Dense Analytic Subspaces in Fractal L 2 -spaces

We consider self-similar measures µ with support in the interval 0 ≤ x ≤ 1 which have the analytic functions e i2πnx : n = 0, 1, 2,. .. span a dense subspace in L 2 (µ). Depending on the fractal dimension of µ, we identify subsets P ⊂ N 0 = {0, 1, 2,. .. } such that the functions {en : n ∈ 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 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 ex-ponentials e i2πλ·x , indexed by vectors λ in R ν , provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.