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l-1 summability of multiple Fourier integrals and positivity

Published online by Cambridge University Press:  01 July 1997

HUBERT BERENS  and
YUAN XU
HUBERT BERENS
Affiliation:
Mathematical Institute, University of Erlangen-Nuremberg, 91054 Erlangen, Germany
YUAN XU
Affiliation:
Department of Mathematics, The University of Oregon, Eugene, OR 97403, U.S.A.
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 122 , Issue 1 , July 1997 , pp. 149 - 172
Copyright
Cambridge Philosophical Society 1997

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