Optimal Quantization for Some Triadic Uniform Cantor Distributions with Exact Bounds

Abstract

Let {Sj:1≤j≤3} be a set of three contractive similarity mappings such that Sj(x)=rx+j−12(1−r) for all x∈R, and 1≤j≤3, where 0 Let {Sj:1≤j≤3} \u3e{Sj:1≤j≤3}{Sj:1≤j≤3} be a set of three contractive similarity mappings such that Sj(x)=rx+j−12(1−r) \u3eSj(x)=rx+j−12(1−r)Sj(x)=rx+j−12(1−r) for all x∈R \u3ex∈Rx∈R, and 1≤j≤3 \u3e1≤j≤31≤j≤3, where 0P has support the Cantor set generated by the similarity mappings Sj \u3eSjSj for 1≤j≤3 \u3e1≤j≤31≤j≤3. Let r0=0.1622776602 \u3er0=0.1622776602r0=0.1622776602, and r1=0.2317626315 \u3er1=0.2317626315r1=0.2317626315 (which are ten digit rational approximations of two real numbers). In this paper, for 00n-means and the nth quantization errors for the triadic uniform Cantor distribution P for all positive integers n≥2 \u3en≥2n≥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=15 \u3er=15r=15. In this paper, we further show that r=r0 \u3er=r0r=r0 is the greatest lower bound, and r=r1 \u3er=r1r=r1 is the least upper bound of the range of r-values to which Roychowdhury formula extends. In addition, we show that for 0

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