# On summability of measures with thin spectra

Maria Roginskaya^{[1]}; Michaël Wojciechowski

- [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)

Annales de l’institut Fourier (2004)

- Volume: 54, Issue: 2, page 413-430
- ISSN: 0373-0956

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topRoginskaya, 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 $p>1$. 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 $p>1$. 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 -

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