Kuttner's problem and a Pólya type criterion for characteristic functions
Author:
Tilmann Gneiting
Journal:
Proc. Amer. Math. Soc. 128 (2000), 17211728
MSC (1991):
Primary 42A82, 60E10; Secondary 42A24, 42A38
Published electronically:
October 27, 1999
MathSciNet review:
1646306
Fulltext PDF Free Access
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Abstract: Let be a continuous function with and . If is convex, then , , 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 , , which gives the minimal value of such that is a characteristic function. Specifically, . Furthermore, improved lower bounds on Kuttner's function are obtained from an inequality due to Boas and Kac.
 1.
Torben
Maack Bisgaard and Zoltán
Sasvári, On the positive definiteness of certain
functions, Math. Nachr. 186 (1997), 81–99. MR 1461214
(99c:42009)
 2.
R.
P. Boas Jr. and M.
Kac, Inequalities for Fourier transforms of positive
functions, Duke Math. J. 12 (1945), 189–206. MR 0012152
(6,265h)
 3.
William
Feller, An introduction to probability theory and its applications.
Vol. II., Second edition, John Wiley & Sons, Inc., New
YorkLondonSydney, 1971. MR 0270403
(42 #5292)
 4.
Julian
Keilson and F.
W. Steutel, Mixtures of distributions, moment inequalities and
measures of exponentiality and normality, Ann. Probability
2 (1974), 112–130. MR 0356180
(50 #8651)
 5.
B.
Kuttner, On the Riesz means of a Fourier series. II, J. London
Math. Soc. 19 (1944), 77–84. MR 0012686
(7,59d)
 6.
Eugene
Lukacs, Characteristic functions, Hafner Publishing Co., New
York, 1970. Second edition, revised and enlarged. MR 0346874
(49 #11595)
 7.
Jolanta
K. Misiewicz and Donald
St. P. Richards, Positivity of integrals of Bessel functions,
SIAM J. Math. Anal. 25 (1994), no. 2, 596–601.
MR
1266579 (95i:33004), http://dx.doi.org/10.1137/S0036141092226934
 8.
G.
Pólya, Remarks on characteristic functions, Proceedings
of the Berkeley Symposium on Mathematical Statistics and Probability, 1945,
1946, University of California Press, Berkeley and Los Angeles, 1949,
pp. 115–123. MR 0028541
(10,463c)
 9.
Zoltán
Sasvári, On a classical theorem in the theory
of Fourier integrals, Proc. Amer. Math.
Soc. 126 (1998), no. 3, 711–713. MR 1469433
(98i:60013), http://dx.doi.org/10.1090/S0002993998046048
 10.
Holger
Wendland, Piecewise polynomial, positive definite and compactly
supported radial functions of minimal degree, Adv. Comput. Math.
4 (1995), no. 4, 389–396. MR 1366510
(96h:41025), http://dx.doi.org/10.1007/BF02123482
 11.
R.
E. Williamson, Multiply monotone functions and their Laplace
transforms, Duke Math. J. 23 (1956), 189–207.
MR
0077581 (17,1061d)
 12.
Aurel
Wintner, On a family of Fourier
transforms, Bull. Amer. Math. Soc. 48 (1942), 304–308. MR 0005920
(3,232a), http://dx.doi.org/10.1090/S000299041942076622
 13.
Zong
Min Wu, Compactly supported positive definite radial
functions, Adv. Comput. Math. 4 (1995), no. 3,
283–292. MR 1357720
(97g:65031), http://dx.doi.org/10.1007/BF03177517
 14.
V. P. Zastavnyi, On positive definiteness of some functions, Manuscript, Donetsk State University, Donetsk, Ukraine, 1998.
 1.
 T. M. Bisgaard and Z. Sasvári, On the positive definiteness of certain functions, Math. Nachr. 186 (1997), 8199. MR 99c:42009
 2.
 R. P. Boas and M. Kac, Inequalities for Fourier transforms of positive functions, Duke Math. J. 12 (1945), 189206. 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), 112130.MR 50:8651
 5.
 B. Kuttner, On the Riesz means of a Fourier series (II), J. London Math. Soc. 19 (1944), 7784. 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), 596601. 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. 115123. MR 10:463c
 9.
 Z. Sasvári, On a classical theorem in the theory of Fourier integrals, Proc. Amer. Math. Soc. 126 (1998), 711713. MR 98i:60013
 10.
 H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995), 389396. MR 96h:41025
 11.
 R. E. Williamson, Multiply monotone functions and their Laplace transforms, Duke Math. J. 23 (1956), 189207. MR 17:1061d
 12.
 A. Wintner, On a family of Fourier transforms, Bull. Amer. Math. Soc. 48 (1942), 304308. MR 3:232a
 13.
 Z. Wu, Compactly supported positive definite radial functions, Adv. Comput. Math. 4 (1995), 283292. MR 97g:65031
 14.
 V. P. Zastavnyi, On positive definiteness of some functions, Manuscript, Donetsk State University, Donetsk, Ukraine, 1998.
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Additional Information
Tilmann Gneiting
Affiliation:
Department of Statistics, Box 354322, University of Washington, Seattle, Washington 98195
Email:
tilmann@stat.washington.edu
DOI:
http://dx.doi.org/10.1090/S0002993999052004
PII:
S 00029939(99)052004
Received by editor(s):
July 13, 1998
Published electronically:
October 27, 1999
Communicated by:
Christopher D. Sogge
Article copyright:
© Copyright 2000
American Mathematical Society
