We show that if log(2 - Δ)f ∞ L<sup>2</sup>(ℝ <sup>d</sup>), 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 ∈ L<sub>2</sub>(ℝ<sup>d</sup>) and log log(4 - Δ)f ∈ L<sub>2</sub>(ℝ<sup>d</sup>). We also consider sequential convergence for general elements of L<sup>2</sup>(ℝ <sup>d</sup>). © 2005 American Mathematical Society.
Colzani, L., Meaney, C., Prestini, E. (2006). Almost everywhere convergence of inverse Fourier transforms. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 134(6), 1651-1660 [10.1090/S0002-9939-05-08329-2].
Almost everywhere convergence of inverse Fourier transforms
COLZANI, LEONARDO;
2006
Abstract
We show that if log(2 - Δ)f ∞ L2(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 √log(2 - Δ)f ∈ L2(ℝd) and log log(4 - Δ)f ∈ L2(ℝd). We also consider sequential convergence for general elements of L2(ℝ d). © 2005 American Mathematical Society.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
