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 Mat. Zametki, 2009, Volume 86, Issue 3, Pages 408–420 (Mi mz8501)

Necessary Conditions for the Weak Generalized Localization of Fourier Series with “Lacunary Sequence of Partial Sums”

O. V. Lifantseva

Moscow State Region University

Abstract: It has been established that, on the subsets $\mathbb{T}^N=[-\pi,\pi]^N$ describing a cross $W$ composed of $N$-dimensional blocks, $W_{x_sx_t}=\Omega_{x_sx_t}\times [-\pi,\pi]^{N-2}$ ($\Omega_{x_sx_t}$ is an open subset of $[-\pi,\pi]^2$) in the classes $L_p(\mathbb{T}^N)$, $p>1$, a weak generalized localization holds, for $N\ge3$, almost everywhere for multiple trigonometric Fourier series when to the rectangular partial sums $S_n(x;f)$ ($x\in\mathbb{T}^N$, $f\in L_p$) of these series corresponds the number $n=(n_1,…,n_N)\in\mathbb Z_{+}^{N}$ some components $n_j$ of which are elements of lacunary sequences. In the present paper, we prove a number of statements showing that the structural and geometric characteristics of such subsets are sharp in the sense of the numbers (generating $W$) of the $N$-dimensional blocks $W_{x_sx_t}$ as well as of the structure and geometry of $W_{x_sx_t}$. In particular, it is proved that it is impossible to take an arbitrary measurable two-dimensional set or an open three-dimensional set as the base of the block.

Keywords: multiple trigonometric Fourier series, $n$-block, lacunary sequence, weak generalized localization, measurable set, Euclidean space, rectangular partial sum

DOI: https://doi.org/10.4213/mzm8501

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English version:
Mathematical Notes, 2009, 86:3, 373–384

Bibliographic databases:

UDC: 517.5
Revised: 17.03.2009

Citation: O. V. Lifantseva, “Necessary Conditions for the Weak Generalized Localization of Fourier Series with “Lacunary Sequence of Partial Sums””, Mat. Zametki, 86:3 (2009), 408–420; Math. Notes, 86:3 (2009), 373–384

Citation in format AMSBIB
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