Positivity and Fourier integrals over regular hexagon

Let f∈L1(R2)f∈L1(R2) and let f̂ be its Fourier integral. We study summability of the partial integral Sρ,H(x)=∫{‖y‖H≤ρ}eix⋅yf̂(y)dy, where ‖y‖H denotes the uniform norm taken over the regular hexagonal domain. We prove that the Riesz (R,δ)(R,δ) means of the inverse Fourier integrals are nonnegative if and only if δ≥2δ≥2. Moreover, we describe a class of ‖⋅‖H-radial functions that are positive definite on R2R2.