Asymptotic Properties of Fourier Transforms of b-Decomposable Distributions

Erdös-Kahane numbers (EK numbers) are introduced in relation to the decay of the Fourier transforms of non-symmetric Bernoulli convolutions. The PV, PS, and EK numbers are characterized by using a certain trigonometric series H b ( u ). The relations between those numbers and the asymptotic properti...

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Bibliographic Details
Published in:The Journal of fourier analysis and applications 2012-08, Vol.18 (4), p.803-827
Main Author: Watanabe, Toshiro
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Approximations and Expansions
Fourier Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Partial Differential Equations
Signal,Image and Speech Processing
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Summary:Erdös-Kahane numbers (EK numbers) are introduced in relation to the decay of the Fourier transforms of non-symmetric Bernoulli convolutions. The PV, PS, and EK numbers are characterized by using a certain trigonometric series H b ( u ). The relations between those numbers and the asymptotic properties of the Fourier transforms of full b -decomposable distributions are shown. A sufficient condition for the absolute continuity of one-dimensional b -decomposable distributions is given. As an application, an open problem on the uniform decay of the Fourier transforms of refinable distributions, raised by Dai et al. (J. Funct. Anal. 250(1):1–20, 2007 ), is solved. Finally, temporal evolution on continuity properties of distributions of some Lévy processes is discussed.
ISSN:1069-5869
1531-5851
DOI:10.1007/s00041-012-9222-9