L-1 SUMMABILITY OF MULTIPLE FOURIER INTEGRALS AND POSITIVITY

Let f is an element of L-1(R-d), and let (f) over cap be its Fourier i ntegral. We study summability of the l-1. partial integral S-R,d((1))( f; x) = integral(\upsilon/1 less than or equal to R)e(iv.x) (f) over c ap)(v)dv, x is an element of R-d; note that the integral ranges over t he l(1)-ball in R-d centred at the origin with radius R > 0. As a cent ral result we prove that for delta greater than or equal to 2d-1 the l -1 Riesz (R, delta) means of the inverse Fourier integral are positive , the lower bound being best possible. Moreover, ave will give an l-1. analogue of Schoenberg's modification of Bochner's theorem. on positi ve definite functions on R-d as well as an extention of Polya's suffic iency condition.