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Almost everywhere convergence of inverse Fourier transforms

Authors:	Leonardo Colzani Christopher Meaney Elena Prestini

Affiliation:	Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Edificio U5, via Cozzi 53, 20125 Milano, Italy ; Department of Mathematics, Macquarie University, North Ryde NSW 2109, Australia Elena Prestini ; Dipartimento di Matematica, Università di Roma ``Tor Vergata', Via della Ricerca Scientifica, 00133 Roma, Italy

Abstract:	We show that if $log(2-Delta)fin L^2({mathbb{R}}^d)$ , then the inverse Fourier transform of converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0. Our proof is based on a simple application of the Rademacher-Menshov Theorem. In the special case of spherical partial integrals, the theorem was proved by Carbery and Soria. We obtain some partial results when $sqrt{log(2-Delta)}fin L^2({mathbb{R}}^d)$ and $loglog(4-Delta)fin L^2({mathbb{R}}^d)$ . We also consider sequential convergence for general elements of $L^2({mathbb{R}}^d)$ .

Keywords:	Rademacher-Menshov Theorem inverse Fourier transform series of orthogonal functions

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