# Dense analytic subspaces in fractal*L*^{2}-spaces

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DOI: 10.1007/BF02788699

- Cite this article as:
- Jorgensen, P.E.T. & Pedersen, S. J. Anal. Math. (1998) 75: 185. doi:10.1007/BF02788699

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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 in*L*^{2} (μ). Moreover, we identify subsets*P* ⊂ ℕ_{0} = {0,1,2,...} such that the functions {*e*_{n}:*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}, obtained from a given expansive*v*-by-*v* matrix and a finite set of translation vectors. We show that the corresponding*L*^{2} (μ) has an orthonormal basis of exponentials*e*^{i2πλ·x}, indexed by vectors λ in ℝ^{ν}, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.