Ursula Molter

Self-similarity and multiwavelets in higher dimensions

Introduction Matrices, tiles, and the joint spectral radius Generalized self-similarity and the refinement equation Multiresolution analysis Examples Bibliography Appendix A. Index of symbols.

Dimension functions of Cantor sets

We estimate the packing measure of Cantor sets associated to non-increasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the dimension functions recently found by Cabrelli et al for these sets.

Sums of Cantor sets

We find conditions on the ratios of dissection of a Cantor set so that the group it generates under addition has positive Lebesgue measure. In particular, we answer affirmatively a special case of a conjecture posed by J. Palis.

Accuracy of Several Multidimensional Refinable Distributions

Compactly supported distributions f1,..., fr on ℝd are fefinable if each fi is a finite linear combination of the rescaled and translated distributions fj(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σk∈Λ ck f(Ax−k), where Λ is a finite subset of Γ, the ck are r×r matrices, and f=(f1,...,fr)T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)<p are exactly reproduced from linear combinations of translates of f1,...,fr along the lattice Γ. We determine the accuracy p from the matrices ck. Moreover, we determine explicitly the coefficients yα,i(k) such that xα=Σi=1rΣk∈Γyα,i(k) fi(x+k). These coefficients are multivariate polynomials yα,i(x) of degree |α| evaluated at lattice points k∈Γ.

CLASSIFYING CANTOR SETS BY THEIR FRACTAL DIMENSIONS

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovich and Taylor. We classify these Cantor sets in terms of their -Hausdorff and -packing measures, for the family of dimension functions , and characterize this classification in terms of the underlying sequences.

Sums of Cantor sets yielding an interval

In this paper we prove that if a Cantor set has ratios of dissection bounded away from zero, then there is a natural number N, such that its N-fold sum is an interval. Moreover, for each element z of this interval, we explicitly construct the N elements of C whose sum yields z. We also extend a result of Mendes and Oliveira showing that when s is irrational Ca + Ca* is an interval if and only ifa/(l— 2a)a /(\ — 2a) > 1. 2000 Mathematics subject classification: primary 28A80, 26A30.

A constructive algorithm to solve "convex recursive deletion" (CoRD) classification problems via two-layer perceptron networks

A sufficient condition that a region be classifiable by a two-layer feedforward neural net (a two-layer perceptron) using threshold activation functions is that either it be a convex polytope or that intersected with the complement of a convex polytope in its interior, or that intersected with the complement of a convex polytope in its interior or ... recursively. These have been called convex recursive deletion (CoRD) regions.We give a simple algorithm for finding the weights and thresholds in both layers for a feedforward net that implements such a region. The results of this work help in understanding the relationship between the decision region of a perceptron and its corresponding geometry in input space. Our construction extends in a simple way to the case that the decision region is the disjoint union of CoRD regions (requiring three layers). Therefore this work also helps in understanding how many neurons are needed in the second layer of a general three-layer network. In the event that the decision region of a network is known and is the union of CoRD regions, our results enable the calculation of the weights and thresholds of the implementing network directly and rapidly without the need for thousands of backpropagation iterations.

PERTURBATION TECHNIQUES IN IRREGULAR SPLINE-TYPE SPACES

In this paper we study various perturbation techniques in the context of irregular spline-type spaces. We first present the sampling problem in this general setting and prove a general result on the possibility of perturbing sampling sets. This result can be regarded as a spline-type space analogue in the spirit of Kadecs Theorem for bandlimited functions (see Refs. 14 and 15). We further derive some quantitative estimates on the amount by which a sampling set can be perturbed, and finally prove a result on the existence of optimal perturbations (with the stability of reconstruction being the optimality criterion). Finally, the techniques developed in the earlier parts of the paper are used to study the problem of disturbing a basis for a spline-type space, in order to derive a sufficient criterion for a space generated by irregular translations to be a spline-type space.

A linear time algorithm for a matching problem on the circle

Given two ordered sets of II points on the line A = (al,. . . ,a,}, B = {bl, , b,}, a matching between them is a bijection from A onto B. Each matching is characterized by a permutation o of { 1, . . . , n} (i.e., to each ai there is a corresponding b,(i)). The cost of the matching pair (ai, b,(i)) is the distance between the points of the pair. The cost of the matching, is the sum of the cost of each pair in the matching. The minimal matching problem is to find a matching which minimizes the cost. It is known that when the sets are on the line, a minimal matching is given by the identity permutation, that is (al, bl), (~72, b2), , (a,, b,) [4,11]. Actually, there could be many minimal matchings, for example, if the point sets are separated. Let us now consider the same problem in the circle. The points here are ordered by its angle, and the distance between two points is the angle between them. This problem was studied in [4,11,7,2]. In [lo] it is shown that when the sets are not necessarily ordered, to find the minimal cost matching requires at least

Local bases for refinable spaces

We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular, these functions include all the homogeneous polynomials that are reproducible by the generator, which links this representation to the accuracy of the space. We completely characterize the class of homogeneous functions in the space and show that they can reproduce the generator. As a result we conclude that the homogeneous functions can be constructed from the vectors associated to the spectrum of the scale matrix (a finite square matrix with entries from the mask of the generator). Furthermore, we prove that the kernel of the transition operator has the same dimension as the kernel of this finite matrix. This relation provides an easy test for the linear independence of the integer translates of the generator. This could be potentially useful in applications to approximation theory, wavelet theory and sampling.