On the positivity of Fourier transform of the stretched Gaußian function

Hanwen Liu

On the positivity of Fourier transform of the stretched Gaußian function

Hanwen Liu

Abstract

The stretched Gaußian 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 Gaußian 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 $\mathbb{R}^d$.

On the positivity of Fourier transform of the stretched Gaußian function

Abstract

The stretched Gaußian function

, as a real function defined on

, has found numerous applications in mathematics and physics. For instance, to describe results from spectroscopy or inelastic scattering, the Fourier transform of the stretched Gaußian function is needed. For

, we prove that the Fourier transform of

is everywhere positive on

Paper Structure (4 theorems, 21 equations)

This paper contains 4 theorems, 21 equations.

Key Result

Proposition 1

Let $\Omega$ be a non-empty open subset of $\mathbb{R}$, and let $g\colon \Omega \rightarrow \mathbb{R}$ be a smooth function. For the smooth function the formula holds for all $n \in \mathbb{N}$ and $x \in \Omega$.

Theorems & Definitions (9)

Proposition 1
proof
Definition 2
Lemma 3
proof
Theorem 4: Bernstein
proof
Corollary 5
proof