  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 $\mathbb{R}^2$.
