# 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

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topKozma, 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 -

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