On some multidimensional quadrature formulas with number-theoretic nets
This paper is devoted to a refinement of Korobov's and Hlawka's method of good lattice points for numerical integration; see [see \textit{L. K. Hua} and \textit{Y. Wang}, Applications of number theory to numerical analysis (Springer 1981; Zbl 0451.10001)]. The author considers the class \(E^ \alpha_ s(C)\) of all periodic functions \(f\) (of period 1) in \(s\) variables such that the Fourier coefficients \(C(m)\) \((m= (m_ 1, \dots, m_ s))\) satisfy a bound \(C(m)\leq C\overline {m}_ 1^{\alpha_ 1} \dots \overline {m}_ s^{\alpha_ s}\), where \(C>0\) is an absolute constant, \(\overline {m}_ j= \max (1, | m_ j|)\), \((j=1,\dots, s)\) and \(\alpha= (\alpha_ 1,\dots, \alpha_ s)\) is a given \(s\)-tuple of positive numbers such that \(1< \alpha_ 1= \dots= \alpha_ \mu< \alpha_{\mu+1}\leq \dots\leq \alpha_ s\). The main result is the following error bound: \[ \sup_{f\in E_ s^ \alpha(C)} \biggl| \underset {[0,1]^ s} \idotsint f(x) dx- {1\over p} \sum^ p_{k=1} f \biggl( {{ka} \over p} \biggr) \biggr|\leq C\cdot c(s,\alpha), \] where \(p\) is a given prime number and \(c(s,\alpha)\) some constant and \(a\) a suitable lattice point.
scientific article; zbMATH DE number 549498
scientific article