Szász-Baskakov type operators based on q-integers

Purshottam N Agrawal,Meenu Goyal,Harun Karsli

doi:10.1186/1029-242x-2014-441

Abstract

Abstract In the present paper, we introduce the q-analog of the Stancu variant of Szász-Baskakov operators. We establish the moments of the operators by using the q-derivatives and then prove the basic convergence theorem. Next, the Voronovskaja type theorem and some direct results for the above operators are discussed. Also, the rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied. Then we obtain a point-wise estimate using the Lipschitz type maximal function. Lastly, we study the A-statistical convergence of these operators and also, in order to obtain a better approximation, we study a King type modification of the above operators. MSC:41A25, 26A15, 40A35.

Highlights

1 Introduction In recent years, there has been intensive research on the approximation of functions by positive linear operators introduced by making use of q-calculus
The purpose of the present paper is to study the basic convergence theorem, Voronovskaja type asymptotic formula, local approximation, rate of convergence, weighted approximation, point-wise estimation and A-statistical convergence of the operators ( . )
To obtain a better approximation we propose a modification of these operators by using a King type approach

Summary

Introduction

There has been intensive research on the approximation of functions by positive linear operators introduced by making use of q-calculus. In [ ], Stancu introduced the positive linear operators Pn(α,β) : C[ , ] → C[ , ] by modifying the Bernstein polynomials as n. To approximate Lebesgue integrable functions defined on [ , ∞), Gupta et al [ ] introduced the following positive linear operators:. By considering the value of the function at zero explicitly, and studied an estimate of error in terms of the higher order modulus of continuity in simultaneous approximation for a linear combination of the operators In [ ], they introduced Kantorovich type generalization of q-Szász-Mirakjan operators and discussed their A-statistical approximation properties. ). The purpose of the present paper is to study the basic convergence theorem, Voronovskaja type asymptotic formula, local approximation, rate of convergence, weighted approximation, point-wise estimation and A-statistical convergence of the operators )tq x) q–t , and (a b)sq si=– (a + qib), s ∈ Z+

Moment estimates

Weighted approximation