Illinois Journal of Mathematics

Non-symmetric convex domains have no basis of exponentials

Mihail N. Kolountzakis

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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.

Article information

Illinois J. Math., Volume 44, Issue 3 (2000), 542-550.

First available in Project Euclid: 20 October 2009

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Primary: 52C22: Tilings in $n$ dimensions [See also 05B45, 51M20]
Secondary: 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 42B05: Fourier series and coefficients 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)


Kolountzakis, Mihail N. Non-symmetric convex domains have no basis of exponentials. Illinois J. Math. 44 (2000), no. 3, 542--550. doi:10.1215/ijm/1256060414.

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