Optimal quantization for the condensation system associated with self-similar measures
Doğan Çömez, Mrinal Kanti Roychowdhury · arXiv (Cornell University) · 2017
Let $S_1, S_2, T_1, T_2$ be contractive similarity mappings such that $S_1(x)=\frac 15 x$, $S_2(x)=\frac 1 5 x+\frac 45$, $T_1(x)=\frac 1{3} x+\frac 4{15}$, and $T_2(x)=\frac 1{3} x+\frac 25$ for all $x\in\mathbb R$. Set $P=\frac 13 P \circ S_1^{-1}+\frac 13 P\circ S_2^{-1}+\frac 13 u$, where $ u=\frac 12 u\circ T_1^{-1} +\frac 12 u\circ T_2^{-1}$. Then, $P$ is a condensation measure associated with the self-similar measure $ u$. For such a measure $P$ we determine the optimal sets of $n$-means and the $n$th quantization errors for all $n\geq 2$. We have also shown that the quantization dimension of the condensation measure $P$ exists and equals the quantization dimension $D( u)$ of the self-similar measure $ u$, but the $D( u)$-dimensional quantization coefficient for the measure $P$ does not exist, and the $D( u)$-dimensional lower and upper quantization coefficients for $P$ are finite and positive.