Positivity and Fourier integrals over regular hexagon

TL;DR

This paper investigates the positivity and summability of Fourier integrals over a regular hexagon, establishing conditions for nonnegativity of Riesz means and characterizing positive definite functions in this geometric setting.

Contribution

It introduces new results on the nonnegativity of Riesz means for Fourier integrals over hexagonal domains and characterizes positive definite functions with respect to the hexagonal norm.

Findings

Riesz $(R, oldsymbol{ heta})$ means are nonnegative if and only if $oldsymbol{ heta} \,\geq\, 2$.

Provides a class of functions that are positive definite with respect to the hexagonal norm.

Establishes conditions for summability and positivity in Fourier analysis over regular hexagons.

Abstract

Let $f \in L^1(\mathbb{R}^2)$ and let $\widehat f$ be its Fourier integral. We study summability of the partial integral $S_{\rho,\mathsf{H}}(x)=\int_{\{\|y\|_\mathsf{H} \le \rho\}} e^{i x\cdot y}\widehat f(y) dy$, where $\|y\|_\mathsf{H}$ denotes the uniform norm taken over the regular hexagonal domain. We prove that the Riesz $(R,\delta)$ means of the inverse Fourier integrals are nonnegative if and if $\delta \ge 2$. Moreover, we describe a class of $\|\cdot\|_\mathsf{H}$-radial functions that are positive definite on $\mathbb{R}^2$.