Self-similar measures and their Fourier transforms. II

Abstract:

A self-similar measure on ${{\mathbf {R}}^n}$ was defined by Hutchinson to be a probability measure satisfying $({\ast })$ \[ \mu = \sum \limits _{j = 1}^m {{a_j}\mu \circ S_j^{ - 1}} \] , where ${S_j}x = {\rho _j}{R_j}x + {b_j}$ is a contractive similarity $(0 < {\rho _j} < 1,{R_j}$ orthogonal) and the weights ${a_j}$ satisfy $0 < {a_j} < 1,\sum \nolimits _{j = 1}^m {{a_j} = 1}$. By analogy, we define a self-similar distribution by the same identity $( {\ast } )$ but allowing the weights ${a_j}$ to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to $( {\ast } )$ among distributions of compact support, and show that the space of such solutions is always finite dimensional. If $F$ denotes the Fourier transformation of a self-similar distribution of compact support, let \[ H(R) = \frac {1}{{{R^{n - \beta }}}}\int _{|x| \leq R} {|F(x){|^2}dx,} \] where $\beta$ is defined by the equation $\sum \nolimits _{j = 1}^m {\rho _j^{ - \beta }|{a_j}{|^2} = 1}$. If $\rho _j^{{\nu _j}} = \rho$ for some fixed $\rho$ and ${\nu _j}$ positive integers we say the $\{ {\rho _j}\}$ are exponentially commensurable. In this case we prove (under some additional hypotheses) that $H(R)$ is asymptotic (in a suitable sense) to a bounded function $\tilde H(R)$ that is bounded away from zero and periodic in the sense that $\tilde H(\rho R) = \tilde H(R)$ for all $R > 0$. If the $\{ {\rho _j}\}$ are exponentially incommensurable then ${\lim _{R \to \infty }}H(R)$ exists and is nonzero.