Quantization for a Condensation System

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