Let f∈L1(ℝd),
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 ]Reiv·xfˆ(v)dv, x∈ℝd;
note that the integral ranges over the l1-ball in
ℝd
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 ℝd as well as
an
extention of Polya's sufficiency condition.