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)

 
 

 

Criteria of Pólya type for radial positive definite functions


Author: Tilmann Gneiting
Journal: Proc. Amer. Math. Soc. 129 (2001), 2309-2318
MSC (2000): Primary 42B10, 60E10, 42A82
DOI: https://doi.org/10.1090/S0002-9939-01-05839-7
Published electronically: January 17, 2001
MathSciNet review: 1823914
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

This article presents sufficient conditions for the positive definiteness of radial functions $f(x) = \varphi(\Vert x\Vert)$, $x \in \mathbb{R}^n$, in terms of the derivatives of $\varphi$. The criterion extends and unifies the previous analogues of Pólya's theorem and applies to arbitrarily smooth functions. In particular, it provides upper bounds on the Kuttner-Golubov function $k_n(\lambda)$ which gives the minimal value of $\kappa$ such that the truncated power function $(1-\Vert x\Vert^\lambda)_+^\kappa$, $x \in \mathbb{R}^n$, is positive definite. Analogous problems and criteria of Pólya type for $\Vert\cdot\Vert _\alpha$-dependent functions, $\alpha > 0$, are also considered.


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

  • 1. Askey, R., Radial Characteristic Functions, University of Wisconsin-Madison, Mathematics Research Center, 1262, 1973.
  • 2. Askey, R., Some characteristic functions of unimodal distributions, J. Math. Anal. Appl., 50 (1975), 465-469. MR 51:6910
  • 3. Berens, H. and Xu, Y., $l$-1 summability of multiple Fourier integrals and positivity, Math. Proc. Cambridge Philos. Soc., 122 (1997), 149-172. MR 98g:42035
  • 4. Bochner, S. and Chandrasekharan, K., Fourier Transforms, Princeton University Press, 1949. MR 11:173d
  • 5. Gneiting, T., On $\alpha$-symmetric multivariate characteristic functions, J. Multivariate Anal., 64 (1998), 131-147. MR 99h:60025
  • 6. Gneiting, T., On the derivatives of radial positive definite functions, J. Math. Anal. Appl., 236 (1999), 86-93. MR 2000k:42012
  • 7. Gneiting, T., Radial positive definite functions generated by Euclid's hat, J. Multivariate Anal., 69 (1999), 88-119. MR 2000g:60022
  • 8. Gneiting, T., A Pólya type criterion for radial characteristic functions in $\mathbb{R}^2$, Exposition. Math., 17 (1999), 181-183. MR 2000e:60024
  • 9. Gneiting, T., Kuttner's problem and a Pólya type criterion for characteristic functions, Proc. Amer. Math. Soc., 128 (2000), 1721-1728. MR 2000j:42016
  • 10. Golubov, B. I., On Abel-Poisson type and Riesz means, Anal. Math., 7 (1981), 161-184. MR 83b:42015
  • 11. Heal, K. M. and Hansen, M. L. and Rickard, K. M., Maple V Learning Guide, Springer, New York, 1996.
  • 12. Koldobskii, A. L., Schoenberg's problem on positive definite functions, St. Petersburg Math. J., 3 (1992), 563-570. MR 93c:42014
  • 13. Koldobsky, A., Positive definite functions, stable measures, and isometries on Banach spaces, Interaction Between Functional Analysis, Harmonic Analysis, and Probability, Lecture Notes in Pure and Appl. Math., 175, Marcel Dekker, 1996, 275-290. MR 96k:46016
  • 14. Kuttner, B., On the Riesz means of a Fourier series (II), J. London Math. Soc., 19 (1944), 77-84. MR 7:59d
  • 15. Letac, G. and Rahman, Q. I., A factorisation of the Askey's characteristic function $(1-\Vert t\Vert _{2n+1})^{n+1}_+$, Ann. Inst. H. Poincaré Probab. Statist., 22 (1986), 169-174. MR 87i:42027
  • 16. Lévy, P. , Extensions d'un théorème de D. Dugué et M. Girault, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 1 (1962), 159-173. MR 26:3096
  • 17. Liflyand, E. R., Ramm, A. G. and Zaslavsky, A. I., Estimates from below for Lebesgue constants, J. Fourier Anal. Appl., 2 (1996), 287-301. MR 97g:42007
  • 18. Matheron, G., Les variables régionalisées et leur estimation, Masson, Paris, 1965.
  • 19. Misiewicz, J. K., Sub-stable and pseudo-isotropic processes, connections with the geometry of sub-spaces of ${L}_\alpha$-spaces, Dissertationes Math., 358 (1996), 1-91. MR 97k:60040b
  • 20. Misiewicz, J. K. and Richards, D. St. P., Positivity of integrals of Bessel functions, SIAM J. Math. Anal., 25 (1994), 596-601. MR 95i:33004
  • 21. Misiewicz, J. K. and Scheffer, C. L., Pseudo isotropic measures, Nieuw Arch. Wisk. IV, 8 (1990), 111-152. MR 92e:60007
  • 22. Mittal, Y., A class of isotropic covariance functions, Pacific J. Math., 64 (1976), 517-538. MR 54:14068
  • 23. Pasenchenko, O. Yu., Sufficient conditions for the characteristic function of a two-dimensional isotropic distribution, Theory Probab. Math. Statist., 53 (1996), 149-152. MR 98b:60033
  • 24. Pólya, G., Remarks on Characteristic Functions, Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1949, 115-123. MR 10:463c
  • 25. Richards, D. St. P., Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform, J. Multivariate Anal., 19 (1986), 280-298. MR 88c:60043a
  • 26. Sasvári, Z., On a classical theorem in the theory of Fourier integrals, Proc. Amer. Math. Soc., 126 (1998), 711-713. MR 98i:60013
  • 27. Schoenberg, I. J., Metric spaces and completely monotone functions, Ann. of Math., 39 (1938), 811-841.
  • 28. Schoenberg, I. J., Metric spaces and positive definite functions, Trans. Amer. Math. Soc., 44 (1938), 522-536. CMP 95:18
  • 29. Trigub, R. M., A criterion for a characteristic function and a Polyá type criterion for radial functions of several variables, Theory Probab. Appl., 34 (1989), 738-742. MR 91f:60033
  • 30. Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math., 4 (1995), 389-396. MR 96h:41025
  • 31. Williamson, R. E., Multiply monotone functions and their Laplace transforms, Duke Math. J., 23 (1956), 189-207. MR 17:1061d
  • 32. Wintner, A., On a family of Fourier transforms, Bull. Amer. Math. Soc., 48 (1942), 304-308. MR 3:232a
  • 33. Wu, Z., Compactly supported positive definite radial functions, Adv. Comput. Math., 4 (1995), 283-292. MR 97g:65031
  • 34. Zastavnyi, V. P., Positive-definite functions that depend on a norm, Russian Acad. Sci. Dokl. Math., 46 (1993), 112-114. MR 94k:42026
  • 35. Zastavnyi, V. P., On positive definiteness of some functions, Dokl. Math., 365 (1999), 159-161. MR 2000i:42005
  • 36. Zastavnyi, V. P., On positive definiteness of some functions, J. Multivariate Anal., 73 (2000), 55-81. CMP 2000:14

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42B10, 60E10, 42A82

Retrieve articles in all journals with MSC (2000): 42B10, 60E10, 42A82


Additional Information

Tilmann Gneiting
Affiliation: Department of Statistics, Box 354322, University of Washington, Seattle, Washington 98195
Email: tilmann@stat.washington.edu

DOI: https://doi.org/10.1090/S0002-9939-01-05839-7
Received by editor(s): November 29, 1999
Published electronically: January 17, 2001
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society