Convergence of Hermite expansions in modulation spaces

2026/01/31 by Philippe Jaming, Michael Speckbacher
Mathematics · Computer Science · #Mathematical Analysis and Transform Methods #Holomorphic and Operator Theory #Digital Filter Design and Implementation

paper · doi:10.1112/blms.70460

Abstract

Abstract The aim of this paper is to give an elementary proof that the Hermite expansion of a function in the modulation space converges to in when and may diverge when , or when . Although related results exist in the context of Fock spaces, their formulation and consequences for modulation spaces appear to have gone unnoticed in time–frequency analysis. More precisely, the result was previously established for by Garling and Wojtaszczyk, for by Lusky, and for by Li in the equivalent setting of Fock spaces, using different methods. Higher dimensional results are also considered.

Citations

• The approximation of almost time- and band-limited functions by their expansion in some orthogonal polynomials bases
• On the Fourier Series of Unbounded Harmonic Functions
• The Multiplier Problem for the Ball
• Mean Convergence of Expansions in Laguerre and Hermite Series
• Summability of Hermite expansions. I
• Summability of Hermite expansions. II
• On the Development of Arbitrary Functions in Series of Hermite's and Laguerre's Polynomials
• A Remark on Stirling's Formula

Related

• Hermite Multipliers on Modulation Spaces
• Pointwise convergence to initial data of heat and Hermite-heat equations in Modulation Spaces
• Hermite expansions for spaces of functions with nearly optimal time-frequency decay
• Wiener estimates on modulation spaces
• Approximation of almost time and band limited functions I: Hermite expansions