Convergence of Hermite expansions in modulation spaces

TL;DR

This paper provides an elementary proof of the convergence behavior of Hermite expansions in modulation spaces, showing convergence for 1<p<+infinity and divergence at the endpoints, with extensions to higher dimensions.

Contribution

It offers a new elementary proof for Hermite expansion convergence in modulation spaces and extends results to higher dimensions, also analyzing divergence at boundary cases.

Findings

Hermite expansions converge in M^p(R) for 1<p<+infinity

Hermite expansions may diverge at p=1 and p=+infinity

Upper bounds for Zak transform of Hermite functions are established

Abstract

The aim of this paper is to give an elementary proof that Hermite expensions of a function $f$ in the modulation space $M^p(R)$ converges to $f$ in $M^p(R)$ when $1< p<+\infty$ and may diverge when $p = 1,\infty$. The result was previously established for $1< p<+\infty$ by Garling and Wojtaszczyk and for $p = 1,\infty$ by Lusky in an equivalent setting of Fock spaces by different methods. Higher dimesional results are also considered. In an appendix, we also establish upper bounds for the Zak transform of Hermite functions.