Increasingly global convergence of Hermite serie

TL;DR

This paper investigates the conditions under which Hermite series converge in various function spaces, providing characterizations for norm and modular convergence.

Contribution

It offers new necessary and sufficient conditions for Hermite series convergence in rearrangement invariant and Orlicz spaces, advancing theoretical understanding.

Findings

Characterization of norm convergence in rearrangement invariant spaces.

Necessary and sufficient conditions for convergence in Orlicz modular.

Analysis of the impact of truncation speed on convergence.

Abstract

We study the convergence of the Hermite series of measurable functions on the real line. We characterize the norm convergence of truncated partial Hermite sums in rearrangement invariant spaces provided that the truncations vanish sufficiently slowly. Moreover, we provide the necessary and sufficient conditions for convergence in the Orlicz modular.