Document Type

Article

Publication Date

10-1993

Abstract

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 ε 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 with Cantor set support. The Hilbert space L2(μ) does not have a total set of orthogonal frequencies, but a harmonic analysis of mu 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).

Comments

First published in the Bulletin of the American Mathematical Society 29.2 (1993), published by the American Mathematical Society.

DOI

10.1090/S0273-0979-1993-00428-2