Quadrature formulas with combined grids
type: Item
instance of: scholarly article
title: Quadrature formulas with combined grids
zbMATH Open document ID: 0832.65021
DOI: 10.1007/BF02113296
author: N. M. Korobov
published in: Mathematical Notes
publication date: 1995-09-24T00:00:00Z
review text: Let \(p\) be a prime, \((n,p) = 1\), \(n \ll \log p\) and \(N = n^sp\). Using the combined grids \((\{k_1/n + a_1k/p\}, \dots, \{k_s/n + a_sk/p\})\) the author constructs the \(s\)-dimensional quadrature formula \[ \begin{multlined} \int^1_0 \cdots \int^1_0 f(x_1, \dots, x_s)dx_1 \dots dx_s =\\ N^{-1} \sum^p_{k = 1} \sum^n_{k_1, \dots, k_s = 1} f(k_1/n + a_1 k/p, \dots, k_s/n + a_s k/p) - R_N [f]\end{multlined} \tag{*} \] for \(f \in E^\alpha_s (C)\), where \(E^\alpha_s(C)\) means the class of functions \(f(x_1,\dots, x_s)\) such that \[ f(x_1, \dots, x_s) = \sum^\infty_{m_1, \dots, m_s = -\infty} C(m_1, \dots, m_s) e^{2\pi i(m_1 x_1 + \cdots + m_s x_s)} \] with \(|C(m_1, \dots, m_s)|\leq C/(\overline {m}_1 \cdots \overline {m}_s)^\alpha\), where \(\alpha > 1\), \(\overline {m} = \max (1, |m|)\) and \(C\) is a constant independent of \(m_1, \dots, m_s\). He proves that there exist optimal coefficients \(a_1, \dots, a_s\) such that \[ \sum^\infty_{m_1, \dots, m_s = -\infty} \delta_p (a_1 m_1 + \dots + a_sm_s)/(\overline {m}_1 \cdots \overline {m}_s)^\alpha \ll (\log p)^{\alpha(r-1)} p^{- \alpha},\quad (r = 1, \dots, s), \] where the sum \(\sum_r\) is over systems of integers \((m_1, \dots, m_s)\) containing precisely \(r\) numbers \(m_j\) different from zero, and \(\delta_p(a) = 1\) or 0 according to \(p \mid a\) or \(p \nmid a\), respectively, and that the error of the corresponding quadrature formula (*) satisfies \[ R_N [f] \ll (\log N)^{\alpha(s-1)} N^{-\alpha}. \tag{**} \] Furthermore, the author also gives an algorithm for finding optimal coefficients \(a_1, \dots, a_s\) for combined grids. The estimate (**) agrees with the best known estimates obtained using optimal parallelepipedal grids, but the number of elementary arithmetic operations needed to calculate the coefficients of the combined grids is at least \(2^s\) times smaller than the corresponding number in the case of using parallelepipedal grids with the same number of nodes.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 11K45
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 796783
zbMATH Keywords: optimal coefficients
zbMATH Keywords: parallelepipedal grids
zbMATH Keywords: combined grids
zbMATH Keywords: quadrature formula
zbMATH Keywords: algorithm
reviewed by: Yaochen Zhu
MaRDI profile type: Publication
cites work: Q3262330
cites work: Q5723455
cites work: Q3994684
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description: scientific article; zbMATH DE number 796783