Advancements in nonlinear exponential sampling: convergence, quantitative analysis and Voronovskaya-type formula

TL;DR

This paper introduces nonlinear exponential Kantorovich sampling series, establishing convergence, asymptotic formulas, and quantitative estimates, extending results to Mellin-Orlicz spaces with applications to functions in Lipschitz classes.

Contribution

It presents new convergence and asymptotic results for nonlinear exponential sampling series, extending analysis to Mellin-Orlicz spaces and providing quantitative convergence estimates.

Findings

Pointwise and uniform convergence established.

A Voronovskaja-type asymptotic formula derived.

Quantitative convergence rates provided for specific function classes.

Abstract

In this paper, we introduce the nonlinear exponential Kantorovich sampling series. We establish pointwise and uniform convergence properties and a nonlinear asymptotic formula of the Voronovskaja-type given in terms of the limsup. Furthermore, we extend these convergence results to Mellin-Orlicz spaces with respect to the logarithmic (Haar) measure. Quantitative results are also given, using the log-modulus of continuity and the log-modulus of smoothness, respectively, for log-uniformly continuous functions and for functions in Mellin-Orlicz spaces. Consequently, the qualitative order of convergence can be obtained in case of functions belonging to suitable Lipschitz (log-H\"olderian) classes.