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

Details Dense analytic subspaces in fractal $L^{2}$-spaces Dense analytic subspaces in fractal $L^{2}$-spaces

1997-09-30

arxiv.org/abs/funct-an/9709007v1

J. Analyse Math. 75 (1998), 185--228

Mathematics

Functional Analysis

41 pages, 5 figures, AMS-LaTeX v1.2b with EPS and LaTeX "picture" graphics. Authors Palle E.T. Jorgensen (The University of Io

Scientific paper

We consider self-similar measures $\mu $ with support in the interval $0\leq x\leq 1$ which have the analytic functions $\left\{e^{i2\pi nx}:n=0,1,2,... \right\} $ span a dense subspace in $L^{2}(\mu) $. Depending on the fractal dimension of $\mu $, we identify subsets $P\subset \mathbb{N}_{0}=\{0,1,2,... \} $ such that the functions $\{e_{n}:n\in P\} $ form an orthonormal basis for $L^{2}(\mu) $. We also give a higher-dimensional affine construction leading to self-similar measures $\mu $ with support in $\mathbb{R}^{\nu}$. It is obtained from a given expansive $\nu $-by-$\nu $ matrix and a finite set of translation vectors, and we show that the corresponding $L^{2}(\mu) $ has an orthonormal basis of exponentials $e^{i2\pi \lambda \cdot x}$, indexed by vectors $\lambda $ in $\mathbb{R}^{\nu}$, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.

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