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

Abstract

top
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 ( n 1 / k ) n = 1 ∞ , then both its singular and absolutely continuous parts share this property.

How to cite

top

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

top
  1. A.B. Aleksandrov, Essays on non locally convex Hardy classes, 864 (1981), 1-89, Springer Zbl0482.46035MR643380
  2. K. de Leeuw, On L p multipliers, Annals of Math 81 (1965), 364-379 Zbl0171.11803MR174937
  3. R.E. Edwards, Fourier series, (1967), Holt, Rinehart and Winston, Inc. Zbl0189.06602
  4. V.P. Havin, B. Jörike, The Uncertainty Principle in Harmonic Analysis, (1994), Springer-Verlag, Berlin Zbl0827.42001MR1303780
  5. E. Hewitt, K. Ross, Abstract Harmonic Analysis, (1963), Springer-Verlag, Berlin-Goettingen-Heidelberg Zbl0115.10603
  6. Y. Meyer, Spectres des mesures et mesures absolument continues, Studia Math. 30 (1968), 87-99 Zbl0159.42501MR227697
  7. M.M. Rao, Z.D. Ren, Theory of Orlicz spaces, (1991), Marcel Dekker Inc., New York Zbl0724.46032MR1113700
  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 ( G ) spanned by characters: 0 l t ; p l t ; 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 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.