Estimates of the approximation error for abstract sampling type operators in Orlicz spaces

Abstract

We get some inequalities concerning the modular distance $I^\varphi_G[Tf -f]$ for bounded functions $f:G\rightarrow \mathbb{R}.$ Here $G$ is a locally compact Hausdorff topological space provided with a regular and $\sigma$-finite measure $\mu_G,$ $I^\varphi_G$ is the modular functional generating the Orlicz spaces $L^\varphi(G)$ and $T$ is a nonlinear integral operator of the form $$(Tf)(s) = \int_H K(s,t, f(t)) d\mu_H(t),$$ where $H$ is a closed subset of $G$ endowed with another regular and $\sigma$-finite measure $\mu_H.$ As a consequence we obtain a convergence theorem for a net of such operators. Some applications to discrete operators are given.