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.

Authors