On a problem of Turán about positive definite functions
Authors:
Mihail N. Kolountzakis and Szilárd Gy. Révész
Journal:
Proc. Amer. Math. Soc. 131 (2003), 34233430
MSC (2000):
Primary 42B10; Secondary 26D15, 52C22, 42A82, 42A05
Published electronically:
March 25, 2003
MathSciNet review:
1990631
Fulltext PDF Free Access
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 halfbody 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. S20S29.
 3.
V.V. Arestov and E.E. Berdysheva, The Turán problem for a class of polytopes, East J. Approx. 8 (2002), 381388.
 4.
Bent
Fuglede, Commuting selfadjoint partial differential operators and
a group theoretic problem, J. Functional Analysis 16
(1974), 101–121. MR 0470754
(57 #10500)
 5.
D.
V. Gorbachev, An extremal problem for periodic functions with
support in a ball, Mat. Zametki 69 (2001),
no. 3, 346–352 (Russian, with Russian summary); English transl.,
Math. Notes 69 (2001), no. 34, 313–319. MR 1846833
(2002e:42006), http://dx.doi.org/10.1023/A:1010275206760
 6.
Alex
Iosevich, Nets
Katz, and Steen
Pedersen, Fourier bases and a distance problem of
Erdős, Math. Res. Lett. 6 (1999), no. 2,
251–255. MR 1689215
(2000j:42013), http://dx.doi.org/10.4310/MRL.1999.v6.n2.a13
 7.
Alex
Iosevich, Nets
Hawk Katz, and Terry
Tao, Convex bodies with a point of curvature do not have Fourier
bases, Amer. J. Math. 123 (2001), no. 1,
115–120. MR 1827279
(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.
Mihail
N. Kolountzakis and Jeffrey
C. Lagarias, Structure of tilings of the line by a function,
Duke Math. J. 82 (1996), no. 3, 653–678. MR 1387688
(97d:11124), http://dx.doi.org/10.1215/S0012709496082277
 10.
Mihail
N. Kolountzakis, Nonsymmetric convex domains have no basis of
exponentials, Illinois J. Math. 44 (2000),
no. 3, 542–550. MR 1772427
(2001h:52019)
 11.
M.
N. Kolountzakis, On the structure of multiple translational tilings
by polygonal regions, Discrete Comput. Geom. 23
(2000), no. 4, 537–553. MR 1753701
(2001c:52025), http://dx.doi.org/10.1007/s004540010014
 12.
Walter
Rudin, An extension theorem for positivedefinite functions,
Duke Math. J. 37 (1970), 49–53. MR 0254514
(40 #7722)
 13.
S.B. Stechkin, An extremal problem for trigonometric series with nonnegative coefficients, Acta Math Acad. Sci. Hung. 23 (1972), 34, pp 289291 (Russian).
 14.
P.
McMullen, Convex bodies which tile space by translation,
Mathematika 27 (1980), no. 1, 113–121. MR 582003
(82c:52016), http://dx.doi.org/10.1112/S0025579300010007
P.
McMullen, Acknowledgement of priority: “Convex bodies which
tile space by translation” [Mathematika 27 (1980), no. 1,
113–121;\ MR 82c:52016], Mathematika 28 (1981),
no. 2, 191 (1982). MR 645098
(83f:52008), http://dx.doi.org/10.1112/S0025579300010238
 15.
B.
A. Venkov, On a class of Euclidean polyhedra, Vestnik
Leningrad. Univ. Ser. Mat. Fiz. Him. 9 (1954), no. 2,
11–31 (Russian). MR 0094790
(20 #1302)
 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. S20S29.
 3.
 V.V. Arestov and E.E. Berdysheva, The Turán problem for a class of polytopes, East J. Approx. 8 (2002), 381388.
 4.
 B. Fuglede, Commuting selfadjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974), 101121. MR 57:10500
 5.
 D.V. Gorbachev, An extremal problem for periodic functions with supports in the ball, Math. Notes 69 (2001), 3, 313319. 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, 251255. 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), 115120. 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, 653678. MR 97d:11124
 10.
 M.N. Kolountzakis, Nonsymmetric convex domains have no basis of exponentials, Illinois J. Math. 44 (2000), 3, 542550. MR 2001h:52019
 11.
 M.N. Kolountzakis, On the structure of multiple translational tilings by polygonal regions, Discr. Comp. Geom. 23 (2000), 537553. MR 2001c:52025
 12.
 W. Rudin, An extension theorem for positive definite functions, Duke Math. J. 37 (1970), 4953. MR 40:7722
 13.
 S.B. Stechkin, An extremal problem for trigonometric series with nonnegative coefficients, Acta Math Acad. Sci. Hung. 23 (1972), 34, pp 289291 (Russian).
 14.
 P. McMullen, Convex bodies which tile space by translation, Mathematika 27 (1980), 113121. 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), 1131 (Russian). MR 20:1302
Similar Articles
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:
http://dx.doi.org/10.1090/S0002993903070230
PII:
S 00029939(03)070230
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 9901080
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
