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The convergence of the Fourier integral of a unimodal distribution
Author(s):
Constantine
Georgakis
Journal:
Proc. Amer. Math. Soc.
126
(1998),
3239-3241.
MSC (1991):
Primary 42A38;
Secondary 60E10
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Abstract:
We show that the characteristic function of a unimodal probability distribution function can be inverted by the Fourier transform a.e. if and only if the distribution is absolutely continuous. The result complements Khintchine's criterion for unimodal distributions.
References:
- 1.
- E. M-J. Bertin, I. Cuculescu, and R. Theodorescu, Unimodality of Probability Measures, Kluwer Academic Publishers, 1997.
- 2.
- S. Dharmadhikari and J-D. Kumar, Unimodality, Convexity, and Applications, Academic Press, 1988. MR 89k:60020
- 3.
- C. Georgakis, The Hausdorff Mean of a Fourier-Stieltjes Transform, Proc. Amer. Math. Soc. 116 (1992), 465-471. MR 92m:42019
- 4.
- R. R. Goldberg, Fourier Transforms, Cambridge University Press, 1962. MR 22:11254
- 5.
- T. Kawata, Fourier Analysis in Probability Theory, Academic Press, 1972. MR 57:4284
- 6.
- A. Ya. Khintchine, On Unimodal Distributions, Izv. Nauchno Issled. Inst. Mat. Mech. Temsk. Gos. Univ. 2, no. 2, 1-7 (1938).
- 7.
- E. Lukacs, Characteristic Functions, Hafner Publishing Co., 1970. MR 49:11595
- 8.
- P. Medgyessy, Decomposition of Superpositions of Density Functions and Discrete Distributions, John Wiley & Sons, 1977. MR 55:13532b
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Additional Information:
Constantine
Georgakis
Affiliation:
Department of Mathematics, DePaul University, Chicago, Illinois 60614
DOI:
10.1090/S0002-9939-98-04385-8
PII:
S 0002-9939(98)04385-8
Keywords:
Khintchine,
unimodal distribution,
Fourier integral,
inversion,
characteristic function
Received by editor(s):
December 14, 1994
Received by editor(s) in revised form:
March 10, 1997
Communicated by:
J. Marshall Ash
Copyright of article:
Copyright
1998,
American Mathematical Society
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