Representation of functions as Walsh series to different bases and an application to the numerical integration of high-dimensional Walsh series

Abstract

We will prove the following theorem on Walsh series, and we will derive from this theorem an effective and constructive method for the numerical integration of Walsh series by number-theoretic methods. Further, concrete computer calculations are given. Theorem. For base $b \geq 2$, dimension $s \geq 1$, and $\alpha > 1,c > 0\;(b,s \in \mathbb {N};c,\alpha \in \mathbb {R})$, let $_b\overline E _s^\alpha (c)$ be the class of all functions $f:[0,1)^s \to \mathbb {C}$ which are representable by absolutely convergent Walsh series to base b with Walsh coefficients $\hat W({h_1}, \ldots ,{h_s})$ with the following property: $|\hat W({h_1}, \ldots ,{h_s})| \leq c \cdot {({\overline h _1} \cdots {\overline h _s})^{ - \alpha }}$ for all ${h_1}, \ldots ,{h_s}$, where $\overline h : = \max (1,|h|)$. We show that if $f \in { _2}\overline E _s^\alpha (c)$, then $f \in { _{{2^h}}}\overline E _s^{\alpha - {\beta _h}}(c \cdot {2^{hs\alpha }})$ for all $h \geq 2$, provided that $\alpha > 1 + {\beta _h}$, where \[ {\beta _h} = \frac {{h - 1}}{{2h}} + \frac {{\sum \nolimits _{k = 0}^{h - 2} {\log \sin \left ( {\frac {\pi }{4} + \frac {\pi }{2}\left \{ {\frac {{{4^{[\frac {h}{2}]}} - 1}}{{3 \cdot {2^{k + 1}}}}} \right \}} \right )} }}{{h \cdot \log 2}}.\] The "exponent" $\alpha - {\beta _h}$ is best possible for all h, and ${\beta _h}$ is monotonically increasing with \[ \beta : = \lim \limits _{h \to \infty } {\beta _h} = \frac {1}{2} + \frac {{\log \sin \frac {{5\pi }}{{12}}}}{{\log 2}} = 0.4499 \ldots .\]