l-1 summability of multiple Fourier integrals and positivity

Abstract

Let f∈L¹(ℝ^d), and let fˆ be its Fourier integral. We study summability of the l-1 partial integral S⁽¹⁾_{R, d}(f; x)=∫_{[mid ]v[mid ][les ]R} e^iv·x fˆ(v)dv, x∈ℝ^d; note that the integral ranges over the l₁-ball in ℝ^d centred at the origin with radius R>0. As a central result we prove that for δ[ges ]2d−1 the l-1 Riesz (R, δ) means of the inverse Fourier integral are positive, the lower bound being best possible. Moreover, we will give an l-1 analogue of Schoenberg's modification of Bochner's theorem on positive definite functions on ℝ^d as well as an extention of Polya's sufficiency condition.