Almost everywhere convergence of inverse Fourier transforms | Macquarie University ResearchOnline

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Title: Almost everywhere convergence of inverse Fourier transforms
Related: Proceedings of the American Mathematical Society, Vol. 134, Issue 6, p.1651-1660
DOI: 10.1090/S0002-9939-05-08329-2
Publisher: American Mathematical Society
Date: 2006
Author/Creator: Colzani, Leonardo
Author/Creator: Meaney, Christopher
Author/Creator: Prestini, Elena
Description: We show that if log(2 − Δ)f ∈ L²(ℝd), 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 √log(2 − Δ)f ∈ L²(ℝd) and log log(4−Δ)f ∈ L²(ℝd). We also consider sequential convergence for general elements of L²(ℝd).
Description: 10 page(s)
Subject Keyword: Fourier transforms
Subject Keyword: Fourier-Stieltjes transforms
Subject Keyword: convergence of Fourier series
Resource Type: journal article
Organisation: Macquarie University. Dept. of Mathematics

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