Journal article
Hilbert spaces built on a similarity and on dynamical renormalization
Journal of Mathematical Physics, Vol.47(5), pp.053504-053504-20
2006
DOI: 10.1063/1.2196750
Abstract
We develop a Hilbert-space framework for a number of general multiscale problems from dynamics. The aim is to identify a spectral theory for a class of systems based on iterations of a non-invertible endomorphism. We are motivated by the more familiar approach to wavelet theory which starts with the two-to-one endomorphism r:z↦z2 in the one-torus T, a wavelet filter, and an associated transfer operator. This leads to a scaling function and a corresponding closed subspace V0 in the Hilbert space L2(R). Using the dyadic scaling on the line R, one has a nested family of closed subspaces Vn, n∊Z, with trivial intersection, and with dense union in L2(R). More generally, we achieve the same outcome, but in different Hilbert spaces, for a class of non-linear problems. In fact, we see that the geometry of scales of subspaces in Hilbert space is ubiquitous in the analysis of multiscale problems, e.g., martingales, complex iteration dynamical systems, graph-iterated function systems of affine type, and subshifts in...
Details
- Title: Subtitle
- Hilbert spaces built on a similarity and on dynamical renormalization
- Creators
- Dorin Ervin DutkayPalle E.T Jorgensen
- Resource Type
- Journal article
- Publication Details
- Journal of Mathematical Physics, Vol.47(5), pp.053504-053504-20
- DOI
- 10.1063/1.2196750
- ISSN
- 0022-2488
- eISSN
- 1089-7658
- Language
- English
- Date published
- 2006
- Academic Unit
- Mathematics
- Record Identifier
- 9983985809002771
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