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arXiv · 2605.07673

Hermite expansions of functions from the weighted Hardy class

Satyajyoti Achar·Manish Chaurasia·Ramesh Manna
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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.

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Satyajyoti Achar, Manish Chaurasia, Ramesh Manna. 2026-05-08. Hermite expansions of functions from the weighted Hardy class. https://arxiv.org/abs/2605.07673

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