Journal article
Harmonic analysis of fractal measures induced by representations of a certain C-algebra
Bulletin of the American Mathematical Society, Vol.29(2), pp.228-234
1993
DOI: 10.1090/S0273-0979-1993-00428-2
Abstract
We describe a class of measurable subsets $\Omega$ in $\br^d$ such that $L^2(\Omega)$ has an orthogonal basis of frequencies $e_\lambda(x)=e^{i2\pi\lambda\cdot x}(x\in\Omega)$ indexed by $\lambda\in\Lambda\subset\br^d$. We show that such spectral pairs $(\Omega ,\Lambda)$ have a self-similarity which may be used to generate associated fractal measures $\mu$ 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 $(\Omega ,\Lambda)$. We show conversely that such a pair may be reconstructed from a certain Cuntz-representation given to act on $L^2(\mu)$.
Details
- Title: Subtitle
- Harmonic analysis of fractal measures induced by representations of a certain C-algebra
- Creators
- Palle E.T JorgensenSteen Pedersen
- Resource Type
- Journal article
- Publication Details
- Bulletin of the American Mathematical Society, Vol.29(2), pp.228-234
- DOI
- 10.1090/S0273-0979-1993-00428-2
- ISSN
- 0273-0979
- eISSN
- 1088-9485
- Language
- English
- Date published
- 1993
- Academic Unit
- Mathematics
- Record Identifier
- 9983985972102771
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