Journal article
Spectral theory for borel sets in Rn of finite measure
Journal of functional analysis, Vol.107(1), pp.72-104
1992
DOI: 10.1016/0022-1236(92)90101-N
Abstract
Let (Ω, Λ) be a pair of subsets in R n such that Ω has finite, positive Lebesgue measure. The pair is called a spectral pair if the functions e iλ · x , for λ ϵ Λ, form an orthogonal, total family in L 2 (Ω) , i.e., when restricted to Ω. Our present work continues earlier papers on simultaneous diagonalization of vector fields, for the special case when Ω is also assumed open . However, the setting here is more general, and the results are stated in the measure theoretic category . (In considering fundamental domains for lattices, for example, we allow distinct lattice point translates of the domain to overlap on sets of measure zero , rather than the traditional condition of empty overlap.) Two classes of examples of spectral pairs are known, fundamental domains and simple factors . The latter may be identified in terms of a certain finite group action. We show that every spectral pair can be factored (in the sense of Cartesian products) as a product of two pairs of lower dimension where the first factor is a fundamental domain, and the second a simple factor. (The dimensions add.) A simple factor yields in a natural way a finite covering of an associated compact torus; but not every such covering corresponds to a spectral pair. We give a complete spectral characterization of the finite coverings and identify those which correspond to simple factors . The latter identification is phrased in terms of generators and relations for certain C ∗ -algebras which are analogous to (but different from) algebras studied earlier by Cuntz and Arveson.
Details
- Title: Subtitle
- Spectral theory for borel sets in Rn of finite measure
- Creators
- Palle E.T JorgensenSteen Pedersen
- Resource Type
- Journal article
- Publication Details
- Journal of functional analysis, Vol.107(1), pp.72-104
- DOI
- 10.1016/0022-1236(92)90101-N
- ISSN
- 1096-0783
- eISSN
- 1096-0783
- Language
- English
- Date published
- 1992
- Academic Unit
- Mathematics
- Record Identifier
- 9983985848602771
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