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Mathematics of Computation

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

Authors: Gerhard Larcher, Wolfgang Ch. Schmid and Reinhard Wolf
Journal: Math. Comp. 63 (1994), 701-716
MSC: Primary 65D30; Secondary 11K45, 42C10
MathSciNet review: 1254146
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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: $ \vert\hat W({h_1}, \ldots ,{h_s})\vert \leq c \cdot {({\overline h _1} \cdots {\overline h _s})^{ - \alpha }}$ for all $ {h_1}, \ldots ,{h_s}$, where $ \overline h : = \max (1,\vert h\vert)$. 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

$\displaystyle {\beta _h} = \frac{{h - 1}}{{2h}} + \frac{{\sum\nolimits_{k = 0}^... ...h}{2}]}} - 1}}{{3 \cdot {2^{k + 1}}}}} \right\}} \right)} }}{{h \cdot \log 2}}.$

The "exponent" $ \alpha - {\beta _h}$ is best possible for all h, and $ {\beta _h}$ is monotonically increasing with

$\displaystyle \beta : = \mathop {\lim }\limits_{h \to \infty } {\beta _h} = \frac{1}{2} + \frac{{\log \sin \frac{{5\pi }}{{12}}}}{{\log 2}} = 0.4499 \ldots .$

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Article copyright: © Copyright 1994 American Mathematical Society