Asymptotics of the quantization errors for condensation measures

TL;DR

This paper explicitly calculates the optimal quantizers, quantization dimension, and coefficients for a specific class of inhomogeneous self-similar measures called condensation measures, advancing understanding of their asymptotic quantization errors.

Contribution

It provides explicit formulas and analysis for quantization properties of condensation measures, a class of inhomogeneous self-similar measures, which was previously less understood.

Findings

Explicit optimal quantizers derived for the measure.

Quantization dimension calculated.

Lower and upper quantization coefficients determined.

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 $(\{S_1, S_2\}, (\frac 13, \frac 13, \frac 13), \gn)$. In this paper, we have explicitly calculated optimal quantizers, quantization dimension, and the lower and upper quantization coefficients for an inhomogeneous self-similar measure.