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Convergence of Hermite expansions in modulation spaces

Philippe Jaming, Michael Speckbacher

TL;DR

The paper addresses the convergence of Hermite expansions in modulation spaces $M^p({\mathbb{R}}^d)$ and identifies a sharp dichotomy: partial Hermite sums are uniformly bounded on $M^p$ if and only if $1<p<\infty$, while convergence fails for $p=1$ and $p=\infty$ due to unbounded operator norms. The authors provide an elementary, multiplier-based proof that connects Hermite multipliers on $M^p$ to Fourier multipliers on $L^p(\mathbb{T}^d)$, via the Bargmann transform to Fock spaces, and extend the results from the real line to higher dimensions. They also show convergence of Bochner–Riesz means in $M^p$ for a range of $\alpha$ and discuss higher-dimensional tensor-product Hermite systems. An appendix gives Zak transform bounds for Hermite functions, enriching time-frequency analysis techniques relevant to Gabor frames.

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.

Convergence of Hermite expansions in modulation spaces

TL;DR

The paper addresses the convergence of Hermite expansions in modulation spaces and identifies a sharp dichotomy: partial Hermite sums are uniformly bounded on if and only if , while convergence fails for and due to unbounded operator norms. The authors provide an elementary, multiplier-based proof that connects Hermite multipliers on to Fourier multipliers on , via the Bargmann transform to Fock spaces, and extend the results from the real line to higher dimensions. They also show convergence of Bochner–Riesz means in for a range of and discuss higher-dimensional tensor-product Hermite systems. An appendix gives Zak transform bounds for Hermite functions, enriching time-frequency analysis techniques relevant to Gabor frames.

Abstract

The aim of this paper is to give an elementary proof that Hermite expensions of a function in the modulation space converges to in when and may diverge when . The result was previously established for by Garling and Wojtaszczyk and for 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.
Paper Structure (6 sections, 12 theorems, 74 equations)

This paper contains 6 sections, 12 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Proof of Main Results
  3. Convergence in higher dimensions
  4. Upper frame and Bessel bounds of Gabor systems associated to Hermite Functions
  5. Statement of results
  6. The proof of Theorem \ref{['thm:app']}

Key Result

Theorem 1.1

There exists $A_\gamma$ such that, for every $f\in C(\gamma)$, $\|f\|_{M^1({\mathbb{R}})}\leq A_\gamma\|f\|_{C(\gamma)}$ if and only if $\gamma\geq 1/{4}$. There exists $B_\gamma>0$ such that, for every $f\in M^1({\mathbb{R}})$, $B_\gamma\|f\|_{C(\gamma)}\leq B_\gamma\|f\|_{M^1({\mathbb{R}})}$ if an

Theorems & Definitions (18)

  • Theorem 1.1: Janssen Jan
  • Theorem 1.2
  • Corollary 1.3: Garling-Wojtaszczyk GaWo, Lusky Lus
  • Proposition 1.4
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['thm:hermiteins0']}
  • Corollary 3.1
  • proof
  • ...and 8 more