Singular distributions, dimension of support, and symmetry of Fourier transform

Gady Kozma[1]; Alexander Olevskiĭ[2]

  • [1] Department of Mathematics, The Weizmann Institute of Science, Rehovot POB 76100, Israel.
  • [2] School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel.

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 4, page 1205-1226
  • ISSN: 0373-0956

Abstract

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We study the “Fourier symmetry” of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are:(i) A one-side extension of Frostman’s theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support;(ii) A construction of compacts of “critical” size, which support distributions (even pseudo-functions) with anti-analytic part belonging to l 2 .We also give examples of non-symmetry which may occur for measures with “small” support. A number of open questions are stated.

How to cite

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Kozma, Gady, and Olevskiĭ, Alexander. "Singular distributions, dimension of support, and symmetry of Fourier transform." Annales de l’institut Fourier 63.4 (2013): 1205-1226. <http://eudml.org/doc/275521>.

@article{Kozma2013,
abstract = {We study the “Fourier symmetry” of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are:(i) A one-side extension of Frostman’s theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support;(ii) A construction of compacts of “critical” size, which support distributions (even pseudo-functions) with anti-analytic part belonging to $l^\{2\}$.We also give examples of non-symmetry which may occur for measures with “small” support. A number of open questions are stated.},
affiliation = {Department of Mathematics, The Weizmann Institute of Science, Rehovot POB 76100, Israel.; School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel.},
author = {Kozma, Gady, Olevskiĭ, Alexander},
journal = {Annales de l’institut Fourier},
keywords = {Hausorff dimension; Frostman’s theorem; Fourier symmetry; Fourier transform; Schwartz distribution; Fourier coefficients; Hausdorff dimension; singular distribution; pseudo-function; Frostman's theorem},
language = {eng},
number = {4},
pages = {1205-1226},
publisher = {Association des Annales de l’institut Fourier},
title = {Singular distributions, dimension of support, and symmetry of Fourier transform},
url = {http://eudml.org/doc/275521},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Kozma, Gady
AU - Olevskiĭ, Alexander
TI - Singular distributions, dimension of support, and symmetry of Fourier transform
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 4
SP - 1205
EP - 1226
AB - We study the “Fourier symmetry” of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are:(i) A one-side extension of Frostman’s theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support;(ii) A construction of compacts of “critical” size, which support distributions (even pseudo-functions) with anti-analytic part belonging to $l^{2}$.We also give examples of non-symmetry which may occur for measures with “small” support. A number of open questions are stated.
LA - eng
KW - Hausorff dimension; Frostman’s theorem; Fourier symmetry; Fourier transform; Schwartz distribution; Fourier coefficients; Hausdorff dimension; singular distribution; pseudo-function; Frostman's theorem
UR - http://eudml.org/doc/275521
ER -

References

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