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Random Menshov spectra

Author(s): Gady Kozma; Alexander Olevskii
Journal: Proc. Amer. Math. Soc. 131 (2003), 1901-1906.
MSC (2000): Primary 42A63, 42A61, 42A55
Posted: January 8, 2003
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Abstract: We show that the spectra $\Lambda$ of frequencies $\lambda$ obtained by random perturbations of the integers allows one to represent any measurable function on $\mathbb{R}$ by an almost everywhere converging sum of harmonics:

$\begin{displaymath}f=\sum _{\Lambda}c_{\lambda}e^{i\lambda t}.\end{displaymath}$

References:

1.

N. Bary, A Treatise on Trigonometric Series, vol. II, Pergamon Press Inc., NY (1964). MR 30:1347

2.

R.S. Davtjan, The representation of measurable functions by Fourier integrals, Akad. Nauk. Armjan. SSR Dokl. 53 (1971), 203-207. (Russian, Armenian abstract) MR 45:4058

3.

A. Olevski ${\v{\i}}\kern.15em$ , Completeness in $L^{2}(\mathbf{R})$ of almost integer translates, C.R. Acad. Sci. Paris, Sèr. I Math., 324

(1997), 987-991. MR 98a:42002

4.

G. Kozma and A. Olevski ${\v{\i}}\kern.15em$ , Representations of non-periodic functions by trigonometric series with almost integer frequencies, C.R. Acad. Sci. Paris, Sèr. I Math., 329 (1999), 275-280. MR 2000e:42001

5.

G. Kozma and A. Olevskii, Menshov Representation Spectra, Journal d'Analyse Mathématique, 84 (2001), 361-393. MR 2002h:42024

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Additional Information:

Gady Kozma
Affiliation: The Weizmann Institute of Science, Rehovot, Israel
Email: gadykozma@hotmail.com, gadyk@wisdom.weizmann.ac.il

Alexander Olevskii
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat-Aviv, Israel 69978
Email: olevskii@math.tau.ac.il

DOI: 10.1090/S0002-9939-03-06879-5
PII: S 0002-9939(03)06879-5
Keywords: Random spectra, representation of functions by trigonometric series
Received by editor(s): February 8, 2002
Posted: January 8, 2003
Additional Notes: Research supported in part by the Israel Science Foundation
Communicated by: Andreas Seeger
Copyright of article: Copyright 2003, American Mathematical Society


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