Hermite expansions of functions from the weighted Hardy class

TL;DR

This paper studies function spaces with Gaussian decay properties, establishing decay estimates for Hermite coefficients and solutions to the harmonic oscillator Schrödinger equation, and analyzing Hermite projections and uncertainty principles.

Contribution

It introduces new decay estimates for Hermite coefficients and solutions, connecting weighted Hardy spaces with Pilipović spaces and improving uncertainty principle results.

Findings

Decay estimate for Hermite coefficients of functions in weighted Hardy spaces

Proven exponential decay of Hermite projection operators on these spaces

Partial improvement on the Vemuri conjecture regarding Hardy uncertainty principles

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.