Quantization for a Condensation System

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 D̲r(μ) and D¯r(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}j=1N, (pj)j=0N,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.