_0:=\mathbb {N}\cup \{ 0\} \) and \(\nu ^{\alpha ,m}_{n, q_n}:=\sup _{x\in \mathcal{I}}\left|\mu ^{\alpha ,m}_{n,q_n}(x)\right|\), \(m\in \mathbb {N}\).

Hence using lemma 2.4 and property (a) of Steklov mean, we have

Since \(\zeta ^{\prime \prime }_{h} \in \mathcal{C}(\mathcal{I})\), by Taylor’s expansion we have

where \(\theta \) lies between \(\hslash \) and \(x\). Then, applying Cauchy-Schwarz inequality

Now, applying property (b) of Steklov mean, we obtain

Choosing \(h=\sqrt{\nu ^{\alpha ,2}_{n,q_n}}\) and combining the equations 3.1–3.3, we get the desired result.

Applying the operator \(\mathcal{G}^{M,1}_{n,\alpha }(.;q_n;x)\) on both sides of the above relation, we get

Using the well known property of modulus of continuity

we obtain

Therefore,

Using Cauchy-Schwarz inequality, we have

Choosing \(\rho =\tfrac {1}{2}\sqrt{\nu ^{\alpha ,2}_{n,q_n}}\), the required assertion is obtained.

The following theorem is concerned with an estimate of error in the approximation by the operators (1.5) by means of the Peetre’s \(K\)-functional.

It is clear that the operator \(\mathcal{G}^{M,1*}_{n,\alpha }(\zeta ;q_n;x)\) is linear, \(\mathcal{G}^{M,1*}_{n,\alpha }(1;q_n;x)=1\) and

For every \(\varphi \in \mathcal{C}^2(\mathcal{I})\) and \(\hslash ,x\in \mathcal{I}\), from the Taylor’s theorem, one can write

Applying \(\mathcal{G}^{M,1*}_{n,\alpha }(.;q;x)\) to the above equation and using 3.0, we obtain

Thus, for all \(x\in \mathcal{I},\) we have

Further, for all \(x\in \mathcal{I},\) in view of lemma 2.4, we get

Now, for \(\zeta \in \mathcal{C}(\mathcal{I})\) and any \( \varphi \in \mathcal{C}^2(\mathcal{I}) \), using 3.0–3.2, we obtain

Now, taking the infimum on the right hand side over all \(\varphi \in \mathcal{C}^2(\mathcal{I})\),

where \(K_2(\zeta ,\rho )=\inf \left\{ \left\Vert \zeta -\varphi \right\Vert + \rho \left\Vert \varphi ^{\prime \prime }\right\Vert :\varphi \in \mathcal{C}^2(\mathcal{I})\right\} .\)
Finally, using the relation between \(K\)-functional and second order modulus of continuity [ 14 ] given by

\(C^{\prime }\) being some constant, we get

This completes the proof.

be the Lipschitz class of continuous functions.

Applying Hölder’s inequality, we get

which proves the required result.

Finally, we investigate Grüss-Voronovskaya type theorem for the operators (1.5).

Now, in view of theorem 3.1, for any \(\zeta \in \mathcal{C}(\mathcal{I})\), it follows that \(\mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)\rightarrow \zeta (x)\), as \(n \rightarrow \infty \), uniformly in \(x\in \mathcal{I}\). Further, following the proof of theorem 3.2, for any \(\zeta \in \mathcal{C}^2(\mathcal{I})\) we get
\([n]_{q_n}\bigg\{ \mathcal{G}^{M,1}_{n,\alpha }(\zeta ,q_n;x)-\zeta (x)-\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x),q_n;x)\zeta ^\prime (x)-\tfrac {\mathcal{G}^{M,1}_{n,\alpha }((\hslash -x)^2,q_n;x)}{2!}\zeta ^{\prime \prime }(x)\bigg\} \rightarrow 0,\, as\, n\rightarrow \infty \), uniformly in \(x\in \mathcal{I}\).

Hence, using remark 2.3, we get

which completes the proof.

4 Bivariate generalization of the operators \(\mathcal{G}^{M,1}_{\lowercase {n},\alpha }(.,\lowercase {q_n};\lowercase {x})\)

For \(\mathcal{I}^2=\mathcal{I}\times \mathcal{I}\), let \(\mathcal{C}(\mathcal{I}^2)\) be the space of all continuous functions on \(\mathcal{I}^2\), equipped with the norm given by \(\| \zeta \| _{\mathcal{C}(\mathcal{I}^2)}=\sup _{(x_1,x_2)\in \mathcal{I}^2}|\zeta (x_1,x_2)|.\)

For \(\zeta \in \mathcal{C}(\mathcal{I}^2)\) and \(0\leq \alpha _1,\alpha _2\leq 1,\) the bivariate generalization of the operator 1.5 is defined by

where \(\{ q_{n_i}\} _{n_i\in \mathbb {N}}\) is a sequence in \((0,1)\) satisfying \(\lim \limits _{n_i\rightarrow \infty }q_{n_i}= 1,\, \lim \limits _{n_i\rightarrow \infty } \, q^{n_i}_{n_i}= c_i,\) \(\lim _{n_i\rightarrow \infty }a_1(n_i,q_{n_i})=\beta _i,\, and\, \lim _{n_i\rightarrow \infty }a_0(n_i,q_{n_i})=\gamma _i,\; \forall \; i=1,2. \)

Further, \(\mathfrak {P}^{\alpha _1,\alpha _2}_{n_1,n_2,k,j}(q_{n_1},q_{n_2};x_1,x_2)=P^{M,1}_{n_1,k,\alpha _1}(q_{n_1};x_1)P^{M,1}_{n_2,j,\alpha _2}(q_{n_2};x_2),\) where \(P^{M,1}_{n_1,k,\alpha _1}(q_{n_1};x_1)\) and \(P^{M,1}_{n_2,k,\alpha _2}(q_{n_2};x_2)\) are defined similarly as \(P^{M,1}_{n,k,\alpha }(q;x)\) in 1.5.

Clearly, \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta ;;x_1,x_2)\) is a linear positive operator. Further, we note that

Let \(e_{pq}(x_1,x_2)=x^{p}_1x^{q}_2,(p,q)\in \mathbb {N}_0\times \mathbb {N}_0\), with \(p+q\leq 2\). In order to establish the results of this section, we require the following Lemmas:

5 Convergence estimates for the bivariate operators

In the following result we show the uniform convergence of \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )\) to \(\zeta \), if \(\zeta \in \mathcal{C}(\mathcal{I}^2).\)

hence applying the well known theorem given by Volkov [ 37 ] , the required result follows.

For \(\zeta \in \mathcal{C}(\mathcal{I}^2)\) and \(\rho {\gt}0\), the total modulus of continuity is given by:

Hence,

whenever \(\sqrt{(\hslash _1-x_1)^2+(\hslash _2-x_2)^2}\leq \rho , \rho {\gt}0\), and for any \(\lambda {\gt}0\),

The partial moduli of continuity with respect to \(x_1\) and \(x_2\) are defined as

For more details on the moduli of continuity for functions of two variables, we refer to [ 24 ] .

Let \(\mathcal{C}^2(\mathcal{I}^2)= \{ \zeta \in \mathcal{C}(\mathcal{I}^2):\frac{\partial ^2 \zeta }{\partial x_{1}^i\partial x_{2}^j}\in \mathcal{C}(\mathcal{I}^2), \, 0\leq i+j\leq 2\, ~ i,j\in \mathbb {N}_0\} \) equipped with the norm

The appropriate Peetre’s \(K\)-functional for the function \(\zeta \in \mathcal{C}(\mathcal{I}^2)\) is given by

From [ 11 , p. 192 ] , we have

where the constant \(M{\gt}0\), is independent of \(\zeta \) and \(\rho \) and \(\overline{\omega _{2}}(\zeta ;\sqrt{\rho })\) is the second order modulus of smoothness.

In our further consideration, let

Now, choosing \(\rho =(\theta ^{q_{n_1}}_{n_1,\alpha _1}+\gamma ^{q_{n_2}}_{n_2,\alpha _2})^\frac {1}{2}\), the required result is proved.

Next, we determine the rate of convergence for the operators \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )\) in terms of Peetre’s \(K\)-functional for any \(\zeta \in \mathcal{C}(\mathcal{I}^2)\).

Then, in view of lemma 4.1, we have \(\overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}(1;x_1,x_2)=1,\)

Let \(\tau \in \mathcal{C}^2(\mathcal{I}^2)\) and \((x_1,x_2)\in \mathcal{I}^2\) be arbitrary. By Taylor’s expansion, we can write

Applying \(\overline{\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}}(.;x_1,x_2)\) on both sides of the equation (5.6) and using (5.5), we obtain

Hence, applying Cauchy-Schwarz inequality

Also, from 5.5, we have

Now, for \(\zeta \in \mathcal{C}(\mathcal{I}^2)\) and any \(\tau \in \mathcal{C}^2(\mathcal{I}^2)\), using 5.7 and 5.8, we may write

Taking the infimum over all \(\tau \in \mathcal{C}^2(\mathcal{I}^2)\) on the right hand side of the above equation and using (5.4), we get

which leads us to the desired assertion.

The next result provides a convergence estimate for functions in \(\mathcal{C}^{1}(\mathcal{I}^2)= \{ \zeta \in \mathcal{C}(\mathcal{I}^2):\frac{\partial \zeta }{\partial x_{1}},\frac{\partial \zeta }{\partial x_{2}}\in \mathcal{C}(\mathcal{I}^2) \} \) by the operators (4.2).

Now, applying \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(.;x_1,x_2)\) on both sides of the above equation, we get

By applying the inequalities

and

we get

Applying Cauchy-Schwarz inequality and lemma 4.2, we obtain

This completes the proof.

In the next result, we discuss the convergence behaviour of \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )\) to \(\zeta \) by virtue of the partial moduli of continuity of the partial derivatives of \(\zeta \).

where \(u\) and \(v\) lie between \(\hslash _1,x_1\) and \(\hslash _2,x_2\) respectively. Applying \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\) \(\left(.;x_1,x_2\right)\) to the above equation, we obtain

Since \(\zeta ^{\prime }_{x_1}\) and \(\zeta ^{\prime }_{x_1}\) are continuous on \(\mathcal{I}^2\), they are bounded therein, therefore there exist positive constants \(C_1 \) and \(C_2\) such that \(|\zeta ^{\prime }_{x_1}|\leq C_1\) and \(|\zeta ^{\prime }_{x_2}|\leq C_2\), for all \((x_1,x_2)\in \mathcal{I}^2\). Hence applying Cauchy-Schwarz inequality, for any \(\rho _1,\rho _2{\gt}0\), we obtain

Choosing \(\rho _i=([n_i]_{q_{n_i}})^{-\frac{1}{2}}, i=1,2\) and applying lemma 4.3, the required result is proved.

Now, we establish a convergence estimate for the operators \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\) with the aid of the Lipschitz class functions.

For \(\zeta :\mathcal{I}^2\rightarrow \mathbb {R}\) and \(0{\lt}\theta \leq 1\), the function \(\zeta \) is said to be in Lipschitz class \(\operatorname {Lip}_{\mathcal{M}}(\theta )\), if \(\exists \) a positive constant \(\mathcal{M}\) such that

\(\forall r=(\hslash _1,\hslash _2), x=(x_1,x_2)\in \mathcal{I}^2,\) where \(\| r-x\| =\{ (\hslash _1-x_1)^2+(\hslash _2-x_2)^2\} ^{\frac{1}{2}}\) is the Euclidean norm.

where \(r=(\hslash _1,\hslash _2), x=(x_1,x_2)\in \mathcal{I}^2\). Applying H\(\ddot{o}\)lder’s inequality , we obtain

hence using lemma 4.2, the required assertion is proved.

Next, we discuss a Voronovskaja type asymptotic theorem.

where \(\varpi (\hslash _1,\hslash _2;x_1,x_2)\in \mathcal{C}(\mathcal{I}^2)\) and \(\varpi (\hslash _1,\hslash _2;x_1,x_2) \rightarrow 0,\) as \((\hslash _1,\hslash _2)\rightarrow (x_1,x_2).\)
Operating \(\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(.;x_1,x_2)\) on both sides of (5.9), we have

Applying Cauchy-Schwarz inequality to the last term of (5.10), we obtain

Since \(\varpi (.,.;x_1,x_2)\in \mathcal{C}(\mathcal{I}^2)\) and \(\varpi (\hslash _1,\hslash _2;x_1,x_2)\rightarrow 0,\) as \((\hslash _1,\hslash _2)\rightarrow (x_1,x_2),\) applying theorem 5.1, we obtain

\(\mbox{uniformly in} ~ ~ ~ ~ (x_1,x_2)\in \mathcal{I}^2.\)

Now, using lemma 4.3 and by the above equation, from (5.10) we reach to the required result.

Grüss [ 20 ] determined the difference between the integral of a product of two functions and the product of integrals of the two functions. Later, Gal and Gonska [ 18 ] , studied the Grüss Voronovskaya type theorem for Bernstein and Paltanea operators with the aid of Grüss inequality which deals with the non-multiplicavity of the operators. For more details in this direction, one can see ( [ 1 ] , [ 28 ] ) and the references therein. In the following theorem, we examine the non-multiplicativity of the operators \(\mathcal{G}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\).

From theorem 5.1, for all \(\zeta \in \mathcal{C}(\mathcal{I}^2)\), it follows that \(\mathcal{G}^{q_{n},q_{n}}_{n,n,\alpha _1,\alpha _2}(\zeta ;x_1,x_2)\rightarrow \zeta (x_1,x_2)\), as \(n\rightarrow \infty \), uniformly in \((x_1,x_2)\in \mathcal{I}^2\), and by theorem 5.8, for every \(\zeta \in \mathcal{C}^2(\mathcal{I}^2)\), we have

uniformly in \((x_1,x_2)\in \mathcal{I}^2.\)

Hence, in view of the fact that \(\zeta ,\tau \in \mathcal{C}^2(\mathcal{I}^2)\), using lemma 4.3, we reach the desired assertion.

6 Construction of GBS operators for the bivariate operators \(\mathcal{G}^{\lowercase {q_{n_1}},\lowercase {q_{n_2}}}_{\lowercase {n_1},\lowercase {n_2},\alpha _1,\alpha _2}(\ \, \cdot \; ;\lowercase {x_1},\lowercase {x_2})\)

In the past decade, the study of GBS (Generalized Boolean Sum) operators associated with the positive linear operators has been an active area of research in the field of approximation theory. The concepts of Bögel continuous and Bögel differentiable functions were first given by Bögel [ 10 ] . Dobrescu and Matei [ 17 ] showed that any Bögel continuous function on a bounded interval can be uniformly approximated by the boolean sum of bivariate Bernstein polynomials. Badea et al. [ 6 ] obtained a Korovkin type theorem for the Bögel continuous functions. Agrawal et al. [ 3 ] studied convergence estimates for the GBS case of the bivariate Lupas-Durrmeyer type operators based on Polya distribution. Barbosu et al. [ 8 ] proposed GBS operators of Durrmeyer-Stancu type based on \(q\)-integers and examined the approximation degree by using Lipschitz class and the mixed modulus of smoothness. Kajla and Miclaus [ 25 ] determined the convergence behaviour of GBS operators of Bernstein-Durrmeyer type for Bögel continuous and Bögel differentiable functions. Agrawal and Chauhan [ 36 ] introduced the sequence of GBS operators of Bernstein-Durrmeyer type on a triangle and investigated the rate of convergence by virtue of the mixed modulus of smoothness for Bögel continuous and Bögel differentiable functions. For a detailed account of the research in this direction, one can see [ 19 ] and the references therein.

A function \(\zeta :\mathcal{I}^2 \rightarrow \mathbb {R}\), is called B-continuous (Bögel continuous) at a point \((x_1,x_2)\in \mathcal{I}^2\) if

where \(\Delta _{(x_1,x_2)} \zeta [(\hslash _1,\hslash _2);(x_1,x_2)]=\zeta (\hslash _1,\hslash _2)-\zeta (\hslash _1,x_2)-\zeta (x_1,\hslash _2)+\zeta (x_1,x_2).\) Further, a function \(\zeta :\mathcal{I}^2 \rightarrow \mathbb {R}\), is said to be \(B\)- continuous on \(\mathcal{I}^2\), if is \(B\)- continuous \(\forall (x_1,x_2)\in \mathcal{I}^2\).

A function \(\zeta :\mathcal{I}^2 \rightarrow \mathbb {R}\), is called \(B\)- differentiable (Bögel differentiable) on \(\mathcal{I}^2\), if for every \((x_1,x_2)\in \mathcal{I}^2\),

The function \(\zeta :\mathcal{I}^2\rightarrow \mathbb {R}\) is said to be B-bounded on \(\mathcal{I}^2\) if \(\exists \) some \(K{\gt}0\), such that \(|\Delta _{(x_1,x_2)} \zeta [(\hslash _1,\hslash _2); (x_1, x_2)]|\leq K\), for every \((\hslash _1,\hslash _2), \, (x_1,x_2)\in \mathcal{I}^2\). The space of B-bounded functions is denoted by \(B_b( \mathcal{I}^2)\), the space of B-continuous functions is denoted by \(C_b( \mathcal{I}^2)\) and the space of all B-differentiable functions is denoted by \(D_b(\mathcal{I}^2).\) Further, let \(B(\mathcal{I}^2)\) be the space of bounded functions on \(\mathcal{I}^2\) endowed with the sup-norm denoted by \(\| .\| _{\infty }\).

For every \(\zeta \in \mathcal{C}_b(\mathcal{I}^2),\) the GBS operator associated with the operators defined in (4.2) is defined as:

The operator \(\mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}\) is well defined on the space \(\mathcal{C}_b(\mathcal{I}^2)\) into \(\mathcal{C}(\mathcal{I}^2)\). We shall analyze the order of approximation of \(\mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )\) to \(\zeta \), for all \(\zeta \in \mathcal{C}_{b}(\mathcal{I}^2)\), using mixed modulus of smoothness.

First, we show the uniform convergence of the operators (6.11) to \(\zeta \), where \(\zeta \in \mathcal{C}_{b}(\mathcal{I}^2)\) by using the following result:

and

hence applying lemma 6.1, we obtain the desired conclusion.

In the next result, we examine convergence estimates of \(\mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(\zeta )\) to \(\zeta \) by virtue of mixed modulus of smoothness.

For \((x_1,x_2)\), \((\hslash _1,\hslash _2)\in \mathcal{I}^2\), the mixed modulus of smoothness of \(\zeta \in \mathcal{C}_b(\mathcal{I}^2)\) is defined by

for any \((\rho _1,\rho _2)\in (0,\infty )\times (0,\infty )\). From definition of \(\widetilde{\omega _B}(\zeta ,\rho _1,\rho _2)\), it follows that

for any \(c_1,c_2{\gt}0\).

for every \((x_1,x_2),(\hslash _1,\hslash _2)\in \mathcal{I}^2\) and for any \(\rho _1, \rho _2 {\gt}0\). Using the definition of \(\Delta _{(x_1,x_2)}\zeta [(\hslash _1,\hslash _2);(x_1,x_2)]\), we may write

Applying the operator \(\mathcal{S}^{q_{n_1},q_{n_2}}_{n_1,n_2,\alpha _1,\alpha _2}(.;x_1,x_2)\) on both sides of the above equality, we get

Hence using 6.11, lemma 4.1 and applying the Cauchy-Schwarz inequality, we get

Now, choosing \(\rho _i=[n_i]_{q_{n_i}}^{\frac{-1}{2}}, i=1,2\), and applying lemma 4.3, the required assertion is proved.

The following result is concerned with the error in the approximation of the B-differentiable functions by the operators 6.11.

and \(\, x_2{\lt}\beta {\lt}\hslash _2.\) Clearly,

Since \(D_B \zeta \in B(\mathcal{I}^2)\), from the above equalities, we have

Considering the properties of mixed modulus of smoothness \(\widetilde{\omega _B}\), for any \(\rho _1,\\ \rho _2 {\gt}0,\) we have

Hence taking into account (6.14), (6.15) and applying the Cauchy-Schwarz inequality, we obtain

Since, for \(i,j\in \{ 1,2\} ,\) we have

uniformly in \((x_1,x_2)\in \mathcal{I}^2\), combining 6.16–6.17 and choosing \(\rho _i=[n_i]_{q_{n_i}}^{\frac{-1}{2}},i=1,2\), we reach the desired result.

Now, we discuss the approximation degree of the operators 6.11 for Lipschitz class of B-continuous functions.

The Lipschitz class \(\operatorname {Lip}_{\mathcal{M},b}^{\theta }\) with \(\theta \in (0,1]\), for B-continuous functions is defined by

for every \(\, (\hslash _1,\hslash _2),\, (x_1,x_2) \in \mathcal{I}^2.\)

Now, applying the Hölder’s inequality with \((p_{1},q_{1})=\left(\tfrac {2}{\theta } ,\tfrac {2}{( 2-\theta )}\right) \) and using lemma 4.1, we get

which yields us the required result. This completes the proof.