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

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
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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• http://mi.mathnet.ru/eng/mz8865
• https://doi.org/10.4213/mzm8865
• http://mi.mathnet.ru/eng/mz/v90/i3/p394

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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
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
3. S. V. Konyagin, I. D. Shkredov, “A quantitative version of the Beurling-Helson theorem”, Funct. Anal. Appl., 49:2 (2015), 110–121
4. Lebedev V., “Quantitative Aspects of the Beurling-Helson Theorem: Phase Functions of a Special Form”, Studia Math., 247:3 (2019), 273–283
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