Abstract
A conjecture of Fuglede states that a bounded measurable set $\Omega \subset \mathbb{R}^{d}$, of measure 1, can tile $\mathbb{R}^{d}$ by translations if and only if the Hilbert space $L^{2}(\Omega)$ has an orthonormal basis consisting of exponentials $e_{\lambda}(x)=\exp 2\pi i \langle \lambda,x \rangle$. If $\Omega$ has the latter property it is called spectral. We generalize a result of Fuglede, that a triangle in the plane is not spectral, proving that every non-symmetric convex domain in $\mathbb{R}^{d}$ is not spectral.
Citation
Mihail N. Kolountzakis. "Non-symmetric convex domains have no basis of exponentials." Illinois J. Math. 44 (3) 542 - 550, Fall 2000. https://doi.org/10.1215/ijm/1256060414
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