Spectral pairs in Cartesian coordinates
| dc.creator | Jorgensen, Palle E. T. | |
| dc.creator | Pedersen, Steen | |
| dc.date | 1999-12-15 | |
| dc.date | 2001-06-28 | |
| dc.date.accessioned | 2026-07-07T05:32:19Z | |
| dc.date.available | 2026-07-07T05:32:19Z | |
| dc.description | Let $ Ω\subset R^d $ 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 functions $ e_λ $ on $ Ω$ given by $ e_λ(x)=e^{i2πλx} $. 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 $ R^d $. In the special case of $ Ω=I^d $, the $ d $-dimensional unit cube, we prove this conjecture, with $ Ω'=I^d $, for $ d \leq 3 $, describing all the tilings by $ I^d $, 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. | |
| dc.description | AMS-LaTeX; 18 pages, 1 figure comprising 2 EPS diagrams; revision provides the graphics files for these figures (no other changes) | |
| dc.identifier | https://arxiv.org/abs/math/9912131 | |
| dc.identifier | http://arxiv.org/abs/math/9912131 | |
| dc.identifier | Journal of Fourier Analysis and Applications 5 (1999), 289--306 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/79619 | |
| dc.subject | Functional Analysis | |
| dc.subject | 42C05, 22D25, 46L55, 47C05 | |
| dc.title | Spectral pairs in Cartesian coordinates | |
| dc.type | text |