A Durrmeyer-type variant of Gr\"unwald Interpolation Operators

TL;DR

This paper introduces a Durrmeyer-type variant of Gr"unwald interpolation operators on $L^p$ spaces, proving their key properties and convergence with quantitative estimates.

Contribution

It constructs and analyzes a new Durrmeyer-type interpolation operator variant, extending classical Gr"unwald operators with proven boundedness and convergence results.

Findings

Operators are bounded in $L^p$-norm.

Convergence is established via a Korovkin-type theorem.

Quantitative convergence estimates are provided.

Abstract

In this paper, we construct a Durrmeyer-type variant of Gr\"unwald interpolation operators on the space $L^p[0,{\pi}]$. We prove their fundamental properties, including boundedness and convergence in the $L^p$-norm. We establish the convergence results using a Korovkin-type theorem in the setting of Banach function spaces. Furthermore, we obtain quantitative estimates for the convergence by means of the modulus of continuity and an appropriate $K$-functional.