Base change problems for generalized Walsh series and multivariate numerical integration

Gerhard Larcher and Gottlieb Pirsic

Vol. 189 (1999), No. 1, 75–105

DOI: 10.2140/pjm.1999.189.75

Abstract

We recall the notion of Walsh functions over a finite abelian group as it was given for example in Larcher, Niederreiter and Schmid, 1996. These function systems play an important role for various “digital lattice rules” in multivariate numerical integration. We consider the following problem:

Assume, that a function f can be represented by a Walsh-series over a group G₁ with a certain speed of convergence. Take another group G₂. What can be said about the speed of convergence of the Walsh-series of f over G₂?

Answers to this question are essential for certain numerical integration error estimates. We are able to give some results, partly best possible ones.

A connection of the above problem to “digital differentiability” of functions and applications to numerical integration are given. Open problems are stated.

Milestones

Received: 29 August 1997

Revised: 29 July 1998

Published: 1 May 1999