Random points in the cube and on the sphere with applications to numerical analysis (Q918109)

Let \(E_ s^{\alpha}(C)\), \(\alpha >1\), \(C>0\), denote the space of all periodic functions \(f(x)=\sum_{h\in Z^ s}c(h)\cdot \exp \{2\pi i<h,x>\}\), \(x=(x_ 1,...,x_ s)\) which satisfy the estimate \(| c(h)| \leq C/\| h\|^{\alpha}\), where \(h=(h_ 1,...,h_ s)\), \(\| h\| =\prod^{s}_{j=1}\max (1,| h_ j|)\) and \(<h,x>=\sum^{s}_{j=1}h_ jx_ j\). The following problem is considered: \(\frac{\partial^ 2}{\partial t^ 2}u(t,x)=-(-D)^ p\cdot u(t,x),\quad u(0,x)=f(x),\partial /\partial t\quad u(t,x)|_{t=0}=g(x),\) where p is a positive integer and \(D=\sum^{s}_{i=1}\sum^{s}_{j=1}a_{ij}\cdot \partial^ 2/(\partial x_ i\partial x_ j)\) a second-order differential operator with positive definite coefficient matrix \(A=(a_{ij})\); f, g are supposed to be in \(E_ s^{\alpha}(C)\) with \(\alpha -2p>1.\) The author presents some numerical experience with the method of good lattice points proposed by \textit{L.-H. Hua} and \textit{Y. Wang} [Recent progress in analytic number theory, Symp. Durham 1979, Vol. 2, 111-118 (1981; Zbl 0465.10045)] to find an approximative solution of the above problem.