Quantization for a condensation system

Abstract

For a given $r\in (0,+\infty )$, the quantization dimension of order $r$, if it exists, denoted by $D_r(\hat{I}¼)$, represents the rate at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $\hat{I}¼$ tends to infinity. If $D_r(\hat{I}¼)$ does not exist, we define ${\underline{D}}_r(\hat{I}¼)$ and ${\overline{D}}_r(\hat{I}¼)$ as the lower and the upper quantization dimensions of $\hat{I}¼$ of order $r$, respectively. In this paper, we investigate the quantization dimension of the condensation measure $\hat{I}¼$ associated with a condensation system $(\{S_j{\}}_{j=1}^N,(p_j{)}_{j=0}^N,\hat{I}½).$ We provide two examples: one where $\hat{I}½$ is an infinite discrete distribution on $\mathbb{R}$, and one where $\hat{I}½$ is a uniform distribution on $\mathbb{R}$. For both the discrete and uniform distributions $\hat{I}½$, we determine the optimal sets of $n$-means, and calculate the quantization dimensions of condensation measures $\hat{I}¼$, and show that the $D_r(\hat{I}¼)$-dimensional quantization coefficients do not exist. Moreover, we demonstrate that the lower and upper quantization coefficients are finite and positive.