Revista Matemática Iberoamericana


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Volume 14, Issue 3, 1998, pp. 519–560
DOI: 10.4171/RMI/244

Published online: 1998-12-31

Average decay of Fourier transforms and geometry of convex sets

Luca Brandolini[1], Marco Rigoli[2] and Giancarlo Travaglini[3]

(1) Università di Bergamo, Dalmine, Italy
(2) Università di Milano, Italy
(3) Università di Milano, Italy

Let $B$ be a convex body in $\mathbb R^2$ with piecewise smooth boundary and let $\widehat {\chi}_B$ denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical $L^p$-averages of $\widehat {\chi}_B$ and we relate our analysis to a problem in the geometry of convex sets. As an application we obtain sharp results on the average number of integer lattice points in large bodies randomly positioned in the plane.

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Brandolini Luca, Rigoli Marco, Travaglini Giancarlo: Average decay of Fourier transforms and geometry of convex sets. Rev. Mat. Iberoamericana 14 (1998), 519-560. doi: 10.4171/RMI/244