Hermite expansions of functions from the weighted Hardy class
Abstract
In this paper, we analyze a function space consisting of functions for which both the function and its Fourier transform exhibit Gaussian decay together with exponential growth governed by suitable weight functions. First, we examine logarithmic-type weights, in which case these function spaces are equivalent to Pilipovi\'c spaces. In this setting, we establish a decay estimate for the Hermite coefficients of functions. Furthermore, by combining these estimates with the asymptotic behavior of Hermite functions, we prove a decay rate for solutions to the harmonic oscillator Schr\"odinger equation. Second, we consider a class of weights and prove the exponential decay of the Hermite projection operators on these spaces by analyzing Laguerre expansions and the short-time Fourier transform. Additionally, we revisit the subcritical Hardy uncertainty principle and obtain a partial improvement toward a conjecture posed by Vemuri.
Keywords
Cite
@article{arxiv.2605.07673,
title = {Hermite expansions of functions from the weighted Hardy class},
author = {Satyajyoti Achar and Manish Chaurasia and Ramesh Manna},
journal= {arXiv preprint arXiv:2605.07673},
year = {2026}
}