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On a problem of Turán about positive definite functions


Authors: Mihail N. Kolountzakis and Szilárd Gy. Révész
Journal: Proc. Amer. Math. Soc. 131 (2003), 3423-3430
MSC (2000): Primary 42B10; Secondary 26D15, 52C22, 42A82, 42A05
DOI: https://doi.org/10.1090/S0002-9939-03-07023-0
Published electronically: March 25, 2003
MathSciNet review: 1990631
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Abstract: We study the following question posed by Turán. Suppose $\Omega$ is a convex body in Euclidean space $\mathbb{R} ^d$ which is symmetric in $\Omega$ and with value $1$ at the origin; which one has the largest integral? It is probably the case that the extremal function is the indicator of the half-body convolved with itself and properly scaled, but this has been proved only for a small class of domains so far. We add to this class of known Turán domains the class of all spectral convex domains. These are all convex domains which have an orthogonal basis of exponentials $e_\lambda(x) = \exp 2\pi i\langle{\lambda}{x}\rangle$, $\lambda \in \mathbb{R} ^d$. As a corollary we obtain that all convex domains which tile space by translation are Turán domains.

We also give a new proof that the Euclidean ball is a Turán domain.


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Additional Information

Mihail N. Kolountzakis
Affiliation: Department of Mathematics, University of Crete, Knossos Ave., 714 09 Iraklio, Greece
Email: kolount@member.ams.org

Szilárd Gy. Révész
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1364 Budapest, Hungary
Email: revesz@renyi.hu

DOI: https://doi.org/10.1090/S0002-9939-03-07023-0
Keywords: Fourier transform, positive definite functions, Tur\'an's extremal problem, convex symmetric domains, tiling of space, lattice tiling, spectral domains
Received by editor(s): May 22, 2002
Published electronically: March 25, 2003
Additional Notes: The first author was supported in part by INTAS grant, project 99-01080
The second author was supported in part by the Hungarian National Foundation for Scientific Research, Grant # T034531 and T032872.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2003 American Mathematical Society

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