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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Harmonic analysis of fractal measures induced by representations of a certain $ C\sp *$-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
MathSciNet review: 1215311
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Abstract: 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^{i2\pi \bullet x}}(x \in \Omega )$ indexed by $ \lambda \in \Lambda \subset {\mathbb{R}^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 $ {{\text{C}}^{\ast}}$-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 )$.

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