Optimal quantization for the Cantor distribution generated by infinite similutudes

Abstract

Let $P$ be a Borel probability measure on $\mathbb R$ generated by an infinite system of similarity mappings $\{S_j : j\in \mathbb N\}$ such that $P=\sum_{j=1}^\infty \frac 1{2^j} P\circ S_j^{-1}$, where for each $j\in \mathbb N$ and $x\in \mathbb R$, $S_j(x)=\frac 1{3^{j}}x+1-\frac 1 {3^{j-1}}$. Then, the support of $P$ is the dyadic Cantor set $C$ generated by the similarity mappings $f_1, f_2 : \mathbb R \to \mathbb R$ such that $f_1(x)=\frac 13 x$ and $f_2(x)=\frac 13 x+\frac 23$ for all $x\in \mathbb R$. In this paper, using the infinite system of similarity mappings $\{S_j : j\in \mathbb N\}$ associated with the probability vector $(\frac 12, \frac 1{2^2}, \cdots)$, for all $n\in \mathbb N$, we determine the optimal sets of $n$-means and the $n$th quantization errors for the infinite self-similar measure $P$. The technique obtained in this paper can be utilized to determine the optimal sets of $n$-means and the $n$th quantization errors for more general infinite self-similar measures.