ISSN:
1531-5851
Keywords:
42C05
;
22D25
;
46L55
;
47C05
;
spectral pair
;
translations
;
tilings
;
Fourier basis
;
operator extensions
;
induced representations
;
spectral resolution
;
Hilbert space
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01259371
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