Revista Matemática Iberoamericana


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

Average decay of Fourier transforms and geometry of convex sets

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

(1) Dip. di Ingegneria Gestionale e dell'Informazione, Università di Bergamo, Viale G. Marconi 5, 24044, DALMINE, ITALY
(2) Dipartimento di Matematica “F. Enriques”, Università di Milano, Via Saldini 50, 20133, MILANO, ITALY
(3) Dipartimento Matematica ‘F. Enriques’, Università di Milano, Via Saldini 50, 20133, 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