Some approximation results in Musielak-Orlicz spaces

TL;DR

This paper investigates the properties of Musielak-Orlicz spaces, providing conditions for translation operator continuity, convergence of approximate identities, and density of smooth functions, thereby enhancing understanding of their topological structure.

Contribution

It establishes new sufficient conditions for translation operator continuity and density of smooth functions in Musielak-Orlicz spaces, with applications to their topological properties.

Findings

Conditions for norm continuity of translation operators

Convergence of approximate identities in Musielak-Orlicz spaces

Density of smooth functions in LM spaces

Abstract

We give sufficient conditions for the continuity in norm of the translation operator in the Musielak-Orlicz LM spaces. An application to the convergence in norm of approximate identities is given, whereby we prove density results of the smooth functions in LM, in both modular and norm topologies. These density results are then applied to obtain basic topological properties.