A note on the quantization error for in-homogeneous self-similar measures

TL;DR

This paper investigates the asymptotic behavior of quantization errors for in-homogeneous self-similar measures, providing new conditions for finiteness and positivity of quantization coefficients and estimating convergence rates.

Contribution

It introduces a new sufficient condition for the finiteness of the upper quantization coefficient and establishes necessary and sufficient conditions for the coefficients to be positive and finite.

Findings

Derived a new sufficient condition for finite upper quantization coefficient.

Established necessary and sufficient conditions for positive and finite quantization coefficients.

Estimated the convergence order of quantization error when the coefficient is infinite.

Abstract

We further study the asymptotics of quantization errors for two classes of in-homogeneous self-similar measures $\mu$. We give a new sufficient condition for the upper quantization coefficient for $\mu$ to be finite. This, together with our previous work, leads to a necessary and sufficient condition for the upper and lower quantization coefficient of $\mu$ to be both positive and finite. Furthermore, we determine (estimate) the convergence order of the quantization error in case that the quantization coefficient is infinite.