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Mat. Zametki, 2011, Volume 90, Issue 3, Pages 394–407 (Mi mz8865)  

This article is cited in 4 scientific papers (total in 4 papers)

Estimates in Beurling–Helson Type Theorems: Multidimensional Case

V. V. Lebedev

Moscow State Institute of Electronics and Mathematics (Technical University)

Abstract: We consider the spaces $A_p(\mathbb T^m)$ of functions $f$ on the $m$-dimensional torus $\mathbb T^m$ such that the sequence of Fourier coefficients $\widehat{f}=\{\widehat{f}(k), k\in\mathbb Z^m\}$ belongs to $l^p(\mathbb Z^m)$, $1\le p<2$. The norm on $A_p(\mathbb T^m)$ is defined by $\|f\|_{A_p(\mathbb T^m)}=\|\widehat{f}\|_{l^p(\mathbb Z^m)}$. We study the rate of growth of the norms $\|e^{i\lambda\varphi}\|_{A_p(\mathbb T^m)}$ as $|\lambda|\to\infty$, $\lambda\in\mathbb R$, for $C^1$-smooth real functions $\varphi$ on $\mathbb T^m$ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogs for the spaces $A_p(\mathbb R^m)$.

Keywords: harmonic analysis, Fourier series, Beurling–Helson theorem

DOI: https://doi.org/10.4213/mzm8865

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English version:
Mathematical Notes, 2011, 90:3, 373–384

Bibliographic databases:

UDC: 517.51
Received: 06.09.2010
Revised: 04.12.2010

Citation: V. V. Lebedev, “Estimates in Beurling–Helson Type Theorems: Multidimensional Case”, Mat. Zametki, 90:3 (2011), 394–407; Math. Notes, 90:3 (2011), 373–384

Citation in format AMSBIB
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\by V.~V.~Lebedev
\paper Estimates in Beurling--Helson Type Theorems: Multidimensional Case
\jour Mat. Zametki
\yr 2011
\vol 90
\issue 3
\pages 394--407
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\crossref{https://doi.org/10.4213/mzm8865}
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\transl
\jour Math. Notes
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\issue 3
\pages 373--384
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    This publication is cited in the following articles:
    1. V. V. Lebedev, “Absolutely Convergent Fourier Series. An Improvement of the Beurling–Helson Theorem”, Funct. Anal. Appl., 46:2 (2012), 121–132  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. V. V. Lebedev, “On the Fourier Transform of the Characteristic Functions of Domains with $C^1$ Boundary”, Funct. Anal. Appl., 47:1 (2013), 27–37  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. S. V. Konyagin, I. D. Shkredov, “A quantitative version of the Beurling-Helson theorem”, Funct. Anal. Appl., 49:2 (2015), 110–121  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. Lebedev V., “Quantitative Aspects of the Beurling-Helson Theorem: Phase Functions of a Special Form”, Studia Math., 247:3 (2019), 273–283  crossref  mathscinet  zmath  isi  scopus
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