Coin-tossing measures and their Fourier transforms

[摘要] A coin-tossing measure mu on [0, 1] is a probability measure satisfying Graphics where P-n is an element of [0, 1], delta(x) denotes the probability atom at x and the convergence is in the weak* sense. We study the asymptotic behavior of averages of the Fourier transform of mu, (mu) over cap (x). For p greater than or equal to 2 and epsilon > 0 we prove that Graphics Where Graphics This extends some results due to R. Strichartz for measures which are not self-similar. We also study the Sobolev exponent of \ (mu) over cap (x) \(p) and its scaling exponent, as well as the asymptotic behavior of sums of the Walsh-Fourier coefficients of mu. (C) 2004 Elsevier Inc. All rights reserved.

[发布日期] 2004-11-15

[关键词] coin-tossing measures;Fourier transformation;Walsh functions