Quadrature formulas with combined grids
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.
scientific article; zbMATH DE number 796783