Optimal Quantization for Some Triadic Uniform Cantor Distributions with Exact Bounds

Abstract

Let $$\{S_j: 1\le j\le 3\}$$ be a set of three contractive similarity mappings such that $$S_j(x)=rx+\frac{j-1}{2}(1-r)$$ for all $$x\in {\mathbb {R}}$$ , and $$1\le j\le 3$$ , where $$0<r<\frac{1}{3}$$ . Let $$P=\sum _{j=1}^3 \frac{1}{3} P\circ S_j^{-1}$$ . Then, P is a unique Borel probability measure on $${\mathbb {R}}$$ such that P has support the Cantor set generated by the similarity mappings $$S_j$$ for $$1\le j\le 3$$ . Let $$r_0=0.1622776602$$ , and $$r_1=0.2317626315$$ (which are ten digit rational approximations of two real numbers). In this paper, for $$0<r\le r_0$$ , we give a general formula to determine the optimal sets of n-means and the nth quantization errors for the triadic uniform Cantor distribution P for all positive integers $$n\ge 2$$ . Previously, Roychowdhury gave an exact formula to determine the optimal sets of n-means and the nth quantization errors for the standard triadic Cantor distribution, i.e., when $$r=\frac{1}{5}$$ . In this paper, we further show that $$r=r_0$$ is the greatest lower bound, and $$r=r_1$$ is the least upper bound of the range of r-values to which Roychowdhury formula extends. In addition, we show that for $$0<r\le r_1$$ the quantization coefficient does not exist though the quantization dimension exists.