Optimal quantizers for a nonuniform distribution on a Sierpinski carpet

The purpose of quantization for a probability distribution is to estimate the probability by a discrete probability with finite support. In this paper, a nonuniform probability measure $P$ on ${\mathbb{R}}^2$ which has support the Sierpi'nski carpet generated by a set of four contractive similarity mappings with equal similarity ratios has been considered. For this probability measure, the optimal sets of $n$-means and the $n$th quantization errors are investigated for all $n\ge 2$.