Quantization for a Condensation System

Authors

Shivam Dubey
Mrinal Kanti Roychowdhury, The University of Texas Rio Grande ValleyFollow
Saurabh Verma

Document Type

Article

Publication Date

4-26-2025

Abstract

For a given r∈(0,+∞), the quantization dimension of order r, if it exists, denoted by Dr(μ), represents the rate at which the nth quantization error of order r approaches zero as the number of elements n in an optimal set of n-means for μ tends to infinity. If Dr(μ) does not exist, we define D−−r(μ) and ¯¯¯Dr(μ) as the lower and the upper quantization dimensions of μ of order r, respectively. In this paper, we investigate the quantization dimension of the condensation measure μ associated with a condensation system ({Sj}Nj=1, (pj)Nj=0,ν). We provide two examples: one where ν is an infinite discrete distribution on R, and one where ν is a uniform distribution on R. For both the discrete and uniform distributions ν, we determine the optimal sets of n-means, calculate the quantization dimensions of condensation measures μ, and show that the Dr(μ)-dimensional quantization coefficients do not exist. Moreover, we demonstrate that the lower and upper quantization coefficients are finite and positive.

Comments

© 2025 by the authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Recommended Citation

Dubey, Shivam, Mrinal Kanti Roychowdhury, and Saurabh Verma. 2025. "Quantization for a Condensation System" Mathematics 13, no. 9: 1424. https://doi.org/10.3390/math13091424

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Publication Title

Mathematics

DOI

10.3390/math13091424