Harmonic Analysis

Convergence of Fourier Expansions

speaker: Leonardo Colzani (Università di Milano Bicocca)

abstract: We study the pointwise convergence of Fourier expansions under appropriate decay conditions on the convolution kernels and differentiability conditions on the functions expanded. We also estimate the Haudorff dimension of the set where divergence may occur. In particular, when the convolution kernel is the Fourier transform of a bounded set in the plane we recover a two dimensional analog of the Dirichlet theorem on the convergence of Fourier series of functions with bounded variation. Beside Fourier integrals on Euclidean spaces, we also consider expansions in eigenfunctions of elliptic operators on manifolds and we present examples of different behaviors between Fourier integrals, Fourier series and spherical harmonic expansions.

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