Hermite expansions of functions from the weighted Hardy class
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| Format: | Preprint |
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2026
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| _version_ | 1866910200977424384 |
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| author | Achar, Satyajyoti Chaurasia, Manish Manna, Ramesh |
| author_facet | Achar, Satyajyoti Chaurasia, Manish Manna, Ramesh |
| contents | 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 Pilipović 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 Schrödinger 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_07673 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hermite expansions of functions from the weighted Hardy class Achar, Satyajyoti Chaurasia, Manish Manna, Ramesh Classical Analysis and ODEs Analysis of PDEs Complex Variables Primary 42C10, Secondary 42C05, 42B35, 33C45, 44A20 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 Pilipović 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 Schrödinger 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. |
| title | Hermite expansions of functions from the weighted Hardy class |
| topic | Classical Analysis and ODEs Analysis of PDEs Complex Variables Primary 42C10, Secondary 42C05, 42B35, 33C45, 44A20 |
| url | https://arxiv.org/abs/2605.07673 |