An Operator-Fractal

TL;DR

This paper explores the spectral properties of Bernoulli convolution measures, demonstrating the existence of multiple Fourier bases, and introduces operator-fractals with self-similar structures that facilitate Fourier analysis.

Contribution

It introduces Cuntz operators to decompose L^2 spaces of spectral measures and constructs operator-fractals that reveal self-similarity in unitary transformations between Fourier bases.

Findings

Existence of multiple Fourier bases for certain spectral measures

Cuntz operators enable orthogonal decompositions for Fourier expansions

Operator-fractals exhibit self-similarity and fractal structure

Abstract

Certain Bernoulli convolution measures (\mu) are known to be spectral. Recently, much work has concentrated on determining conditions under which orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure known to be spectral, the orthonormal basis need not be unique; indeed, there are often families of such spectral bases. Let \lambda = 1/(2n) for a natural number n and consider the Bernoulli measure (\mu) with scale factor \lambda. It is known that L^2(\mu) has a Fourier basis. We first show that there are Cuntz operators acting on this Hilbert space which create an orthogonal decomposition, thereby offering powerful algorithms for computations for Fourier expansions. When L^2(\mu) has more than one Fourier basis, there are natural unitary operators U, indexed by a subset of odd scaling factors p; each U is defined by mapping one ONB to another. We show that the unitary operator U can also be orthogonally decomposed according to the Cuntz relations. Moreover, this operator-fractal U exhibits its own self-similarity.