Kernels for composition of positive linear operators

Abstract

This paper investigates the composition of Bernstein–Durrmeyer operators and Szász–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. Like the well-known representation in terms of Legendre polynomials, they show the commutativity of these operators naturally. While this Legendre representation contains all possible products $p_{k,i}\left(x\right)p_{k,j}\left(y\right),0\le i,j\le k\le n$, of Bernstein basis polynomials, the new representation has the beautiful property to contain only products $p_{k,\ell }\left(x\right)p_{k,\ell }\left(y\right),0\le \ell \le k\le n$, where n is the smallest degree of the Bernstein–Durrmeyer polynomials involved. This fact immediately implies that the composition can be written as a linear combination of the operators themselves. Building on the eigenstructure of the Bernstein–Durrmeyer operator $M_n$, we obtain a representation of its r-th iterate as a linear combination of the operators $M_k$, for $k=0,1,\dots ,n$. We also address the composition of Szász–Mirakjan–Durrmeyer operators and revisit a known result giving an elementary proof.