Spectral measures and Cuntz algebras

Abstract

We consider a family of measures μ \mu supported in R d \mathbb {R}^d and generated in the sense of Hutchinson by a finite family of affine transformations. It is known that interesting sub-families of these measures allow for an orthogonal basis in L 2 ( μ ) L^2(\mu ) consisting of complex exponentials, i.e., a Fourier basis corresponding to a discrete subset Γ \Gamma in R d \mathbb {R}^d . Here we offer two computational devices for understanding the interplay between the possibilities for such sets Γ \Gamma (spectrum) and the measures μ \mu themselves. Our computations combine the following three tools: duality, discrete harmonic analysis, and dynamical systems based on representations of the Cuntz C ∗ C^* -algebras O N \mathcal O_N .