On the Fourier transforms of inhomogeneous self-similar measures

Abstract

The inhomogeneous self-similar measure $\mu$ is defined by the relation \[ \mu= \sum_{j=1}^N p_j \mu oS_j^{-1} + p \nu, \] where $(p_1, \ldots, p_N, p)$ is a probability vector, $ S_j : \mathbb R^n \rightarrow\mathbb R^n, j = 1, \ldots, N$ are contracting similarities and $\nu$ is a probability measure on $\mathbb R^n$ with compact support. The existence of such measures is well known, see Math. Proc. Cambridge Philos. Soc. 144 (2008) 465-493) and the references therein. In Math. Proc. Cambridge Philos. Soc. 144 (2008) 465-493), the authors have studied the Fourier transforms of inhomogeneous self-similar measures and they give relations about the asymptotic behavior of the Fourier transform of $\nu$ and $\mu$. Some constructions which are given with precise asymptotic behavior arise from a discrete measure $\nu$. Here we will see that these constructions can be extended with purely continuous measures $\nu$. In order to prove this, we will construct suitable symmetric Bernoulli convolution measures Essays in Commutative Harmonic Analysis (1979) Springer) and will use the results of J. Math. Anal. Appl. 299 (2004) 550–562).