Some recent developments in quantization of fractal measures

Abstract

We give an overview on the quantization problem for fractal measures, including some related results and methods which have been developed in the last decades. Based on the work of Graf and Luschgy, we propose a three-step procedure to estimate the quantization errors. We survey some recent progress, which makes use of this procedure, including the quantization for self-affine measures, Markov-type measures on graph-directed fractals, and product measures on multiscale Moran sets. Several open problems are mentioned.

13 pages

References in corpus (2)

• On the quantization for self-affine measures on Bedford-McMullen carpets
• Convergence order of the geometric mean errors for Markov-type measures

Cited by in corpus (12)

• On the quantization for self-affine measures on Bedford-McMullen carpets
• Quantization dimensions of compactly supported probability measures via Rényi dimensions
• Asymptotic order of the quantization errors for self-affine measures on Bedford-McMullen carpets
• Best finite constrained approximations of one-dimensional probabilities
• Asymptotic order of the geometric mean error for self-affine measures on Bedford-McMullen carpets
• On the convergence rate of the chaos game
• On the optimal Voronoi partitions for Ahlfors-David measures with respect to the geometric mean error
• Quantization dimensions of negative order
• On the asymptotic quantization error for the doubling measures on Moran sets
• Asymptotics of the quantization problem on metric measure spaces
• Convergence order of the quantization error for self-affine measures on Lalley-Gatzouras carpets
• Asymptotic uniformity of the quantization error for Moran measures on ${\mathbb{R}}^1$