Abstract :
A coin-tossing measure μ on [0, 1] is a probability measure satisfying
μ =
∞ ∗
n=1
pnδ(0) +
(
1− pn)δ(1/2n)
where pn ∈ [0, 1], δ(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 μ, μˆ(x). For p 2 and
ε >0 we prove that
|x| R μˆ (x)
p
dx = O(R1−βp+ε), R→+∞,
where
βp = 1 −lim sup
N→∞
1
N
N
n=1
log2 1 + |an|p , an = 2pn − 1.
This extends some results due to R. Strichartz for measures which are not self-similar.We also study
the Sobolev exponent of | ˆ μ(x)|p and its scaling exponent, as well as the asymptotic behavior of sums
of the Walsh–Fourier coefficients of μ.
2004 Elsevier Inc. All rights reserved.
