Harmonic Analysis of Fractal Measures Induced by Representations of a Certain C-algebra

We describe a class of measurable subsets Ω in R such that L2(Ω) has an orthogonal basis of frequencies eλ(x) = e i2πλ·x(x ∈ Ω) indexed by λ ∈ Λ ⊂ R. 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 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 Calgebra 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).