Resumen de Optimal Quantization for Some Triadic Uniform Cantor Distributions with Exact Bounds

Mrinal Kanti Roychowdhury

• Let {Sj : 1 ≤ j ≤ 3} be a set of three contractive similarity mappings such that Sj(x) = r x + j−1 2 (1 − r) for all x ∈ R, and 1 ≤ j ≤ 3, where 0 < r < 1 3 .

  Let P = 3 j=1 1 3 P ◦ S−1 j . Then, P is a unique Borel probability measure on R such that P has support the Cantor set generated by the similarity mappings Sj for 1 ≤ j ≤ 3. Let r0 = 0.1622776602, and r1 = 0.2317626315 (which are ten digit rational approximations of two real numbers). In this paper, for 0 < r ≤ r0, 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 ≥ 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 = 1 5 . In this paper, we further show thatr = r0 is the greatest lower bound, and r = r1 is the least upper bound of the range of r-values to which Roychowdhury formula extends. In addition, we show that for 0 < r ≤ r1 the quantization coefficient does not exist though the quantization dimension exists.