JournalsrmiVol. 14, No. 3pp. 519–560

Average decay of Fourier transforms and geometry of convex sets

  • Luca Brandolini

    Università di Bergamo, Dalmine, Italy
  • Marco Rigoli

    Università di Milano, Italy
  • Giancarlo Travaglini

    Università di Milano, Italy
Average decay of Fourier transforms and geometry of convex sets cover
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Abstract

Let BB be a convex body in R2\mathbb R^2 with piecewise smooth boundary and let χ^B\widehat {\chi}_B denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical LpL^p-averages of χ^B\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.

Cite this article

Luca Brandolini, Marco Rigoli, Giancarlo Travaglini, Average decay of Fourier transforms and geometry of convex sets. Rev. Mat. Iberoam. 14 (1998), no. 3, pp. 519–560

DOI 10.4171/RMI/244