Simultaneous approximation by operators of exponential type

TL;DR

This paper investigates the asymptotic behavior of exponential-type positive linear operators, demonstrating that their asymptotic expansions can be differentiated term-by-term, generalizing several classical approximation results.

Contribution

It introduces a general theorem on differentiability of asymptotic expansions for a broad class of exponential-type operators, extending previous specific results.

Findings

Asymptotic expansions can be differentiated term-by-term under certain conditions.

The theorem encompasses classical operators like Bernstein polynomials.

It generalizes the Voronovskaja formula for exponential-type operators.

Abstract

There are many results on the simultaneous approximation by sequences of special positive linear operators. In the year 1978, Ismail and May as well as Volkov independently studied operators of exponential type covering the most classical approximation operators. In this paper we study asymptotic properties of these class of operators. We prove that under certain conditions, asymptotic expansions for sequences of operators belonging to a slightly larger class of operators, can be differentiated term-by-term. This general theorem contains several results which were previously obtained by several authors for concrete operators. One corollary states, that the complete asymptotic expansion for the Bernstein polynomials can be differentiated term-by-term. This implies a well-known result on the Voronovskaja formula obtained by Floater.