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On obtaining higher order convergence for smooth periodic functions. (English)
J. Complexity 24, No. 3, 328-340 (2008).
Consider the problem of approximating integrals of the form $$I=\int_{[0,1)^{s}}f(x)\,dx,$$ where $f$ is a function on the $s$-dimensional unit cube $[0,1)^{s}$ with an absolutely convergent Fourier series. Of particular interest are cubature rules of the form $$Q_{N}=\sum_{k=1}^{N}w_{k}f(x_{k}),$$ with sample points of an infinite sequence $x_{1}, x_{2}, \dots$ and weights $w_{k}$. The authors show how to construct a general and open algorithm for {\it H. Niederreiter}’s method [Aequationes Math. 8 (1972), 304‒311 (1971; Zbl 0252.65023); Diophantine Approx. Appl., Proc. Conf. Washington 1972; 129‒199 (1973; Zbl 0268.65014)] and for {\it M. Sugihara} and {\it K. Murota}’s [Math. Comput. 39, 549‒554 (1982; Zbl 0502.65009)] method, so that one can use them to obtain general higher order error convergence for smooth periodic functions. The memory cost of these algorithms is low and they are linear in the number of steps. The absolute errors as a function of the number of sample points for both methods are comparable. Recall that Niederreiter considers the following algorithm $$Q_{N}^{(q)}=N^{-q}\sum_{n=0}^{q(N-1)}a_{N,n}^{(q)}f(\{nα\}),$$ where $α$ is an $s$-dimensional vector of irrational components, $\{\cdot\}$ is the fractional parts and the weights $a_{N,n}^{(q)}$ are uniquely determined from some polynomial identity. Sugihara and Murota consider the algorithm $$\widetilde{Q}_{\widetilde{N}}^{(q)}=\widetilde{N}^{-1}\sum_{k=1}^{\widetilde{N}}w_{q} \left(\frac{k}{\widetilde{N}}\right)f(\{kα\}),$$ where $w_{q}(x)=A_{q}x^{q}(1-x)^{q}$ with $A_{q}=\frac{(2q+1)!}{(q!)^{2}}$.
Reviewer: Ana-Maria Acu (Sibiu)