We consider self-similar measures $\mu $ with support in the interval $0\leq x\leq 1$ which have the analytic functions $\left\{ e^{ i2\pi nx}:n=0,1,2,... \right\} $ span a dense subspace in $L^{ 2}(\mu) $. Depending on the fractal dimension of $\mu $, we identify subsets $P\subset \mathbb{ N}_{ 0}=\{ 0,1,2,... \} $ such that the functions $\{ e_{ n}:n\in P\} $ form an orthonormal basis for $L^{ 2}(\mu) $. We also give a higher-dimensional affine construction leading to self-similar measures $\mu $ with support in $\mathbb{ R}^{ \nu}$. It is obtained from a given expansive $\nu $-by-$\nu $ matrix and a finite set of translation vectors, and we show that the corresponding $L^{ 2}(\mu) $ has an orthonormal basis of exponentials $e^{ i2\pi \lambda \cdot x}$, indexed by vectors $\lambda $ in $\mathbb{ R}^{ \nu}$, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.
Categories: funct-an math.FA