Properties of convergence for the q-Meyer-König and Zeller operators

By Wang Heping. 2007

52 citations recorded by the source corpus.

In this paper, we discuss properties of convergence for the q-Meyer-Konig and Zeller operators Mn,q. Based on an explicit expression for Mn,q(t2,x) in terms of q-hypergeometric series, we show that for qn∈(0,1], the sequence (Mn,qn(f))n⩾1 converges to f uniformly on [0,1] for each f∈C[0,1] if and only if limn→∞qn=1. For fixed q∈(0,1), we prove that the sequence (Mn,q(f)) converges for each f∈C[0,1] and obtain the estimates for the rate of convergence of (Mn,q(f)) by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. We also give explicit formulas of Voronovskaya type for the q-Meyer-Konig and Zeller operators for fixed 0<q<1. If 0<q<1, f∈C1[0,1], we show that the rate of convergence for the Meyer-Konig and Zeller operators is o(qn) if and only if f(1−qk−1)−f(1−qk)(1−qk−1)−(1−qk)=f′(1−qk),k=1,2,….

Subfields: Applied Mathematics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/218f24679d67242053a1babbdb7e7e4259fa8a4e

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In Applied Mathematics

In Numerical Analysis

In Statistics and Probability

Citations in the corpus

On q-Parametric Szász-Mirakjan Operators (2010, background): “[1]-[27]). Recently, an intensive research has been conducted on polynomials and operators based on q-integers, see [3], [4], [5], [9], [15], [17], [21]-[ 14 ].”