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.