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 \[ \beta : = \lim \limits _{h \to \infty } {\beta _h} = \frac {1}{2} + \frac {{\log \sin \frac {{5\pi }}{{12}}}}{{\log 2}} = 0.4499 \ldots .\]