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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.03858 |
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| _version_ | 1866916778426236928 |
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| author | Imekraz, Rafik Latocca, Mickaël |
| author_facet | Imekraz, Rafik Latocca, Mickaël |
| contents | We continue the analysis of random series associated to the multidimensional harmonic oscillator $-Δ+ |x|^2$ on $\mathbb{R}^d$ with d \geq 2$$. More precisely we obtain a necessary and sufficient condition to get the almost sure uniform convergence on the whole space $\mathbb{R}^d$ . It turns out that the same condition gives the almost sure uniform convergence on the sphere $\mathbb{S}^{d-1}$ (despite $\mathbb{S}^{d-1}$ is a zero Lebesgue measure of $\mathbb{R}^d$). From a probabilistic point of view, our proof adapts a strategy used by the first author for boundaryless Riemannian compact manifolds. However, our proof requires sharp off-diagonal estimates of the spectral function of $-Δ+ |x|^2$ . Such estimates are obtained using elementary tools. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03858 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost Sure Uniform Convergence Of Random Hermite Series Imekraz, Rafik Latocca, Mickaël Functional Analysis We continue the analysis of random series associated to the multidimensional harmonic oscillator $-Δ+ |x|^2$ on $\mathbb{R}^d$ with d \geq 2$$. More precisely we obtain a necessary and sufficient condition to get the almost sure uniform convergence on the whole space $\mathbb{R}^d$ . It turns out that the same condition gives the almost sure uniform convergence on the sphere $\mathbb{S}^{d-1}$ (despite $\mathbb{S}^{d-1}$ is a zero Lebesgue measure of $\mathbb{R}^d$). From a probabilistic point of view, our proof adapts a strategy used by the first author for boundaryless Riemannian compact manifolds. However, our proof requires sharp off-diagonal estimates of the spectral function of $-Δ+ |x|^2$ . Such estimates are obtained using elementary tools. |
| title | Almost Sure Uniform Convergence Of Random Hermite Series |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2506.03858 |