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)



Kuttner's problem and a Pólya type criterion
for characteristic functions

Author: Tilmann Gneiting
Journal: Proc. Amer. Math. Soc. 128 (2000), 1721-1728
MSC (1991): Primary 42A82, 60E10; Secondary 42A24, 42A38
Published electronically: October 27, 1999
MathSciNet review: 1646306
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\varphi : [0,\infty) \to \mathbb{R}$ be a continuous function with $\varphi(0) = 1$ and $\lim _{t \to \infty} \varphi(t)$ $= 0$. If $t^{-1} (\sqrt{t} \, \varphi''(\sqrt{t}) - \varphi'(\sqrt{t}))$ is convex, then $\psi(t) = \varphi(|t|)$, $t \in \mathbb{R}$, is the characteristic function of an absolutely continuous probability distribution. The criterion complements Pólya's theorem and applies to characteristic functions with various types of behavior at the origin. In particular, it provides upper bounds on Kuttner's function $k(\lambda)$, $\lambda \in (0,2)$, which gives the minimal value of $\kappa$ such that $(1-|t|^\lambda)_+^\kappa$ is a characteristic function. Specifically, $k(5/3) \leq 3$. Furthermore, improved lower bounds on Kuttner's function are obtained from an inequality due to Boas and Kac.

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

  • 1. T. M. Bisgaard and Z. Sasvári, On the positive definiteness of certain functions, Math. Nachr. 186 (1997), 81-99. MR 99c:42009
  • 2. R. P. Boas and M. Kac, Inequalities for Fourier transforms of positive functions, Duke Math. J. 12 (1945), 189-206. MR 6:265h
  • 3. W. Feller, An introduction to probability theory and its applications, vol. II, second ed., John Wiley, New York, 1971. MR 42:5292
  • 4. J. Keilson and W. Steutel, Mixtures of distributions, moment inequalities and measures of exponentiality and normality, Ann. Probability 2 (1974), 112-130.MR 50:8651
  • 5. B. Kuttner, On the Riesz means of a Fourier series (II), J. London Math. Soc. 19 (1944), 77-84. MR 7:59d
  • 6. E. Lukacs, Characteristic functions, second ed., Griffin, London, 1970. MR 49:11595
  • 7. J. K. Misiewicz and D. St. P. Richards, Positivity of integrals of Bessel functions, SIAM J. Math. Anal. 25 (1994), 596-601. MR 95i:33004
  • 8. G. Pólya, Remarks on characteristic functions, Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability (J. Neyman, ed.), University of California Press, 1949, pp. 115-123. MR 10:463c
  • 9. Z. Sasvári, On a classical theorem in the theory of Fourier integrals, Proc. Amer. Math. Soc. 126 (1998), 711-713. MR 98i:60013
  • 10. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995), 389-396. MR 96h:41025
  • 11. R. E. Williamson, Multiply monotone functions and their Laplace transforms, Duke Math. J. 23 (1956), 189-207. MR 17:1061d
  • 12. A. Wintner, On a family of Fourier transforms, Bull. Amer. Math. Soc. 48 (1942), 304-308. MR 3:232a
  • 13. Z. Wu, Compactly supported positive definite radial functions, Adv. Comput. Math. 4 (1995), 283-292. MR 97g:65031
  • 14. V. P. Zastavnyi, On positive definiteness of some functions, Manuscript, Donetsk State University, Donetsk, Ukraine, 1998.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42A82, 60E10, 42A24, 42A38

Retrieve articles in all journals with MSC (1991): 42A82, 60E10, 42A24, 42A38

Additional Information

Tilmann Gneiting
Affiliation: Department of Statistics, Box 354322, University of Washington, Seattle, Washington 98195

Received by editor(s): July 13, 1998
Published electronically: October 27, 1999
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society