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 ℝν, 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 ℝν, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.
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Work supported by the National Science Foundation.
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Jorgensen, P.E.T., Pedersen, S. Dense analytic subspaces in fractalL 2-spaces. J. Anal. Math. 75, 185–228 (1998). https://doi.org/10.1007/BF02788699
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DOI: https://doi.org/10.1007/BF02788699