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- W1971290932 abstract "We describe a class of measurable subsets $Omega$ in ${mathbb {R}^d}$ such that ${L^2}(Omega )$ has an orthogonal basis of frequencies ${e_lambda }(x) = {e^{i2pi bullet x}}(x in Omega )$ 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 ${L^2}(mu )$." @default.
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- W1971290932 date "1993-09-01" @default.
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- W1971290932 title "Harmonic Analysis of Fractal Measures Induced by Representations of a Certain C$^*$-Algebra" @default.
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- W1971290932 doi "https://doi.org/10.1090/s0273-0979-1993-00428-2" @default.
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