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 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.