Let $(X,d)$ be a compact metric space, and let an iterated function system (IFS) be given on $X$, i.e., a finite set of continuous maps $\sigma_{ i}$: $ X\to X$, $i=0,1,..., N-1$. The maps $\sigma_{ i}$ transform the measures $\mu $ on $X$ into new measures $\mu_{ i}$. If the diameter of $ \sigma_{ i_{ 1}}\circ >... \circ \sigma_{ i_{ k}}(X)$ tends to zero as $ k\to \infty $, and if $p_{ i}>0$ satisfies $\sum_{ i}p_{ i}=1$, then it is known that there is a unique Borel probability measure $\mu $ on $X$ such that $\mu =\sum_{ i}p_{ i} \mu_{ i} \tag{ *}$. In this paper, we consider the case when the $p_{ i}$s are replaced with a certain system of sequilinear functionals. This allows us to study the variable coefficient case of (*), and moreover to understand the analog of (*) which is needed in the theory of wavelets.
Categories: math.CA