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Gady Kozma; Alexander Olevskiĭ
Singular distributions, dimension of support, and symmetry of Fourier transform
(Distributions singulières, dimension du support et symétrie de la transformation de Fourier.)
Annales de l'institut Fourier, 63 no. 4 (2013), p. 1205-1226, doi: 10.5802/aif.2801
Article PDF | Reviews MR 3137353 | Zbl 06359587
Class. Math.: 42A63, 42A50, 42A20, 28A80
Keywords: Hausorff dimension, Frostman’s theorem, Fourier symmetry
[1] Robert D. Berman, “Boundary limits and an asymptotic Phragmén-Lindelöf theorem for analytic functions of slow growth”, Indiana University Mathematics Journal 41/2 (1992), p. 465-481 Article | MR 1183354 | Zbl 0759.30015
[2] Arne Beurling, Sur les spectres des fonctions, [French, on the spectrum of functions], Analyse Harmonique, Colloq. Internat. CNRS 15, Paris, 1949, p. 9-29 MR 33367 | Zbl 0040.21102
[3] B. E. J. Dahlberg, “On the radial boundary values of subharmonic functions”, Math. Scand. 40 (1977), p. 301-317 MR 460668 | Zbl 0371.31001
[4] Kenneth Falconer, Fractal geometry, Mathematical foundations and applications, John Wiley & Sons, Inc., Hoboken, New Jersey, 2003 MR 2118797 | Zbl 0689.28003
[5] S. V. Hruščev & V. V. Peller, Hankel operators of Schatten-von Neumann class and their application to stationary processes and best approximations, Appendix to the English edition of: N. K. Nikol’skiĭ,
[6] J.-P. Kahane, Some random series of functions, Cambridge Studies in Advanced Mathematics 5, Cambridge University Press, Cambridge, 1985 MR 833073 | Zbl 0571.60002
[7] J.-P. Kahane & Raphaël Salem, Ensembles parfaits et séries trigonométriques, [French, Perfect sets and trigonometric series], Second ed., With notes by Kahane, Thomas W. Körner, Russell Lyons and Stephen William Drury. Hermann, Paris, 1994 MR 1303593 | Zbl 0856.42001
[8] Yitzhak Katznelson, An introduction to harmonic analysis, Dover Publications, Inc., New York, 1976 MR 422992 | Zbl 0352.43001
[9] Robert Kaufman, “On the theorem of Jarník and Besicovitch”, Acta Arith. 39:3 (1981), p. 265-267 MR 640914 | Zbl 0468.10031
[10] Alexander S. Kechris & Alain Louveau, Descriptive set theory and the structure of sets of uniqueness, London Mathematical Society Lecture Note Series 128, Cambridge University Press, Cambridge, 1987 MR 953784 | Zbl 0642.42014
[11] Gady Kozma & Alexander Olevskiĭ, “A null series with small anti-analytic part”, Comptes Rendus de l’Académie des Sciences Paris, Série I Mathématique 336:6 (2003), p. 475-478 Article | MR 1975082 | Zbl 1035.42001
[12] Gady Kozma & Alexander Olevskiĭ, “Analytic representation of functions and a new quasi-analyticity threshold”, Annals of Math. 164:3 (2006), p. 1033-1064 Article | MR 2259252 | Zbl 1215.42012
[13] Gady Kozma & Alexander Olevskiĭ, “Is PLA large?”, Bull. Lond. Math. Soc. 39:2 (2007), p. 173-180 Article | MR 2323445 | Zbl 1124.42006
[14] K. de Leeuw & Yitzhak Katznelson, “The two sides of a Fourier-Stieltjes transform and almost idempotent measures”, Israel J. Math. 8 (1970), p. 213-229 Article | MR 275060 | Zbl 0198.47901
[15] Nir Lev & Alexander Olevskiĭ, “Wiener’s ‘closure of translates’ problem and Piatetski-Shapiro’s uniqueness phenomenon”, To appear in Ann. Math.
[16] Pertti Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, Cambridge, 1995 MR 1333890 | Zbl 0819.28004
[17] Gerd Mockenhaupt, “Salem sets and restriction properties of Fourier transforms”, Geom. Funct. Anal. 10:6 (2000), p. 1579-1587 Article | MR 1810754 | Zbl 0974.42013
[18] I. I. Pyateckiĭ-Šapiro, “Дополнение к работе “К проблеме единственности разложения функции в тригонометрический ряд””, Moskov. Gos. Univ. Uč. Zap. Mat. 165 (1954), p. 79-97, [Russian, Supplement to the work “On the problem of uniqueness of expansion of a function in a trigonometric series”] English translation in Selected Works of Ilya Piatetski-Shapiro, AMS Collected Works, vol. 15, 2000
[19] Alexandre Rajchman, “Une classe de séries trigonométriques qui convergent presque partout vers zéro”, [French, A class of trigonometric series converging almost everywhere to zero] Math. Ann. 101:1 (1929), p. 686-700 MR 1512561 | JFM 55.0162.04
[20] Masakazu Tamashiro, “Dimensions in a separable metric space”, Kyushu J. Math. 49:1 (1995), p. 143-162 Article | MR 1339704 | Zbl 0905.54023
[21] Antoni Zygmund, Trigonometric series. Vol. I, II, With a foreword by Robert A. Fefferman. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002 MR 1963498 | Zbl 1084.42003
Gady Kozma; Alexander Olevskiĭ
Singular distributions, dimension of support, and symmetry of Fourier transform
(Distributions singulières, dimension du support et symétrie de la transformation de Fourier.)
Annales de l'institut Fourier, 63 no. 4 (2013), p. 1205-1226, doi: 10.5802/aif.2801
Article PDF | Reviews MR 3137353 | Zbl 06359587
Class. Math.: 42A63, 42A50, 42A20, 28A80
Keywords: Hausorff dimension, Frostman’s theorem, Fourier symmetry
Résumé - 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.
Bibliography
[2] Arne Beurling, Sur les spectres des fonctions, [French, on the spectrum of functions], Analyse Harmonique, Colloq. Internat. CNRS 15, Paris, 1949, p. 9-29 MR 33367 | Zbl 0040.21102
[3] B. E. J. Dahlberg, “On the radial boundary values of subharmonic functions”, Math. Scand. 40 (1977), p. 301-317 MR 460668 | Zbl 0371.31001
[4] Kenneth Falconer, Fractal geometry, Mathematical foundations and applications, John Wiley & Sons, Inc., Hoboken, New Jersey, 2003 MR 2118797 | Zbl 0689.28003
[5] S. V. Hruščev & V. V. Peller, Hankel operators of Schatten-von Neumann class and their application to stationary processes and best approximations, Appendix to the English edition of: N. K. Nikol’skiĭ,
Treatise on the shift operator, Translated from the Russian by Jaak Peetre. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 273, Springer-Verlag, 1986, p. 399–454[6] J.-P. Kahane, Some random series of functions, Cambridge Studies in Advanced Mathematics 5, Cambridge University Press, Cambridge, 1985 MR 833073 | Zbl 0571.60002
[7] J.-P. Kahane & Raphaël Salem, Ensembles parfaits et séries trigonométriques, [French, Perfect sets and trigonometric series], Second ed., With notes by Kahane, Thomas W. Körner, Russell Lyons and Stephen William Drury. Hermann, Paris, 1994 MR 1303593 | Zbl 0856.42001
[8] Yitzhak Katznelson, An introduction to harmonic analysis, Dover Publications, Inc., New York, 1976 MR 422992 | Zbl 0352.43001
[9] Robert Kaufman, “On the theorem of Jarník and Besicovitch”, Acta Arith. 39:3 (1981), p. 265-267 MR 640914 | Zbl 0468.10031
[10] Alexander S. Kechris & Alain Louveau, Descriptive set theory and the structure of sets of uniqueness, London Mathematical Society Lecture Note Series 128, Cambridge University Press, Cambridge, 1987 MR 953784 | Zbl 0642.42014
[11] Gady Kozma & Alexander Olevskiĭ, “A null series with small anti-analytic part”, Comptes Rendus de l’Académie des Sciences Paris, Série I Mathématique 336:6 (2003), p. 475-478 Article | MR 1975082 | Zbl 1035.42001
[12] Gady Kozma & Alexander Olevskiĭ, “Analytic representation of functions and a new quasi-analyticity threshold”, Annals of Math. 164:3 (2006), p. 1033-1064 Article | MR 2259252 | Zbl 1215.42012
[13] Gady Kozma & Alexander Olevskiĭ, “Is PLA large?”, Bull. Lond. Math. Soc. 39:2 (2007), p. 173-180 Article | MR 2323445 | Zbl 1124.42006
[14] K. de Leeuw & Yitzhak Katznelson, “The two sides of a Fourier-Stieltjes transform and almost idempotent measures”, Israel J. Math. 8 (1970), p. 213-229 Article | MR 275060 | Zbl 0198.47901
[15] Nir Lev & Alexander Olevskiĭ, “Wiener’s ‘closure of translates’ problem and Piatetski-Shapiro’s uniqueness phenomenon”, To appear in Ann. Math.
http://arxiv.org/abs/0908.0447MR 2811607 | Zbl 1231.42003[16] Pertti Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, Cambridge, 1995 MR 1333890 | Zbl 0819.28004
[17] Gerd Mockenhaupt, “Salem sets and restriction properties of Fourier transforms”, Geom. Funct. Anal. 10:6 (2000), p. 1579-1587 Article | MR 1810754 | Zbl 0974.42013
[18] I. I. Pyateckiĭ-Šapiro, “Дополнение к работе “К проблеме единственности разложения функции в тригонометрический ряд””, Moskov. Gos. Univ. Uč. Zap. Mat. 165 (1954), p. 79-97, [Russian, Supplement to the work “On the problem of uniqueness of expansion of a function in a trigonometric series”] English translation in Selected Works of Ilya Piatetski-Shapiro, AMS Collected Works, vol. 15, 2000
[19] Alexandre Rajchman, “Une classe de séries trigonométriques qui convergent presque partout vers zéro”, [French, A class of trigonometric series converging almost everywhere to zero] Math. Ann. 101:1 (1929), p. 686-700 MR 1512561 | JFM 55.0162.04
[20] Masakazu Tamashiro, “Dimensions in a separable metric space”, Kyushu J. Math. 49:1 (1995), p. 143-162 Article | MR 1339704 | Zbl 0905.54023
[21] Antoni Zygmund, Trigonometric series. Vol. I, II, With a foreword by Robert A. Fefferman. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002 MR 1963498 | Zbl 1084.42003
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