Some Recent Developments in Quantization of Fractal Measures

Abstract

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.

Best finite constrained approximations of one-dimensional probabilities

“…Lebesgue measure). Results in a similar spirit have been established for important classes of singular measures, notably selfsimilar and -conformal probabilities [18,20,30]. While these classical results crucially employ Voronoi partitions (as developed in some detail, e.g., in [17]), alternative tools and extensions to other metrics have recently been studied also [5,7,10].…”

“…For instance, Proposition 5.27 implies that D r (µ) = 1 whenever µ a = 0. The relations of D r (µ) to various other concepts of dimension have attracted considerable attention [17,20,29,32]. From this, it is clear that D r (µ) = log 2 log 3 , which is independent of r and coincides with the Hausdorff dimension of supp µ.…”

“…If µ ∈ P s is singular then Proposition 5.27 only yields lim n→∞ nd r (δ •,n • , µ) = 0. The detailed analysis of d r (δ •,n • , µ) in this case is an active research area, for which already a substantial literature exists, notably for important classes of singular probabilities such as self-similar and -conformal measures; see, e.g., [17,18,20,29,30,32]. A key notion in this context is the so-called quantization dimension of µ ∈ P r of order r, defined as D r (µ) = lim n→∞ log n − log d r (δ •,n • , µ)…”

“…Since d r (δ •,n • , µ) ≤ d r (δ un • , µ) for every µ ∈ P r and n ∈ N, clearly lim n→∞ d r (δ •,n • , µ) = 0. The rate of convergence of d r (δ •,n • , µ) has been, and continues to be, studied extensively; see, e.g., [17,19,20,21,23,27] and the references therein. Arguably the simplest situation occurs if µ ∈ P r has a non-trivial absolutely continuous part and satisfies a mild moment condition.…”

On the Asymptotic Uniformity of the Quantization Error for Moran Measures on ℝ1

“…One of the major obstacles is that, for a given cylinder set A (see Definition 1.1), we are unable to estimate the number of the cylinder sets B, with A, B non-overlapping and E r (B) ≍ E r (A), whose ǫ-neighborhoods intersect that of A, no matter how small ǫ is. Hence, a significant direction of effort is to seek some conditions, under which the abovementioned numbers are bounded by some constant and then manage to apply the covering technique as descried in [17] by Kesseböhmer and Zhu. In the present paper, we will prove that, (1.2) holds for the doubling measures on Moran sets in R q . We will assume a version of the open set condition which allows cylinder sets to touch one another.…”