Journal article
Dense analytic subspaces in fractal L2-spaces
Journal d'analyse mathématique (Jerusalem), Vol.75, pp.185-228
1998
DOI: 10.1007/BF02788699
Abstract
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.
Details
- Title: Subtitle
- Dense analytic subspaces in fractal L2-spaces
- Creators
- Palle E.T JorgensenSteen Pedersen
- Resource Type
- Journal article
- Publication Details
- Journal d'analyse mathématique (Jerusalem), Vol.75, pp.185-228
- DOI
- 10.1007/BF02788699
- ISSN
- 0021-7670
- eISSN
- 1565-8538
- Language
- English
- Date published
- 1998
- Academic Unit
- Mathematics
- Record Identifier
- 9983985811302771
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