Quantization for a Condensation System

For a given r ∈ ( 0 , + ∞ ) , the quantization dimension of order r , if it exists, denoted by D r ( μ ) , represents the rate at which the n th 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 D r ( μ ) does not exist, we define D ̲ r ( 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 ( { S j } j = 1 N , ( p j ) j = 0 N , 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 D r ( μ ) -dimensional quantization coefficients do not exist. Moreover, we demonstrate that the lower and upper quantization coefficients are finite and positive.