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

Published electronically:
March 25, 2003

MathSciNet review:
1990631

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Abstract: We study the following question posed by Turán. Suppose is a convex body in Euclidean space which is symmetric in 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 , . 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