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Almost everywhere convergence of inverse Fourier transforms

Authors: Leonardo Colzani, Christopher Meaney and Elena Prestini
Journal: Proc. Amer. Math. Soc. 134 (2006), 1651-1660
MSC (2000): Primary 42B10, 43A50; Secondary 42C15
Published electronically: October 18, 2005
MathSciNet review: 2204276
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Abstract: We show that if $ \log(2-\Delta)f\in L^2({\mathbb{R}}^d)$, then the inverse Fourier transform of $ f$ 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)}f\in L^2({\mathbb{R}}^d)$ and $ \log\log(4-\Delta)f\in L^2({\mathbb{R}}^d)$. We also consider sequential convergence for general elements of $ L^2({\mathbb{R}}^d)$.

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Additional Information

Leonardo Colzani
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Edificio U5, via Cozzi 53, 20125 Milano, Italy

Christopher Meaney
Affiliation: Department of Mathematics, Macquarie University, North Ryde NSW 2109, Australia

Elena Prestini
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy

Keywords: Rademacher-Menshov Theorem, inverse Fourier transform, series of orthogonal functions
Received by editor(s): December 27, 2004
Published electronically: October 18, 2005
Additional Notes: The second author was partially supported by Progetto cofinanziato MIUR “Analisi Armonica”. We are grateful to Fulvio Ricci and the Centro di Ricerca Matematica Ennio De Giorgi for their hospitality
The third author was partially supported by Progetto cofinanziato MIUR “Analisi Armonica”.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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