Fourier frequencies in affine iterated function systems

Abstract

We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in R d , and the “IFS” refers to such a finite system of transformations, or functions. The iteration limits are pairs ( X , μ ) where X is a compact subset of R d (the support of μ), and the measure μ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L 2 ( X , μ ) ; and (2) explicit constructions of Fourier bases from the given data defining the IFS.