Harmonic Analysis and Fractal Limit-Measures Induced by Representations of a Certain C*-Algebra
✍ Scribed by P.E.T. Jorgensen; S. Pedersen
- Publisher
- Elsevier Science
- Year
- 1994
- Tongue
- English
- Weight
- 585 KB
- Volume
- 125
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
We describe a class of measurable subsets (\Omega) in (\mathbb{R}^{d}) such that (L^{2}(\Omega)) has an orthogonal basis of frequencies (e_{\lambda}(x)=e^{i 2 \pi \lambda \cdot x}(x \in \Omega)) indexed by (\lambda \in A \subset \mathbb{R}^{d}). We show that such spectral pairs ((\Omega, A)) have a self-similarity which may be used to generate associated fractal measures (\mu) (typically with Cantor set support). The Hilbert space (L^{2}(\mu)) 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 ( (\Omega, A) ). We show conversely that such a pair may be reconstructed from a certain Cuntz-representation given to act on (L^{2}(\mu)). 1994 Academic Press. Inc.