, Volume 75, Issue 1, pp 185-228

Dense analytic subspaces in fractalL 2-spaces

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Abstract

We show that for certain self-similar measures μ with support in the interval 0≤x≤1, the analytic functions {e i2πnx :n=0,1,2, …} contain an orthonormal basis inL 2 (μ). Moreover, we identify subsetsP ⊂ ℕ0 = {0,1,2,...} such that the functions {e n :n ∈ P} form an orthonormal basis forL 2 (μ). We also give a higher-dimensional affine construction leading to self-similar measures μ with support in ℝ ν , obtained from a given expansivev-by-v matrix and a finite set of translation vectors. We show that the correspondingL 2 (μ) has an orthonormal basis of exponentialse i2πλ·x , indexed by vectors λ in ℝ ν , provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.

Work supported by the National Science Foundation.