Harmonic Analysis and Fractal Limit-Measures Induced by Representations of a Certain C*-Algebra

We describe a class of measurable subsets Ω in Rd such that L2(Ω) has an orthogonal basis of frequencies eλ(x) = ei2πλ · x(x ∈ Ω) indexed by λ ∈ Λ ⊂ Rd. We show that such spectral pairs (Ω, Λ) have a self-similarity which may be used to generate associated fractal measures μ (typically with Cantor set support). The Hilbert space L2(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 L2(μ).