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 ℝd, 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.
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Communicated by Henry J. Landau
Dedicated to the memory of Irving E. Segal
Acknowledgements and Notes, Work supported by the National Science Foundation.
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Jorgensen, P.E.T., Pedersen, S. Spectral pairs in cartesian coordinates. The Journal of Fourier Analysis and Applications 5, 285–302 (1999). https://doi.org/10.1007/BF01259371
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DOI: https://doi.org/10.1007/BF01259371