Harmonic analysis of fractal measures induced by representations of a certain $C^ *$-algebra

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 ${L^2}(\mu )$.