Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 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

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

Szilárd Gy. Révész
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1364 Budapest, Hungary

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

American Mathematical Society