#### Document Type

Article

#### Publication Date

10-1993

#### Abstract

We describe a class of measurable subsets Ω in **R**^{d} such that L^{2}(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

*L*

^{2}(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

*L*

^{2}(μ).

#### 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

## Comments

First published in the

Bulletin of the American Mathematical Society29.2 (1993), published by the American Mathematical Society.