Dense analytic subspaces in fractal L2-spaces

Abstract

We show that for certain self-similar measures μ with support in the interval 0 ≤ x ≤ 1, the analytic functions {ei2πnx : n = 0, 1,2,...}contain an orthonormal basis in L2 (μ). Moreover, we identify subsets P ⊂ ℕ0 = {0, 1,2,...} such that the functions {en : n ∈ P} form an orthonormal basis for L2 (μ). We also give a higher-dimensional affine construction leading to self-similar measures μ with support in ℝν, obtained from a given expansive ν-by-ν matrix and a finite set of translation vectors. We show that the corresponding L2 (μ) has an orthonormal basis of exponentials ei2πλ·x, indexed by vectors λ in ℝν, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.