Commutativity and spectral properties for a general class of Szasz-Mirakjan-Durrmeyer operators

TL;DR

This work analyzes a general class of Szász-Mirakjan-Durrmeyer-type operators $S_{n,j}$ and their interaction with associated differential operators. It establishes a precise commutativity structure: the composition $S_{m,j}\circ S_{n,j}$ equals $S_{\frac{mn}{m+n},j}$, leading to commuting iterates and explicit iterative representations. The authors derive derivative formulas, introduce the operator $\mathcal{D}^{2}_j$ with $\mathcal{D}^{2}_j=(1-j)D+e_{1}D^{2}$, and prove commutativity with $S_{n,j}$ for sufficiently smooth functions, along with higher-order generalizations $\mathcal{D}^{2l}_j$. They construct eigenfunctions $g_{j,p}$ of $S_{n,j}$ in closed form, linked to modified Bessel functions $I_j$, and show these yield eigenvalues $e^{p/n}$; this yields a spectral picture for $S_{n,j}$. Finally, the eigenstructure of $\mathcal{D}^{2}_j$ is tied to the Bessel equation, establishing a correspondence between eigenfunctions of $S_{n,j}$ and $\mathcal{D}^{2}_j$ through a Voronovskaja-type connection and asymptotic results, thereby unifying operator commutativity, differential relations, and spectral properties for this broad operator family.

Abstract

In this paper we present commutativity results for a general class of Szasz-Mirakjan-Durrmeyer type operators and associated differential operators and investigate their eigenfunctions.

Theorems & Definitions (16)

• Definition 1
• Theorem 2
• Corollary 3
• Remark 4
• Proposition 5
• Remark 6
• Remark 7
• Theorem 8
• Proposition 9
• Corollary 10