Optimal quantization for the Cantor distribution generated by infinite similutudes

Authors

Mrinal Kanti Roychowdhury, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

6-2019

Abstract

Let P be a Borel probability measure on ℝ generated by an infinite system of similarity mappings {Sj : j ∈ ℕ} such that P=Σ∞j=112jP∘S−1j, where for each j ∈ ℕ and x ∈ ℝ, Sj(x)=13jx+1−13j−1. Then, the support of P is the dyadic Cantor set C generated by the similarity mappings f1, f2 : ℝ → ℝ such that f1(x) = 1/3x and f2(x) = 1/3x+ 2/3 for all x ∈ ℝ. In this paper, using the infinite system of similarity mappings {Sj : j ∈ ℕ} associated with the probability vector (12,122,...), for all n ∈ ℕ, we determine the optimal sets of n-means and the nth quantization errors for the infinite self-similar measure P. The technique obtained in this paper can be utilized to determine the optimal sets of n-means and the nth quantization errors for more general infinite self-similar measures.

Comments

© 2019, Springer Nature. Original published version available at https://doi.org/10.1007/s11856-019-1859-5

Recommended Citation

Roychowdhury, M.K. Optimal quantization for the Cantor distribution generated by infinite similutudes. Isr. J. Math. 231, 437–466 (2019). https://doi.org/10.1007/s11856-019-1859-5

First Page

437

Last Page

466

Publication Title

Israel Journal of Mathematics

DOI

10.1007/s11856-019-1859-5