Spectral measures and Cuntz algebras

Abstract

We consider a family of measures $\mu$ supported in $\br^d$ 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 $L^2(\mu )$ consisting of complex exponentials, i.e., a Fourier basis corresponding to a discrete subset $\Gamma$ in $\br^d$. Here we offer two computational devices for understanding the interplay between the possibilities for such sets $\Gamma$ (spectrum) and the measures $\mu$ themselves. Our computations combine the following three tools: duality, discrete harmonic analysis, and dynamical systems based on representations of the Cuntz $C^{\ast }$-algebras ${\mathcal{O}}_N$.

References in corpus (1)

• Iterated function systems, Ruelle operators, and invariant projective measures

Cited by in corpus (1)

• Iterative approximations of exponential bases on fractal measures