Harmonic analysis of fractal measures induced by representations of a certain C$^*$-algebra

Abstract

We describe a class of measurable subsets $Ω$ in $\br^d$ such that $L^2(Ω)$ has an orthogonal basis of frequencies $e_λ(x)=e^{i2πλ\cdot x}(x\inΩ)$ indexed by $λ\inΛ\subset\br^d$. We show that such spectral pairs $(Ω,Λ)$ have a self-similarity which may be used to generate associated fractal measures $μ$ with Cantor set support. The Hilbert space $L^2(μ)$ does not have a total set of orthogonal frequencies, but a harmonic analysis of $μ$ may be built instead from a natural representation of the Cuntz C$^*$- algebra which is constructed from a pair of lattices supporting the given spectral pair $(Ω,Λ)$. We show conversely that such a pair may be reconstructed from a certain Cuntz-representation given to act on $L^2(μ)$.