L. OLSENa1 and N. SNIGIREVAa2
a1 Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland. e-mail: firstname.lastname@example.org
a2 Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland. e-mail: email@example.com
Let Sj: d → d for j = 1, . . ., N be contracting similarities. Also, let (p1,. . ., pN, p) be a probability vector and let ν be a probability measure on d with compact support. We show that there exists a unique probability measure μ such that
The measure μ is called an in-homogenous self-similar measure. In this paper we study the asymptotic behaviour of the Fourier transforms of in-homogenous self-similar measures. Finally, we present a number of applications of our results. In particular, non-linear self-similar measures introduced and investigated by Glickenstein and Strichartz are special cases of in-homogenous self-similar measures, and as an application of our main results we obtain simple proofs of generalizations of Glickenstein and Strichartz's results on the asymptotic behaviour of the Fourier transforms of non-linear self-similar measures.
(Received April 24 2006)
(Revised May 08 2007)
(Online publication February 11 2008)