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

Authors

Palle Jorgensen
Steen Pedersen, Wright State University - Main CampusFollow

Document Type

Article

Publication Date

10-1993

Abstract

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

Repository Citation

Jorgensen, P., & Pedersen, S. (1993). Harmonic-Analysis of Fractal Measures Induced by Representations of a Certain C*-Algebra. Bulletin of the American Mathematical Society, 29 (2), 228-234.
https://corescholar.libraries.wright.edu/math/26

DOI

10.1090/S0273-0979-1993-00428-2