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Approximation degree of Durrmeyer-Bézier type operators.

Purshottam N Agrawal1, Serkan Araci2, Martin Bohner3

  • 11Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India.

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Summary

This study introduces Bézier variants of mixed hybrid operators, generalizing existing mathematical operators. The research investigates approximation properties and convergence rates for these new operators.

Keywords:
Baskakov–Szász type operatorsBounded variationDitzian–Totik modulus of smoothnessRate of convergence

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Area of Science:

  • Approximation Theory
  • Numerical Analysis
  • Mathematical Operators

Background:

  • Mixed hybrid operators generalize Phillips and Baskakov-Szász operators.
  • Bézier variants offer new avenues for approximation analysis.

Purpose of the Study:

  • To introduce and analyze Bézier variants of mixed hybrid operators.
  • To investigate the approximation capabilities of these novel operators.
  • To establish convergence rates and direct approximation theorems.

Main Methods:

  • Utilizing Lipschitz class functions and modulus of continuity.
  • Applying a weighted space approach.
  • Employing the unified Ditzian-Totik modulus of smoothness.
  • Analyzing functions with derivatives of bounded variation.

Main Results:

  • Established degree of approximation for the new operators.
  • Developed a direct approximation theorem.
  • Determined the rate of convergence for specific function classes.

Conclusions:

  • The Bézier variant of mixed hybrid operators exhibits significant approximation properties.
  • The study provides a comprehensive analysis of convergence rates and approximation theorems.
  • These findings contribute to the understanding of advanced mathematical operators in approximation theory.