Lehrstuhl VIII Approximationstheorie
Permanent URI for this collection
Browse
Recent Submissions
Item Construction principles of tight wavelet frames in connection with linear system theory(2025) Pastoors, Nicolai; Stöckler, Joachim; Putinar, MihaiThis dissertation explores the connection between two mathematical fields: wavelet analysis, a subfield of time-frequency analysis, and linear system theory, which has its origins in the modeling of real-world systems using dynamical systems. In the former, we focus on the construction of multivariate tight wavelet frames via the Unitary and Oblique Extension Principles. In the latter, we are particularly interested in the state-space approach and its ties to the Nevanlinna-Pick interpolation and operator theory. In 2015, Scheiderer, Charina, Putinar, and Stöckler linked wavelet frames constructed via the Unitary Extension Principle to so-called realizations from linear system theory. This dissertation aims to generalize this connection to frames based on the Oblique Extension Principle, potentially aiding the construction and study of such frames. After brief introductions to the two fields mentioned above and to results on the factorization of trigonometric polynomials, we first consider the problem in the univariate case. Here, we can solve the problem both by bringing it back to the setting of the Unitary Extension Principle and by using a generalized version of the Nevanlinna-Pick interpolation. These univariate results lead to a multivariate result: Using the Kronecker product, we construct so-called separable multivariate frames and connect them to linear system theory for any dimension. Finally, in the bivariate case, we have to restrict ourselves to a specific subset of frames. For this subset, we use a parameterized version of the univariate results to solve the bivariate problem and again describe connections to the Nevanlinna-Pick interpolation. We illustrate all constructions with examples.Item An isogeometric mortar method for the coupling of multiple NURBS domains with optimal convergence rates(2021-11-13) Dornisch, Wolfgang; Stöckler, JoachimWe investigate the mortar finite element method for second order elliptic boundary value problems on domains which are decomposed into patches Ωk with tensor-product NURBS parameterizations. We follow the methodology of IsoGeometric Analysis (IGA) and choose discrete spaces Xh,k on each patch Ωk as tensor-product NURBS spaces of the same or higher degree as given by the parameterization. Our work is an extension of Brivadis et al. (Comput Methods Appl Mech Eng 284:292–319, 2015) and highlights several aspects which did not receive full attention before. In particular, by choosing appropriate spaces of polynomial splines as Lagrange multipliers, we obtain a uniform infsup-inequality. Moreover, we provide a new additional condition on the discrete spaces Xh,k which is required for obtaining optimal convergence rates of the mortar method. Our numerical examples demonstrate that the optimal rate is lost if this condition is neglected.Item Total positive Funktionen und exponentielle B-Splines in der Zeit-Frequenz-Analyse(2015) Kloos, Tobias; Stöckler, Joachim; Christensen, OleDie vorliegende Arbeit behandelt die Anwendung von Schoenbergs total positiven Funktionen, sowie exponentieller B-Splines in der Zeit-Frequenz-Analyse. Wir werden aufzeigen, dass sich diese Funktionen sehr gut als Fenster der Gabor-Transformation eignen und darüber hinaus anwendungsorientierte Algorithmen zur Implementierung angeben. Nach einer kurzen Einführung in die Thematik betrachten wir zunächst die Zak-Transformierten der genannten Funktionen und charakterisieren für eine Teilklasse der total positiven Funktionen ihre Nullstellenmengen. Dies liefert bereits Gabor-Frames mit ganzzahligem oversampling und gibt Hinweise über die Existenz im Fall von rationalem oversampling. Anschließend beschäftigen wir uns mit Gabor-Systemen auf beliebigen separablen Gittern und legen einige Situationen dar, in welchen die Systeme der betrachteten Funktionen einen Frame liefern. In diesen Fällen beschreiben wir Algorithmen zur Konstruktion unendlich vieler verschiedener Duale mit kompakten Trägern, welche gegen den kanonischen Dual konvergieren. Weiter geben wir einen kurzen Einblick in die sich ergebenden Möglichkeiten zur Bildung von Gabor-Frames über nicht-separablen Gittern. Abschließend erläutern wir, wie die gewonnenen Erkenntnisse genutzt werden können, um diskrete Gabor-Frames und deren Duale zu konstruieren.Item Total frame potential and its applications in data clustering(2013-12-06) Springer, Tobias; Stöckler, Joachim; Ickstadt, KatjaShort time series arise in a variety of fields such as biology or social sciences. For the statistical analysis of microarray gene expression data, the clustering of short time series is an important objective in order to identify subsets of genes sharing a temporal expression pattern. An established method, the Short Time Series Expression Miner (STEM) by Ernst et al. (2005), assigns time series data to the closest of suitably selected prototypes followed by the selection of significant clusters and eventual grouping. For the clustering of normalized d-dimensional data we propose to minimize the penalized frame potential by Springer et al. (2011). The functional contains the "Total Frame Potential" of Finite Unit Norm Tight Frames (FUNTFs), see Benedetto and Fickus (2003), and includes a data-driven component for the selection of prototypes. The idea of using the frame potential in combination with a data-dependent term for optimization was originally proposed by Benedetto, Czaja and Ehler (2010) for finding sparse representations. We show that the solution of the corresponding constrained optimization problem is naturally connected to the spherical Dirichlet cells of the given normalized data. Furthermore, the minimizers of the functional are located in the interior of the Dirichlet cells. The objective function is differentiable in the minimum and satisfies a matrix-valued extremal condition. The general problem is closely related to the search for point configurations on the unit sphere like in Tammes' (1930) or Thomson's Problem (1904). Moreover, the minimization contains connections to problems in matrix completion (see e.g. Candes and Tao (2009) or Mazumder, Hastie and Tibshirani (2010)). The work contains an exhaustive analysis of the proposed functional using methods from calculus, linear algebra and nonlinear programming. Numerical results from the application on real and artificial data are included.Item Nicht-negative Polynome und nicht-negative Funktionale(2012-09-19) Siemko, Katrin; Möller, Hans-Michael; Stöckler, JoachimItem Erzeugung hierarchischer Gitter mittels globaler Parametrisierung von NURBS-Flächenverbänden(2010-06-30) Scharfschwerdt, Michael; Stockler, Joachim; Turek, StefanItem Construction of Hilbert transform pairs of MRA tight frames and its application(2008-01-16T08:30:42Z) Lee, Kyoung-Yong; Stöckler, Joachim; Christensen, OleItem Beiträge zu globalen Fragen in der NURBS-Technik(2007-08-15T10:38:24Z) Selimovic, Ilijas; Stöckler, Joachim; Walter, RolfDie vorliegende Dissertation befasst sich mit den Eigenschaften von NURBS-Kurven und -Flächen und deren Anwendung in praktischen Algorithmen zur Lösung von geometrischen Problemen wie z.B. der Berechnung von Projektions- oder Schnittpunkten. Weiterhin ist eine differentialgeometrische Aussage enthalten, die Aussagen über das freie Rollen von Kugeln auf offenen Kurven erlaubt.