Asymptotic Properties of Fourier Transforms of b-Decomposable Distributions

Erdos-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 L,vy processes is discussed.