Quantitative estimates for nonlinear sampling Kantorovich operators

TL;DR

This paper provides quantitative approximation estimates for nonlinear sampling Kantorovich operators within Orlicz spaces, extending classical results and improving bounds in $L^p$-spaces, with applications to specific kernels.

Contribution

It introduces new quantitative estimates for nonlinear sampling Kantorovich operators in Orlicz spaces, including $L^p$, and improves existing bounds in $L^p$-spaces using a direct approach.

Findings

Quantitative approximation estimates in Orlicz spaces.

Qualitative order of approximation for Lipschitz functions.

Improved bounds in $L^p$-spaces using a direct method.

Abstract

In this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of continuity in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $L^{p}$-spaces, $1\leq p<\infty $, and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces. Further, the qualitative order of approximation has been obtained assuming $f$ in suitable Lipschitz classes. The above estimates achieved in the general setting of Orlicz spaces, have been also improved in the $L^p$-case, using a direct approach suitable to this context. At the end, we consider the particular cases of the nonlinear sampling Kantorovich operators constructed by using some special kernels.