Canonical sequences of optimal quantization for condensation measures

Abstract

We consider condensation measures of the form $P:=\frac{1}{3}P\circ S_1^{-1}+\frac{1}{3}P\circ S_2^{-1}+\frac{1}{3}\nu$ associated with the system $(\mathcal{S},(\frac{1}{3},\frac{1}{3},\frac{1}{3}),\nu ),$ where $\mathcal{S}=\{S_i{\}}_{i=1}^2$ are contractions and $\nu$ is a Borel probability measure on $\mathbb{R}$ with compact support. Let $D(\mu )$ denote the quantization dimension of a measure $\mu$ if it exists. In this paper, we study self-similar measures $\nu$ satisfying $D(\nu )>\kappa$, $D(\nu )<\kappa$, and $D(\nu )=\kappa ,$ respectively, where $\kappa$ is the unique number satisfying $[\frac{1}{3}(\frac{1}{5}{)}^2{]}^{\frac{\kappa }{2+\kappa }}=\frac{1}{2}.$ For each case we construct two sequences $a(n)$ and $F(n)$, which are utilized in determining the optimal sets of $F(n)$-means and the $F(n)$th quantization errors for $P.$ We also show that for each measure $\nu$ the quantization dimension $D(P)$ of $P$ exists and satisfies $D(P)=\max \{\kappa ,D(\nu )\}.$ Moreover, we show that for $D(\nu )>\kappa$, the $D(P)$-dimensional lower and upper quantization coefficients are finite, positive and unequal; and for $D(\nu )\le \kappa$, the $D(P)$-dimensional lower quantization coefficient is infinity.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1610.07490