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On the Fourier transform of the symmetric decreasing rearrangements

Philippe Jaming[1]

  • [1] Université d’Orléans - Faculté des Sciences MAPMO UMR CNRS 6628 Fédération Denis Poisson, FR CNRS 2964 BP 6759 45067 Orléans Cedex 2 (France) and Université Bordeaux 1 Institut de Mathématiques de Bordeaux UMR CNRS 5251 351, cours de la Libération 33405 TALENCE cedex (France)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 1, page 53-77
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Jaming, Philippe. "On the Fourier transform of the symmetric decreasing rearrangements." Annales de l’institut Fourier 61.1 (2011): 53-77. <http://eudml.org/doc/219848>.

@article{Jaming2011,
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 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.},
affiliation = {Université d’Orléans - Faculté des Sciences MAPMO UMR CNRS 6628 Fédération Denis Poisson, FR CNRS 2964 BP 6759 45067 Orléans Cedex 2 (France) and Université Bordeaux 1 Institut de Mathématiques de Bordeaux UMR CNRS 5251 351, cours de la Libération 33405 TALENCE cedex (France)},
author = {Jaming, Philippe},
journal = {Annales de l’institut Fourier},
keywords = {Fourier transform; rearrangement inequalities; Bessel functions},
language = {eng},
number = {1},
pages = {53-77},
publisher = {Association des Annales de l’institut Fourier},
title = {On the Fourier transform of the symmetric decreasing rearrangements},
url = {http://eudml.org/doc/219848},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Jaming, Philippe
TI - On the Fourier transform of the symmetric decreasing rearrangements
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 1
SP - 53
EP - 77
AB - 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.
LA - eng
KW - Fourier transform; rearrangement inequalities; Bessel functions
UR - http://eudml.org/doc/219848
ER -

References

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