Kernels for composition of positive linear operators

TL;DR

This paper explores the composition and kernel properties of Bernstein--Durrmeyer and Szász--Mirakjan--Durrmeyer operators, revealing new identities, eigenstructure representations, and proofs of known results.

Contribution

It introduces new kernel identities for compositions, eigenstructure-based representations, and simplified proofs for operator properties, advancing understanding of these positive linear operators.

Findings

Established new identities for composed kernels.

Derived eigenstructure-based representations of operator iterates.

Provided an elementary proof for a known composition result.

Abstract

This paper investigates the composition of Bernstein--Durrmeyer operators and Sz\'asz--Mirakjan--Durrmeyer operators, focusing on the structure and properties of the associated kernel functions. In the case of the Bernstein--Durrmeyer operators, we establish new identities for the kernel arising from the composition of two and three operators, from which the commutativity of these operators follows naturally. Building on the eigenstructure of the Bernstein--Durrmeyer operator $M_n$, we obtain a representation of the iterate $M_n^r$ as a linear combination of the operators $M_k$, for $k=0,1,\dots,n$. We also address the composition of Sz\'asz--Mirakjan--Durrmeyer operators and revisit a known result giving an elementary proof.