Marcinkiewicz Sampling Theorem for Orlicz Spaces

TL;DR

This paper extends the Marcinkiewicz sampling theorem to Orlicz spaces, providing new conditions involving $N$-functions, Hilbert transform boundedness, and Matuszewska-Orlicz indices.

Contribution

It introduces a generalized sampling theorem in Orlicz spaces with novel criteria for Hilbert transform boundedness.

Findings

Sampling theorem holds under specific $N$-function conditions

New criterion established for Hilbert transform boundedness

Conditions involve restricted submultiplicativity and supermultiplicativity

Abstract

In the article we generalize the Marcinkiewicz sampling theorem in the context of Orlicz spaces. We establish conditions under which sampling theorem holds in terms of restricted submultiplicativity and supermultiplicativity of an $N$-function $\varphi$, boundedness of the Hilbert transform and Matuszewska-Orlicz indices. In addition we give a new criterion for boundedness of Hilbert transform on Orlicz space.