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The impact of the limit $q$-Durrmeyer operator on continuous functions

Övgü Gürel Yılmaz, Sofiya Ostrovska, Mehmet Turan

Abstract

The limit $q$-Durrmeyer operator, $D_{\infty,q},$ was introduced and its approximation properties were investigated by V. Gupta in 2008 during a study of $q$-analogues for the Bernstein-Durrmeyer operator. In the present work, this operator is investigated from a different perspective. More precisely, the growth estimates are derived for the entire functions comprising the range of $D_{\infty,q}$. The interrelation between the analytic properties of a function $f$ and the rate of growth for $D_{\infty,q}f$ are established, and the sharpness of the obtained results are demonstrated.

The impact of the limit $q$-Durrmeyer operator on continuous functions

Abstract

The limit -Durrmeyer operator, was introduced and its approximation properties were investigated by V. Gupta in 2008 during a study of -analogues for the Bernstein-Durrmeyer operator. In the present work, this operator is investigated from a different perspective. More precisely, the growth estimates are derived for the entire functions comprising the range of . The interrelation between the analytic properties of a function and the rate of growth for are established, and the sharpness of the obtained results are demonstrated.
Paper Structure (4 sections, 12 theorems, 59 equations)

This paper contains 4 sections, 12 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. Auxiliary Results
  4. Proofs of Main Results

Key Result

Theorem 2.1

For each $f \in C[0,1]$, the function $(D_{\infty,q}f)(x)$ admits an analytic continuation from $[0,1]$ as an entire function given by

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Corollary 2.7
  • Theorem 2.8
  • Theorem 2.9
  • ...and 10 more