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Dal'nevost. Mat. Zh., 2019, Volume 19, Number 1, Pages 10–19 (Mi dvmg391)  

Extremal cubature formulas for anisotropic classes

V. A. Bykovskii

Khabarovsk Division of the Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences

Abstract: Let $E^{(\alpha; s)}$ be a class of periodical functions
$$ f(x_1, …, x_s)=\sum_{(m_1, …, m_s)\in \mathbb{Z}^s} c(m_1, …, m_s)\exp(2\pi i(m_1 x_1+…+ m_s x_s)) $$
with $ |c(m_1, …, m_s)|\leq \prod_{j=1} (max (1, |m_j|))^{-\alpha}, $ and $1< \alpha < \infty$. In this work for all natural numbers $1< N < \infty$ we prove best possible estimation
$$ R_N(E^{(\alpha; s)})\ll_{\alpha, s} \frac{(\log N)^{s-1}}{N^\alpha} $$
for the error of the best cubature formula on the class $E^{(\alpha; s)}$ with $N$ nodes and weights. Similar results are proved for other classes of functions.

Key words: cubature formulas, anisotropic classes of functions.

Full text: PDF file (530 kB)
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UDC: 519.68
MSC: 65J01
Received: 21.05.2019

Citation: V. A. Bykovskii, “Extremal cubature formulas for anisotropic classes”, Dal'nevost. Mat. Zh., 19:1 (2019), 10–19

Citation in format AMSBIB
\Bibitem{Byk19}
\by V.~A.~Bykovskii
\paper Extremal cubature formulas for anisotropic classes
\jour Dal'nevost. Mat. Zh.
\yr 2019
\vol 19
\issue 1
\pages 10--19
\mathnet{http://mi.mathnet.ru/dvmg391}


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