Quantization for a Condensation System - Shivam Dubey; Mrinal Kanti Roychowdhury; Saurabh Verma (Mathematics)
For a given <semantics>r∈(0,+∞)</semantics>, the quantization dimension of order r, if it exists, denoted by <semantics>Dr(μ)</semantics>, 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 <semantics>μ</semantics> tends to infinity. If <semantics>Dr(μ)</semantics> does not exist, we define <semantics><munder>D̲</munder>r(μ)</semantics> and <semantics><mover>D¯</mover>r(μ)</semantics> as the lower and the upper quantization dimensions of <semantics>μ</semantics> of order r, respectively. In this paper, we investigate the quantization dimension of the condensation measure <semantics>μ</semantics> associated with a condensation system <semantics>(<msubsup>Sjj=1N</msubsup>, <msubsup>(pj)j=0N</msubsup>,ν).</semantics> We provide two examples: one where <semantics>ν</semantics> is an infinite discrete distribution on <semantics>R</semantics>, and one where <semantics>ν</semantics> is a uniform distribution on <semantics>R</semantics>. For both the discrete and uniform distributions <semantics>ν</semantics>, we determine the optimal sets of n-means, calculate the quantization dimensions of condensation measures <semantics>μ</semantics>, and show that the <semantics>Dr(μ)</semantics>-dimensional quantization coefficients do not exist. Moreover, we demonstrate that the lower and upper quantization coefficients are finite and positive.