Resumen de Canonical Sequences of Optimal Quantization for Condensation Measures

Dogan Çömez, Mrinal Kanti Roychowdhury

• We consider condensation measures of the form P := 1 3 P ◦S−1 1 + 1 3 P ◦S−1 2 + 1 3 ν associated with the system (S, ( 1 3 , 1 3 , 1 3 ), ν), where S = {Si}2 i=1 are contractions and ν is a Borel probability measure on R with compact support. Let D(μ) denote the quantization dimension of a measure μ if it exists. In this paper, we study self-similar measures ν satisfying D(ν) > κ, D(ν) < κ, and D(ν) = κ, respectively, where κ is the unique number satisfying [ 1 3 ( 1 5 )2] κ 2+κ = 1 2 . For each case we construct two sequences a(n) and F(n), which are utilized in determining the optimal sets of F(n)-means and the F(n)th quantization errors for P. We also show that for each measure ν the quantization dimension D(P) of P exists and satisfies D(P) = max{κ, D(ν)}. Moreover, we show that for D(ν) > κ, the D(P)-dimensional lower and upper quantization coefficients are finite, positive and unequal; and for D(ν) ≤ κ, the D(P)-dimensional lower quantization coefficient is infinity.