Spectral measures and Cuntz algebras

TL;DR

This paper explores the relationship between spectral measures generated by affine transformations and their Fourier bases, using duality, harmonic analysis, and Cuntz algebra representations for computational insights.

Contribution

It introduces two computational methods to analyze the connection between spectral sets and measures via Cuntz algebra representations.

Findings

New computational tools for spectral measure analysis

Enhanced understanding of Fourier bases in fractal measures

Application of Cuntz algebras to harmonic analysis

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 $C^*$-algebras $\mathcal O_N$.