Numerical integration of Hölder continuous, absolutely convergent Fourier-, Fourier cosine-, and Walsh series

Abstract

We introduce quasi-Monte Carlo rules for the numerical integration of functions $f$ defined on $[0,1{]}^s$, $s\ge 1$, which satisfy the following properties: the Fourier-, Fourier cosine- or Walsh coefficients of $f$ are absolutely summable and $f$ satisfies a Hölder condition of order $\alpha$, for some $0<\alpha \le 1$. We show a convergent rate of the integration error of order $\max ((s-1)N^{-1/2},s^{\alpha /2}{N}^{-\alpha })$. The construction of the quadrature points is explicit and is based on Weil sums.

arXiv commentAdded literature review and tractability discussion; Minor corrections

References in corpus (1)

• Lattice rules for nonperiodic smooth integrands