Optimal Quantization for Some Triadic Uniform Cantor Distributions with Exact Bounds

Authors

Mrinal Kanti Roychowdhury, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

10-17-2022

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}">{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)">Sj(x)=rx+j−12(1−r)Sj(x)=rx+j−12(1−r) for all x∈R">x∈Rx∈R, and 1≤j≤3">1≤j≤31≤j≤3, where 0P has support the Cantor set generated by the similarity mappings Sj">SjSj for 1≤j≤3">1≤j≤31≤j≤3. Let r0=0.1622776602">r0=0.1622776602r0=0.1622776602, and r1=0.2317626315">r1=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">n≥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">r=15r=15. In this paper, we further show that r=r0">r=r0r=r0 is the greatest lower bound, and r=r1">r=r1r=r1 is the least upper bound of the range of r-values to which Roychowdhury formula extends. In addition, we show that for 00

Comments

Copyright © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG. Original published version available at https://doi.org/10.1007/s12346-022-00678-8

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Recommended Citation

Roychowdhury, M.K. Optimal Quantization for Some Triadic Uniform Cantor Distributions with Exact Bounds. Qual. Theory Dyn. Syst. 21, 149 (2022). https://doi.org/10.1007/s12346-022-00678-8

Publication Title

Qual. Theory Dyn. Syst.

DOI

10.1007/s12346-022-00678-8