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Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

In-homogenous self-similar measures and their Fourier transforms

L. OLSENa1 and N. SNIGIREVAa2

a1 Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland. e-mail: lo@st-and.ac.uk

a2 Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland. e-mail: ns93@st-and.ac.uk

Abstract

Let Sj: xs211Ddxs211Dd for j = 1, . . ., N be contracting similarities. Also, let (p1,. . ., pN, p) be a probability vector and let ν be a probability measure on xs211Dd with compact support. We show that there exists a unique probability measure μ such that

\begin{eqnarray}$$\[
\mu = \sum_{j}p_{j}\mu\circ S_{j}^{-1} \,+\, p\nu.
\]$$\end{eqnarray}]

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)