# On summability of measures with thin spectra

• [1] Chalmers TH Göteborg University, Department of Mathematics, Eklandagatan 86, 41296 Göteborg (Suède), Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, I p., 00-950 Warszawa, (Pologne)
• Volume: 54, Issue: 2, page 413-430
• ISSN: 0373-0956

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## Abstract

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We study different conditions on the set of roots of the Fourier transform of a measure on the Euclidean space, which yield that the measure is absolutely continuous with respect to the Lebesgue measure. We construct a monotone sequence in the real line with this property. We construct a closed subset of ${ℝ}^{d}$ which contains a lot of lines of some fixed direction, with the property that every measure with spectrum contained in this set is absolutely continuous. We also give examples of sets with such property that every measure with spectrum contained in them is locally ${L}^{p}$ summable for suitable $p>1$. We discuss some related problems; among them we show that if a measure on the real line is such that its Fourier transform vanishes on the sequence ${\left({n}^{1/k}\right)}_{n=1}^{\infty }$, then both its singular and absolutely continuous parts share this property.

## How to cite

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Roginskaya, Maria, and Wojciechowski, Michaël. "On summability of measures with thin spectra." Annales de l’institut Fourier 54.2 (2004): 413-430. <http://eudml.org/doc/116116>.

@article{Roginskaya2004,
abstract = {We study different conditions on the set of roots of the Fourier transform of a measure on the Euclidean space, which yield that the measure is absolutely continuous with respect to the Lebesgue measure. We construct a monotone sequence in the real line with this property. We construct a closed subset of $\{\mathbb \{R\}\} ^d$ which contains a lot of lines of some fixed direction, with the property that every measure with spectrum contained in this set is absolutely continuous. We also give examples of sets with such property that every measure with spectrum contained in them is locally $L^p$ summable for suitable $p&gt;1$. We discuss some related problems; among them we show that if a measure on the real line is such that its Fourier transform vanishes on the sequence $(n^\{1/k\})_\{n=1\}^\infty$, then both its singular and absolutely continuous parts share this property.},
affiliation = {Chalmers TH Göteborg University, Department of Mathematics, Eklandagatan 86, 41296 Göteborg (Suède), Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, I p., 00-950 Warszawa, (Pologne)},
author = {Roginskaya, Maria, Wojciechowski, Michaël},
journal = {Annales de l’institut Fourier},
keywords = {Riesz sets; singular measures; support of Fourier transform; de Leeuw transference method; Young function; Hausdorff measure},
language = {eng},
number = {2},
pages = {413-430},
publisher = {Association des Annales de l'Institut Fourier},
title = {On summability of measures with thin spectra},
url = {http://eudml.org/doc/116116},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Roginskaya, Maria
AU - Wojciechowski, Michaël
TI - On summability of measures with thin spectra
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 2
SP - 413
EP - 430
AB - We study different conditions on the set of roots of the Fourier transform of a measure on the Euclidean space, which yield that the measure is absolutely continuous with respect to the Lebesgue measure. We construct a monotone sequence in the real line with this property. We construct a closed subset of ${\mathbb {R}} ^d$ which contains a lot of lines of some fixed direction, with the property that every measure with spectrum contained in this set is absolutely continuous. We also give examples of sets with such property that every measure with spectrum contained in them is locally $L^p$ summable for suitable $p&gt;1$. We discuss some related problems; among them we show that if a measure on the real line is such that its Fourier transform vanishes on the sequence $(n^{1/k})_{n=1}^\infty$, then both its singular and absolutely continuous parts share this property.
LA - eng
KW - Riesz sets; singular measures; support of Fourier transform; de Leeuw transference method; Young function; Hausdorff measure
UR - http://eudml.org/doc/116116
ER -

## References

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8. M. Roginskaya, Two multidimensional analogs of the F. and M. Riesz theorem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 255 (1998), 164-176 Zbl0981.42007MR1692912
9. J. H. Shapiro, Subspaces of ${L}^{p}\left(G\right)$ spanned by characters:$0lt;plt;1$, Israel J. Math. 29 (1978), 248-264 Zbl0382.46015MR477605
10. E.M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, (1993), Princeton University Press, Princeton, NJ Zbl0821.42001MR1232192
11. E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, (1971), Princeton Univ. Press, Princeton Zbl0232.42007MR304972
12. M. Wojciechowski, On the roots of the Fourier transform of Singular measures

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