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

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 $L^2(\mu)$.