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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**N.N. Andreev, personal communication.**2.**V.V. Arestov and E.E. Berdysheva, Turán's problem for positive definite functions with supports in a hexagon, Proc. Steklov Inst. Math., Suppl. 1, 2001, pp. S20-S29.**3.**V.V. Arestov and E.E. Berdysheva, The Turán problem for a class of polytopes, East J. Approx.**8**(2002), 381-388.**4.**B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal.**16**(1974), 101-121. MR**57:10500****5.**D.V. Gorbachev, An extremal problem for periodic functions with supports in the ball, Math. Notes**69**(2001), 3, 313-319. MR**2002e:42006****6.**A. Iosevich, N. Katz and S. Pedersen, Fourier bases and a distance problem of Erdos, Math. Res. Lett.**6**(1999), no. 2, 251-255. MR**2000j:42013****7.**A. Iosevich, N. Katz and T. Tao, Convex bodies with a point of curvature do not admit exponential bases, Amer. J. Math.**123**(2001), 115-120. MR**2002g:42011****8.**A. Iosevich, N. Katz and T. Tao, Fuglede's conjecture for convex bodies in the plane, Math. Res. Lett., to appear.**9.**M.N. Kolountzakis and J.C. Lagarias, Structure of tilings of the line by a function, Duke Math. J.**82**(1996), 3, 653-678. MR**97d:11124****10.**M.N. Kolountzakis, Non-symmetric convex domains have no basis of exponentials, Illinois J. Math.**44**(2000), 3, 542-550. MR**2001h:52019****11.**M.N. Kolountzakis, On the structure of multiple translational tilings by polygonal regions, Discr. Comp. Geom.**23**(2000), 537-553. MR**2001c:52025****12.**W. Rudin, An extension theorem for positive definite functions, Duke Math. J.**37**(1970), 49-53. MR**40:7722****13.**S.B. Stechkin, An extremal problem for trigonometric series with nonnegative coefficients, Acta Math Acad. Sci. Hung.**23**(1972), 3-4, pp 289-291 (Russian).**14.**P. McMullen, Convex bodies which tile space by translation, Mathematika**27**(1980), 113-121. MR**82c:52016**; acknowledgement of priority MR**83f:52008****15.**B.A. Venkov, On a class of Euclidean polyhedra, Vestnik Leningrad Univ. Ser. Math. Fiz. Him.**9**(1954), 11-31 (Russian). MR**20:1302**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
42B10,
26D15,
52C22,
42A82,
42A05

Retrieve articles in all journals with MSC (2000): 42B10, 26D15, 52C22, 42A82, 42A05

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