Dense analytic subspaces in fractal $L^{2}$-spaces

Abstract

We consider self-similar measures $μ$ with support in the interval $0\leq x\leq 1$ which have the analytic functions $\left\{e^{i2πnx}:n=0,1,2,... \right\} $ span a dense subspace in $L^{2}(μ) $. Depending on the fractal dimension of $μ$, 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}(μ) $. We also give a higher-dimensional affine construction leading to self-similar measures $μ$ with support in $\mathbb{R}^ν$. It is obtained from a given expansive $ν$-by-$ν$ matrix and a finite set of translation vectors, and we show that the corresponding $L^{2}(μ) $ has an orthonormal basis of exponentials $e^{i2πλ\cdot x}$, indexed by vectors $λ$ in $\mathbb{R}^ν$, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.