Trudy Instituta matematiki i mehaniki, Function theory, Tome 11 (2005) no. 2, pp. 10-29
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N. Yu. Antonov. Growth rate of sequences of multiple rectangular Fourier sums. Trudy Instituta matematiki i mehaniki, Function theory, Tome 11 (2005) no. 2, pp. 10-29. http://geodesic.mathdoc.fr/item/TIMM_2005_11_2_a1/
@article{TIMM_2005_11_2_a1,
author = {N. Yu. Antonov},
title = {Growth rate of sequences of multiple rectangular {Fourier} sums},
journal = {Trudy Instituta matematiki i mehaniki},
pages = {10--29},
year = {2005},
volume = {11},
number = {2},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TIMM_2005_11_2_a1/}
}
TY - JOUR
AU - N. Yu. Antonov
TI - Growth rate of sequences of multiple rectangular Fourier sums
JO - Trudy Instituta matematiki i mehaniki
PY - 2005
SP - 10
EP - 29
VL - 11
IS - 2
UR - http://geodesic.mathdoc.fr/item/TIMM_2005_11_2_a1/
LA - ru
ID - TIMM_2005_11_2_a1
ER -
%0 Journal Article
%A N. Yu. Antonov
%T Growth rate of sequences of multiple rectangular Fourier sums
%J Trudy Instituta matematiki i mehaniki
%D 2005
%P 10-29
%V 11
%N 2
%U http://geodesic.mathdoc.fr/item/TIMM_2005_11_2_a1/
%G ru
%F TIMM_2005_11_2_a1
In the case when a sequence of $d$-dimensional vectors $\mathbf n_k=(n_k^1,n_k^2,\dots,n_k^d)$ with nonnegative integral coordinates satisfies the condition $$ n_k^j=\alpha_jm_k+O(1),\quad k\in\mathbb N,\quad1\le j\le d, $$ where $\alpha_1\dots,\alpha_d$ are nonnegative real numbers and $\{m_k\}_{k=1}^\infty$ is a sequence of positive integers, the following estimate of the rate of growth of sequences $S_{\mathbf n_k}(f,\mathbf x)$ of rectangular partial sums of multiple trigonometric Fourier series is obtained: if $f\in L(\ln^+L)^{d-1}([-\pi,\pi)^d)$, then $$ S_{\mathbf n_k}(f,\mathbf x)=o(\ln k)\quad\text{a.e.} $$ Analogous estimates are valid for conjugate series as well.
