Dense analytic subspaces in fractalL 2-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 inL2 (μ). Moreover, we identify subsetsP ⊂ ℕ0 = {0,1,2,...} such that the functions {en:n ∈ P} form an orthonormal basis forL2 (μ). We also give a higher-dimensional affine construction leading to self-similar measures μ with support in ℝν, obtained from a given expansivev-by-v matrix and a finite set of translation vectors. We show that the correspondingL2 (μ) has an orthonormal basis of exponentialsei2πλ·x, indexed by vectors λ in ℝν, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.