On the Fourier transform of the symmetric decreasing rearrangements
JAMING, Philippe
Mathématiques - Analyse, Probabilités, Modélisation - Orléans [MAPMO]
Institut de Mathématiques de Bordeaux [IMB]
Mathématiques - Analyse, Probabilités, Modélisation - Orléans [MAPMO]
Institut de Mathématiques de Bordeaux [IMB]
JAMING, Philippe
Mathématiques - Analyse, Probabilités, Modélisation - Orléans [MAPMO]
Institut de Mathématiques de Bordeaux [IMB]
< Reduce
Mathématiques - Analyse, Probabilités, Modélisation - Orléans [MAPMO]
Institut de Mathématiques de Bordeaux [IMB]
Language
en
Article de revue
This item was published in
Annales de l'Institut Fourier. 2011, vol. 61, p. 53-77
Association des Annales de l'Institut Fourier
English Abstract
Inspired by work of Montgomery on Fourier series and Donoho-Strak in signal processing, we investigate two families of rearrangement inequalities for the Fourier transform. More precisely, we show that the $L^2$ behavior ...Read more >
Inspired by work of Montgomery on Fourier series and Donoho-Strak in signal processing, we investigate two families of rearrangement inequalities for the Fourier transform. More precisely, we show that the $L^2$ behavior of a Fourier transform of a function over a small set is controlled by the $L^2$ behavior of the Fourier transform of its symmetric decreasing rearrangement. In the $L^1$ case, the same is true if we further assume that the function has a support of finite measure. As a byproduct, we also give a simple proof and an extension of a result of Lieb about the smoothness of a rearrangement. Finally, a straightforward application to solutions of the free Shrödinger equation is given.Read less <
English Keywords
Fourier transform
rearrangement inequalities
Bessel functions
ANR Project
Analyse Harmonique et Problèmes Inverses - ANR-07-BLAN-0247
Origin
Hal imported