## Functiones et Approximatio Commentarii Mathematici

- Funct. Approx. Comment. Math.
- Volume 36 (2006), 45-70.

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

Carlo Bordaro and Ilaria Mantellini

#### 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.

#### Article information

**Source**

Funct. Approx. Comment. Math., Volume 36 (2006), 45-70.

**Dates**

First available in Project Euclid: 18 December 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.facm/1229616441

**Digital Object Identifier**

doi:10.7169/facm/1229616441

**Mathematical Reviews number (MathSciNet)**

MR2296638

**Subjects**

Primary: 47G10: Integral operators [See also 45P05]

Secondary: 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) [See also 45Gxx, 45P05] 26D15: Inequalities for sums, series and integrals 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

**Keywords**

Sampling operators discrete operators Orlicz spaces moduli of continuity

#### Citation

Bordaro, Carlo; Mantellini, Ilaria. Estimates of the approximation error for abstract sampling type operators in Orlicz spaces. Funct. Approx. Comment. Math. 36 (2006), 45--70. doi:10.7169/facm/1229616441. https://projecteuclid.org/euclid.facm/1229616441