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 $f_1(x)=\frac{1}{3}x$ and $f_2(x)=\frac{1}{3}x+\frac{2}{3}$ 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 $(\frac{1}{2},\frac{1}{2^2},\cdots )$, 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.