On the positivity of Fourier transform of the stretched Gau{\ss}ian function

Hanwen Liu

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arXiv · 2403.12843

On the positivity of Fourier transform of the stretched Gau{\ss}ian function

Hanwen Liu

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Abstract

The stretched Gau{\ss}ian function $f(\mathbf{x})=\exp \left(-\|\mathbf{x}\|^s\right)$, as a real function defined on $\mathbb{R}^d$, has found numerous applications in mathematics and physics. For instance, to describe results from spectroscopy or inelastic scattering, the Fourier transform of the stretched Gau{\ss}ian function is needed. For $s \in(0,2]$, we prove that the Fourier transform of $f(\mathbf{x})=\exp \left(-\|\mathbf{x}\|^s\right)$ is everywhere positive on $\mathbb{R}^d$.

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Hanwen Liu. 2024-03-19. On the positivity of Fourier transform of the stretched Gau{\ss}ian function. https://arxiv.org/abs/2403.12843

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