Academic Journal
Packing, Tiling, Orthogonality and Completeness: Packing, tiling, orthogonality and completeness
| Τίτλος: | Packing, Tiling, Orthogonality and Completeness: Packing, tiling, orthogonality and completeness |
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| Συγγραφείς: | Mihail N. Kolountzakis |
| Πηγή: | Bulletin of the London Mathematical Society. 32:589-599 |
| Publication Status: | Preprint |
| Στοιχεία εκδότη: | Wiley, 2000. |
| Έτος έκδοσης: | 2000 |
| Θεματικοί όροι: | Metric Geometry (math.MG), 0102 computer and information sciences, Harmonic analysis and almost periodicity in probabilistic number theory, 01 natural sciences, Harmonic analysis in several variables, Mathematics - Metric Geometry, Mathematics - Classical Analysis and ODEs, Tilings in \(n\) dimensions (aspects of discrete geometry), tiling, Classical Analysis and ODEs (math.CA), FOS: Mathematics, tight orthogonal packing region, 0101 mathematics |
| Περιγραφή: | Let $��\subseteq {\bf R}^d$ be an open set of measure 1. An open set $D \subseteq {\bf R}^d$ is called a ``tight orthogonal packing region'' for $��$ if $D-D$ does not intersect the zeros of the Fourier Transform of the indicator function of $��$ and $D$ has measure 1. Suppose that $��$ is a discrete subset of ${\bf R}^d$. The main contribution of this paper is a new way of proving the following result (proved by different methods by Lagarias, Reeds and Wang and, in the case of $��$ being the cube, by Iosevich and Pedersen: $D$ tiles ${\bf R}^d$ when translated at the locations $��$ if and only if the set of exponentials $E_��= \{\exp 2��i ��\cdot x: ��\in��\}$ is an orthonormal basis for $L^2(��)$. (When $��$ is the unit cube in ${\bf R}^d$ then it is a tight orthogonal packing region of itself.) In our approach orthogonality of $E_��$ is viewed as a statement about ``packing'' ${\bf R}^d$ with translates of a certain nonnegative function and, additionally, we have completeness of $E_��$ in $L^2(��)$ if and only if the above-mentioned packing is in fact a tiling. We then formulate the tiling condition in Fourier Analytic language and use this to prove our result. |
| Τύπος εγγράφου: | Article |
| Περιγραφή αρχείου: | application/xml |
| Γλώσσα: | English |
| ISSN: | 0024-6093 |
| DOI: | 10.1112/s0024609300007281 |
| DOI: | 10.48550/arxiv.math/9904066 |
| Σύνδεσμος πρόσβασης: | http://arxiv.org/pdf/math/9904066 http://arxiv.org/abs/math/9904066 http://doi.wiley.com/10.1112/S0024609300007281 https://academic.oup.com/blms/article/32/5/589/596019 http://ui.adsabs.harvard.edu/abs/1999math......4066K/abstract https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024609300007281 https://www.cambridge.org/core/journals/bulletin-of-the-london-mathematical-society/article/packing-tiling-orthogonality-and-completeness/0AC746F7217C80F7F485E97C108E5437 |
| Rights: | Wiley Online Library User Agreement arXiv Non-Exclusive Distribution |
| Αριθμός Καταχώρησης: | edsair.doi.dedup.....f35c6e0b5ea74df75eba4ae29a9de839 |
| Βάση Δεδομένων: | OpenAIRE |
| ISSN: | 00246093 |
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| DOI: | 10.1112/s0024609300007281 |