Orthonormal bases of exponentials for the \(n\)-cube. (English) Zbl 0978.42007
Consider the unit cube \(Q\) in \({\mathbb R}^d\). It is proved here that a set \(\Lambda \subseteq {\mathbb R}^d\) is a tiling set for \(Q\) (that is, the sets \(Q+\lambda\), \(\lambda\in\Lambda\), are non-overlapping and fill the space) if and only if the set \(E_\Lambda = \{\exp{(2\pi i \lambda\cdot x)}: \lambda\in\Lambda\}\) is a complete orthonormal basis in \(L^2(Q)\).
This is related to a conjecture of Fuglede which states that for a domain \(\Omega\) in the space of volume 1, such a basis for \(L^2(\Omega)\) exists if and only if \(\Omega\) can be translated so as to tile the space.
This is related to a conjecture of Fuglede which states that for a domain \(\Omega\) in the space of volume 1, such a basis for \(L^2(\Omega)\) exists if and only if \(\Omega\) can be translated so as to tile the space.
Reviewer: Mihail Kolountzakis (Crete)
MSC:
| 42B05 | Fourier series and coefficients in several variables |
| 52C22 | Tilings in \(n\) dimensions (aspects of discrete geometry) |
| 11K70 | Harmonic analysis and almost periodicity in probabilistic number theory |
| 47A13 | Several-variable operator theory (spectral, Fredholm, etc.) |
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