# Optimal quantization for the Cantor distribution generated by infinite similutudes

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## Abstract

Let *P* be a Borel probability measure on ℝ generated by an infinite system of similarity mappings {*S*_{j} : *j* ∈ ℕ} such that \(P=\Sigma_{j=1}^{\infty}\frac{1}{2^{j}}P\circ{S}_j^{-1}\), where for each *j* ∈ ℕ and *x* ∈ ℝ, \(S_j(x)=\frac{1}{3^j}x+1-\frac{1}{3^{j-1}}\). Then, the support of *P* is the dyadic Cantor set *C* generated by the similarity mappings *f*_{1}, *f*_{2} : ℝ → ℝ such that *f*_{1}(*x*) = 1/3*x* and *f*_{2}(*x*) = 1/3*x*+ 2/3 for all *x* ∈ ℝ. In this paper, using the infinite system of similarity mappings {*S*_{j} : *j* ∈ ℕ} associated with the probability vector \((\frac{1}{2},\frac{1}{{{2^2}}},...)\), for all *n* ∈ ℕ, we determine the optimal sets of *n*-means and the *n*th quantization errors for the infinite self-similar measure *P*. The technique obtained in this paper can be utilized to determine the optimal sets of *n*-means and the *n*th quantization errors for more general infinite self-similar measures.

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