A Nice Representation for a Link between Baskakov- and Sz\'asz-Mirakjan-Durrmeyer Operators and their Kantorovich Variants

TL;DR

This paper establishes a mathematical link between Baskakov-Durrmeyer and Szász-Mirakjan-Durrmeyer operators and their Kantorovich variants, providing new representations and solving open problems in approximation theory.

Contribution

It introduces a new representation for Kantorovich variants of arbitrary order, proving convexity properties and solving an open problem in the field.

Findings

Derived a useful representation for Kantorovich variants

Proved convexity properties of linking operators

Solved an open problem in approximation theory

Abstract

In this paper we consider a link between Baskakov-Durrmeyer type operators and corresponding Kantorovich type modifications of their classical variants. We prove a useful representation for Kantorovich variants of arbitrary order which leads to a simple proof of convexity properties for the linking operators. This also solves an open problem. Another open problem is presented at the end of the paper.