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On a problem of Turán about positive definite functions

Authors:	Mihail N Kolountzakis Szilá rd Gy Ré vé sz

Institution:	Department of Mathematics, University of Crete, Knossos Ave., 714 09 Iraklio, Greece ; Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1364 Budapest, Hungary

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 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.

Keywords:	Fourier transform positive definite functions Tur\'an's extremal problem convex symmetric domains tiling of space lattice tiling spectral domains

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