Positivity and Fourier integrals over regular hexagon

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 $\mathbb{R}^2$.