Elsevier

Applied Mathematics and Computation

Volume 242, 1 September 2014, Pages 931-944
Applied Mathematics and Computation

On generalized integral Bernstein operators based on q-integers

https://doi.org/10.1016/j.amc.2014.05.134Get rights and content

Abstract

The purpose of this paper is to study a simple integral generalization of q-analog of well known Bernstein operators with two parameters α and β, which was introduced in Mishra and Patel (2013) [17]. We give detail proof of the rate of convergence and Voronovskaja type asymptotic formula for the operators Dn,qα,β. Also, we establish statistical convergence for these operators and discuss modification of the operators Dn,qα,β for better approximation results over compact interval.

Introduction

In 1885, Weierstrass proved Weierstrass approximation theorem: ‘Every continuous function on [a,b], can be uniformly approximated by a sequence of polynomial functions’.

In 1912, Bernstein gave an alternative proof of the Weierstrass’s theorem using the Bernstein polynomials. Since Bernstein polynomial plays an important role in approximation theory, many researchers have been studied Bernstein polynomial and its different generalizations [1], [2], [3], [4].

In recent years, due to the intensive generalizations of q-Calculus and its applications in many fields such as mathematics, physics and computer science, generalizations of classical operators related to q-Calculus have emerged. Since last decades, q-calculus has been studied rigorously because its latent application in Mathematics, Mechanics and Physics. After Philips [5] introduced and studied q analog of Bernstein polynomials, the applications of q-calculus in the approximation theory has become one of the major areas of research. Sofia and Ankara [6] discussed that, the q-Bernstein polynomials inherit some properties of the classical Bernstein polynomials. Among those properties they mention the end-point interpolation property, the shape-preserving properties in the case 0<q<1. Like the classical Bernstein polynomials, the q-Bernstein polynomials reproduce linear functions, and they are degree-reducing on the set of polynomials. Alternatively, the examination of the convergence properties of the q-Bernstein polynomials reveals that these properties are essentially different from those of the classical ones. Since, q-generalization of some positive operators have attracted much interest, and a great number of interesting results related to these operators have been obtained [7], [9], [11], [12], [10], [13], [8], [14], [15].

To approximate Lebesgue integrable functions on the interval [0,1], Durrmeyer introduced the integral modification of the well known Bernstein polynomials. In 1981 Derriennic [4] first studied these operators in details, q-generalization of this operators introduced by Gupta and Heping [16]. In the present article we study convergence behavior of the q-analog of the Stancu generalization of Durrmeyer operators, which was introduced by Mishra and Patel [17].

We recall some concept of q-calculus. All of the results can be found in [18], [19]. In what follows, q is a real number satisfying 0<q<1.

For nN,[n]q1+q+q2++qn-1,q1,n,q=1,and(1+x)qn(1+x)(1+qx)(1+qn-1x),n=1,2,,1,n=0.Also for any real number α, we have (1+x)qα=(1+x)q(1+qαx)q. In special case when α is a whole number this definition coincides with the above definition.

The q-analog of integration, discovered by Thomae [20] in the interval [0,a] is defined by0af(x)dqx=(1-q)an=0f(aqn)qn,a>0.For any m,n>0, the q-Beta function is defined asBq(m,n)01tm-1(1-qt)qn-1dqt.In particular, for m,nN, we haveBq(m,n)[m-1]q![n-1]q![m+n-1]q!.

Section snippets

Construction of operators

We setpnk(q;x)=nkqxk(1-x)qn-k,pk(q;x)=xk(1-q)k[k]q!(1-x)q.After development of quantum calculus, Phillips [5] proposed the following q-Bernstein polynomials, which for each positive integer n and fC[0,1] are defined asBn,q(f;x)=k=0nf[k]q[n]qpnk(q;x).In 2005, Derriennic [21] introduced a q-analog of the integral summation operators and established some approximation properties of q-Durrmeyer operators. Recently, Gupta [22] studied following q-analog of Durrmeyer operators and discussed

Estimation of moments

In this section we shall obtain Dn,qα,β(ti;x),i=0,1,.

Note that for s=0,1, and by the definition of q-Beta function (see [19]), we have01tspnk(q;qt)dqt=qk[n]q![k+s]q![n+s+1]q![k]q!and01tspk(q;qt)dqt=(1-q)s+1qk[k+s]q![k]q!.

Lemma 1

[22]

We haveDn,q(1;x)=1,Dn,q(t;x)=1+qx[n]q[n+2]qandDn,q(t2;x)=q3x2[n]q([n]q-1)+(1+q)2qx[n]q+1+q[n+3]q[n+2]q.

Lemma 2

We haveDn,qα,β(1;x)=1,Dn,qα,β(t;x)=[n]q+α[n+2]q+qx[n]q2[n+2]q[n]q+βandDn,qα,β(t2;x)=q3[n]q3[n]q-1x2+q(1+q)2+2αq4[n]q3+2αq[3]q[n]q2x([n]q+β)2[n+2]q[n+3]q+(1+q+2αq3)[n]q2+2α

Convergence of q-Durrmeyer–Stancu operators

Theorem 1

Let qn(0,1]. Then the sequence {Dn,qnα,β(f)} converges to f uniformly on [0,1] for each fC[0,1] if and only if limnqn=1.

Proof

Since the operator Dn,qnα,β are positive linear operators on C[0,1] by well known theorem due to Korovkin, it implies that Dn,qnα,β(f;x)f as n(x[0,1]) for fC[0,1] if and only ifDn,qnα,β(ti;x)xiasn,i=0,1,2forx[0,1].If qn1, then [n]qn and for s=1,2,3,limn[n+s]qn[n]qn=1, hence (4.1) follows from Lemma 2.

On the other hand, if we assume that for any fC[0,1],Dn,qnα

Statistical convergence

A sequence (xn)n is said to be statistically convergent to a number L, denoted by st-limnxn=L, if for every >0δ{nN:|xn-L|}=0,whereδ(K)=limnj=1nχK(j)is the natural density of set KN and χK is the characteristic function of K. For instantxn=lgn,n{10k,kN},1,otherwiseseries (xn)nN converges statistically, but limnxn does not exist. We note that every convergent sequence is a statistical convergent, but converse need not be true (details can be found in [38]).

As an application of the

Voronovskaja type theorem

In this section we establish a Voronovskaja type asymptotic formula for the operators Dn,qnα,β:

Lemma 4

Assume that qn(0,1),qn1 as n. Then, for every x[0,1], we havelimn[n]qnDn,qnα,β(t-x;x)=-(2+β)x+1+α,limn[n]qnDn,qnα,β((t-x)2;x)=2x(1-x).

Theorem 6

Let f be bounded and integrable on the interval [0,1], second derivative of f exists at a fixed point x[0,1] and q=qn(0,1) such that qn1 as n, thenlimn[n]qnDn,qnα,β(f;x)-f(x)=(α+1-(2+β)x)f(x)+x(1-x)f(x).

Proof

Using Taylor’s expansion of f, we can writef(t)=f(

Better estimation

To make the convergence faster, King [40] proposed an approach to modify the Bernstein polynomials, so that sequence preserve test functions e0 and e2. As the operators Dn,qα,β(f;x) reproduce only constant functions, this motivated us to propose the modification of the operators (2.1), so that they reproduce constant as well as linear functions, for this purpose we propose the modification of the operators (2.1) asD̃n,qα,β(f;x)=[n+1]qk=0nq-kpnk(q;rn,q(x))01f[n]qt+α[n]q+βpnk(q;qt)dqt,where rn,q

Conclusion

The results of our lemmas and theorems are more general rather than the results of any other previous proved lemmas and theorems, which will be enrich the literate of Applications of quantum calculus in operator theory and convergence estimates in the theory of approximations by linear operators. The researchers and professionals working or intend to work in areas of analysis and its applications will find this research article to be quite useful. Consequently, the results so established may be

Acknowledgements

The authors would like to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this research article. Special thanks are due to Prof. Theodore E. Simos, Editor in chief and Prof. Melvin R. Scott, Emeritus Editor of Applied Mathematics and Computation, for kind cooperation, smooth behavior during communication and for their efforts to send the reports of the

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