Harmonic analysis of fractal measures induced by representations of a certain -algebra

Authors:
Palle E. T. Jorgensen and Steen Pedersen

Journal:
Bull. Amer. Math. Soc. **29** (1993), 228-234

MSC (2000):
Primary 46L55; Secondary 28A80, 42C05

DOI:
https://doi.org/10.1090/S0273-0979-1993-00428-2

MathSciNet review:
1215311

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Abstract | References | Similar Articles | Additional Information

Abstract: We describe a class of measurable subsets in such that has an orthogonal basis of frequencies indexed by . We show that such spectral pairs have a self-similarity which may be used to generate associated fractal measures with Cantor set support. The Hilbert space does not have a total set of orthogonal frequencies, but a harmonic analysis of may be built instead from a natural representation of the Cuntz -algebra which is constructed from a pair of lattices supporting the given spectral pair . We show conversely that such a pair may be reconstructed from a certain Cuntz-representation given to act on .

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Additional Information

DOI:
https://doi.org/10.1090/S0273-0979-1993-00428-2

Article copyright:
© Copyright 1993
American Mathematical Society