Asymptotic properties for a general class of Szasz-Mirakjan-Durrmeyer operators

TL;DR

This work introduces a unifying family of Szász--Mirakjan--Durrmeyer type operators $S_{n,j}$ depending on an integer $j$, which preserves constants and $x^{j}$ for $j\in\mathbb{N}$ and provides local approximation for functions of exponential growth on the positive half-line. It develops a thorough asymptotic theory by expressing moments and central moments through the differential operator $\mathcal{D}_{j}^{2}$ and its powers, yielding a complete asymptotic expansion $S_{n,j}f(x)=f(x)+\sum_{k\ge1}\frac{1}{k!n^{k}}(\mathcal{D}_{j}^{2k}f)(x)$ under suitable smoothness, with special cases recovering classical and Phillips-type operators. The paper also proves a localization property enabling exponential decay when $f$ vanishes near $x$, and provides rate-of-convergence bounds via modulus of continuity. Together, these results unify and extend known operator families, offer detailed simultaneous approximations, and broaden the applicability to exponentially growing functions on $[0,\infty)$.

Abstract

In this paper we introduce a general family of Szász--Mirakjan--Durrmeyer type operators depending on an integer parameter $j \in \mathbb{Z}$. They can be viewed as a generalization of the Szász--Mirakjan--Durrmeyer operators [9], Phillips operators [11] and corresponding Kantorovich modifications of higher order. For $j\in {\mathbb{N}}$, these operators possess the exceptional property to preserve constants and the monomial $x^{j}$. It turns out, that an extension of this family covers certain well-known operators studied before, so that the outcoming results could be unified. We present the complete asymptotic expansion for the sequence of these operators. All its coefficients are given in a concise form. In order to prove the expansions for the class of locally integrable functions of exponential growth on the positive half-axis, we derive a localization result which is interesting in itself.

Theorems & Definitions (26)

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