Harmonic Analysis and Fractal Limit-measures Induced by Representations of a Certain C-algebra

Palle E.T. Jorgensen and Steen Pedersen Abstra t. We describe a class of measurable subsets Ω in Rd such that L(Ω) has an orthogonal basis of frequencies eλ(x) = e (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 L(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 L(μ).