Journal article
Spectral pairs in cartesian coordinates
The Journal of Fourier Analysis and Applications, Vol.5(4), pp.285-302
07/1999
DOI: 10.1007/BF01259371
Abstract
Let Ω ⊂ℝd have finite positive Lebesgue measure, and let $$\mathcal{L}^2$$ (Ω) be the corresponding Hilbert space of $$\mathcal{L}^2$$ -functions on Ω. We shall consider the exponential functionse λ on Ω given bye λ(x)=e i2πλ·x . If these functions form an orthogonal basis for $$\mathcal{L}^2$$ (Ω), when λ ranges over some subset Λ in ℝ d , then we say that (Ω, Λ) is a spectral pair, and that Λ is a spectrum. We conjecture that (Ω, Λ) is a spectral pair if and only if the translates of some set Ω′ by the vectors of Λ tile ℝd. In the special case of Ω=Id, the d-dimensional unit cube, we prove this conjecture, with Ω′=Id, for d≤3, describing all the tilings by Id, and for all d when Λ is a discrete periodic set. In an appendix we generalize the notion of spectral pair to measures on a locally compact abelian group and its dual.
Details
- Title: Subtitle
- Spectral pairs in cartesian coordinates
- Creators
- Palle Jorgensen - Department of Mathematics Wright State University 45435 Dayton OHSteen Pedersen - Department of Mathematics Wright State University 45435 Dayton OH
- Resource Type
- Journal article
- Publication Details
- The Journal of Fourier Analysis and Applications, Vol.5(4), pp.285-302
- DOI
- 10.1007/BF01259371
- ISSN
- 1069-5869
- eISSN
- 1531-5851
- Publisher
- Birkhäuser-Verlag; Boston
- Language
- English
- Date published
- 07/1999
- Academic Unit
- Mathematics
- Record Identifier
- 9983985801402771
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