Canonical sequences of optimal quantization for condensation measures

Abstract

Let $P:=\frac 1 3 P\circ S_1^{-1}+\frac 13 P\circ S_2^{-1}+\frac 13\nu$, where $S_1(x)=\frac 15 x$, $S_2(x)=\frac 1 5 x+\frac 45$ for all $x\in \mathbb R$, and $\nu$ be a Borel probability measure on $\mathbb R$ with compact support. Such a measure $P$ is called a condensation measure, or an an inhomogeneous self-similar measure, associated with the condensation system $(\{S_1, S_2\}, (\frac 13, \frac 13, \frac 13), \nu)$. Let $D(\mu)$ denote the quantization dimension of a measure $\mu$ if it exists. Let $\kappa$ be the unique number such that $(\frac 13 (\frac 15)^2)^{\frac {\kappa}{2+\kappa}}+(\frac 13 (\frac 15)^2)^{\frac {\kappa}{2+\kappa}}=1$. In this paper, we have considered four different self-similar measures $\nu:=\nu_1, \nu_2, \nu_3, \nu_4$ satisfying $D(\nu_1)>\kappa$, $D(\nu_2)\kappa$, and $D(\nu_4)=\kappa$. For each measure $\nu$ we show that there exist two sequences $a(n)$ and $F(n)$, which we call as canonical sequences. With the help of the canonical sequences, we obtain a closed formula to determine the optimal sets of $F(n)$-means and $F(n)$th quantization errors for the condensation measure $P$ for each $\nu$. Then, we show that for each measure $\nu$ the quantization dimension $D(P)$ of the condensation measure $P$ exists, and satisfies: $D(P)=\max\{\kappa, D(\nu)\}$. Moreover, we show that for $D(\nu_1)>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients are finite, positive and unequal; on the other hand, for $\nu=\nu_2, \nu_3, \nu_4$, the $D(P)$-dimensional lower quantization coefficient is infinity. This shows that for $D(\nu)>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients can be either finite, positive and unequal, or it can be infinity