@unpublished{Kreitmeier2008, author = {Kreitmeier, Wolfgang}, title = {Optimal quantization for uniform distributions on Cantor-like sets}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-12449}, year = {2008}, abstract = {In this paper, the problem of optimal quantization is solved for uniform distributions on some higher dimensional, not necessarily self-similar \$N-\$adic Cantor-like sets. The optimal codebooks are determined and the optimal quantization error is calculated. The existence of the quantization dimension is characterized and it is shown that the quantization coefficient does not exist. The special case of self-similarity is also discussed. The conditions imposed are a separation property of the distribution and strict monotonicity of the first \$N\$ quantization error differences. Criteria for these conditions are proved and as special examples modified versions of classical fractal distributions are discussed.}, subject = {Maßtheorie}, language = {en} } @unpublished{Kreitmeier2005, author = {Kreitmeier, Wolfgang}, title = {Optimal Quantization for Dyadic Homogeneous Cantor Distributions}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-3845}, year = {2005}, abstract = {For a large class of dyadic homogeneous Cantor distributions in \mathbb{R}, which are not necessarily self-similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the non-existence of the quantization coefficient. The class contains all self-similar dyadic Cantor distributions, with contraction factor less than or equal to \frac{1}{3}. For these distributions we calculate the quantization errors explicitly.}, subject = {Maßtheorie}, language = {en} } @unpublished{Kreitmeier2007, author = {Kreitmeier, Wolfgang}, title = {Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-7374}, year = {2007}, abstract = {For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the quantization dimension of the uniform distribution on these Cantor sets. Moreover for a special sub-class of these sets we present a linkage between the Hausdorff and the Packing measure of these sets and the high-rate asymptotics of the quantization error.}, subject = {Maßtheorie}, language = {en} } @unpublished{Kreitmeier2007, author = {Kreitmeier, Wolfgang}, title = {Optimal quantization of probabilities concentrated on small balls}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-12010}, year = {2007}, abstract = {We consider probability distributions which are uniformly distributed on a disjoint union of balls with equal radius. For small enough radius the optimal quantization error is calculated explicitly in terms of the ball centroids. We apply the results to special self-similar measures.}, subject = {Maßtheorie}, language = {en} } @unpublished{Kreitmeier2009, author = {Kreitmeier, Wolfgang}, title = {Hausdorff measure of uniform self-similar fractals}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-17948}, year = {2009}, abstract = {Let d \&\#8805; 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on Rd. Denote by D the Hausdorff dimension, by HD(E) the Hausdorff measure and by diam (E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set \ω of fixed points of the UIFS does not depend on c, we will show the existence of a positive constant depending only on \ω, such that the Hausdorff dimension is smaller than one and HD = (E) D if c is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover we present a parametrised UIFS where \ω depends on c and HD < diam(E)D, if c is small enough.}, subject = {Maßtheorie}, language = {en} } @phdthesis{Kreitmeier2006, author = {Kreitmeier, Wolfgang}, title = {Optimale Quantisierung verallgemeinerter Cantor-Verteilungen}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-913}, school = {Universit{\"a}t Passau}, year = {2006}, abstract = {F{\"u}r verallgemeinerte Cantor-Verteilungen, die im Eindimensionalen mittels klassischer Wischkonstruktion bzw. in h{\"o}heren Dimensionen {\"u}ber iterierte Funktionensysteme definiert werden, wird das Problem der optimalen Quantisierung unter bestimmten Voraussetzungen vollst{\"a}ndig gel{\"o}st. Es werden die optimalen Codeb{\"u}cher bestimmt und Formeln f{\"u}r den optimalen Quantisierungsfehler bewiesen. Im eindimensionalen Fall wird eine Existenzcharakterisierung der Quantisierungsdimension gegeben und unter bestimmten Voraussetzungen die Nichtexistenz des Quantisierungskoeffizienten gezeigt. Auch in h{\"o}heren Dimensionen wird f{\"u}r die betrachteten Verteilungen bewiesen, dass der Quantisierungskoeffizient, bei existenter Quantisierungsdimension, nicht existiert. Die gewonnenen Resultate werden auf die Gleichverteilungen von modifizierten klassischen fraktalen Mengen, wie das Sierpinski-Dreieck, die Cantormenge und den Cantor-Staub angewandt.}, subject = {Maßtheorie}, language = {de} }