Contributions to the theory of boundedness in uniform spaces and topological groups

First, we discuss the behavior of boundedness in uniform spaces with respect to subspaces, projective limits, and suprema in relation to precompactness. A special uniformly isomorphic embedding of un arbitrary uniform space in a bounded uniform space is presented and examined in 2.6. Hejcman’s characterization (by B-conservativity) of uniform spaces in which boundedness can be tested by a single pseudometric is proved in a new way, see 3.13, using a version 3.1 of the metrization lemma. We comment briefly on boundedness in topological vector spaces. In topological groups we investigate a hierarchy of partly new notions of boundedness, strongly interrelated among themselves, and exhibit various situations in which certain of these notions coincide. "Boundedness respecting subspaces" of a uniform space prove useful. Many examples illustrate and complement the general theory, see, e.g., Example 6.4

Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings

The class of generalized shearlet dilation groups has recently been developed to allow the unified treatment of various shearlet groups and associated shearlet transforms that had previously been studied on a case-by-case basis. We consider several aspects of these groups: First, their systematic construction from associative algebras, secondly, their suitability for the characterization of wavefront sets, and finally, the question of constructing embeddings into the symplectic group in a way that intertwines the quasi-regular representation with the metaplectic one. For all questions, it is possible to treat the full class of generalized shearlet groups in a comprehensive and unified way, thus generalizing known results to an infinity of new cases. Our presentation emphasizes the interplay between the algebraic structure underlying the construction of the shearlet dilation groups, the geometric properties of the dual action, and the analytic properties of the associated shearlet transforms.Comment: 28 page

Die Bedeutung von Verhaltensannahmen in der wissenschaftlichen Beratung: am Beispiel der Förderschwerpunkte :[riw] und Ina und im Hinblick auf Ausschreibungen im BMBF-Rahmenprogramm "Forschung für Nachhaltigkeit" (FONA)

"Forschungsvorhaben im Rahmen von FONA zielen darauf ab, Empfehlungen für das Verhalten von Akteuren oder für die Gestaltung von Rahmenbedingungen zu entwickeln. Dabei geht jeder Forschungsantrag – zumindest implizit – von Annahmen darüber aus, welche Faktoren das Verhalten der Akteure bestimmen, die Gegenstand des jeweiligen Forschungsvorhabens sind. Welche Verhaltensannahmen dies sind, ist für das Forschungsergebnis und daraus resultierende Handlungsempfehlungen oftmals von ausschlaggebender Bedeutung, wie sich etwa anhand der Studien in den Förderschwerpunkten Ina und :[riw] deutlich machen lässt. Voneinander abweichende Gestaltungsempfehlungen in verschiedenen Gutachten haben nicht selten ihre Ursache in divergierenden Verhaltensannahmen. Die 'Abnehmer' der Forschungsvorhaben – und dies gilt für private Akteure (etwa in Unternehmen) ebenso wie für die öffentliche Hand – stehen vor dem Problem, die erzielten Ergebnisse einzuordnen. Dafür ist es notwendig, die Verhaltensannahmen nachvollziehen zu können, die dem Projekt zugrunde lagen. Dies zu ermöglichen, ist eine Bringschuld der Wissenschaftler. Indem sie Verhaltensannahmen nachvollziehbar machen, leisten sie einen Beitrag zur wissenschaftlichen Qualitätssicherung und erhöhen damit zugleich die Prognosefähigkeit ihrer Aussagen. Insgesamt steigt damit die Überzeugungskraft wissenschaftlicher Aussagen." (Autorenreferat

Pruning Algorithms for Pretropisms of Newton Polytopes

Pretropisms are candidates for the leading exponents of Puiseux series that represent solutions of polynomial systems. To find pretropisms, we propose an exact gift wrapping algorithm to prune the tree of edges of a tuple of Newton polytopes. We prefer exact arithmetic not only because of the exact input and the degrees of the output, but because of the often unpredictable growth of the coordinates in the face normals, even for polytopes in generic position. We provide experimental results with our preliminary implementation in Sage that compare favorably with the pruning method that relies only on cone intersections.Comment: exact, gift wrapping, Newton polytope, pretropism, tree pruning, accepted for presentation at Computer Algebra in Scientific Computing, CASC 201

Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

Using coherent-state techniques, we prove a sampling theorem for Majorana's (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of $N$ samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of over- and under-sampling. Sample points are roots of unity, a fact which allows explicit inversion formulas for resolution and overlapping kernel operators through the theory of Circulant Matrices and Rectangular Fourier Matrices. The case of band-limited functions on the Riemann sphere, with spins up to $J$, is also considered. The connection with the standard Euler angle picture, in terms of spherical harmonics, is established through a discrete Bargmann transform.Comment: 26 latex pages. Final version published in J. Fourier Anal. App

Splines and Wavelets on Geophysically Relevant Manifolds

Analysis on the unit sphere $\mathbb{S}^{2}$ found many applications in seismology, weather prediction, astrophysics, signal analysis, crystallography, computer vision, computerized tomography, neuroscience, and statistics. In the last two decades, the importance of these and other applications triggered the development of various tools such as splines and wavelet bases suitable for the unit spheres $\mathbb{S}^{2}$, $\>\>\mathbb{S}^{3}$ and the rotation group $SO(3)$. Present paper is a summary of some of results of the author and his collaborators on generalized (average) variational splines and localized frames (wavelets) on compact Riemannian manifolds. The results are illustrated by applications to Radon-type transforms on $\mathbb{S}^{d}$ and $SO(3)$.Comment: The final publication is available at http://www.springerlink.co

Operator-Valued Continuous Gabor Transforms over Non-unimodular Locally Compact Groups

In this article, we present the abstract harmonic analysis aspects of the operator-valued continuous Gabor transform (CGT) on second countable, non-unimodular, and type I locally compact groups. We show that the operator-valued continuous Gabor transform CGT satisfies a Plancherel formula and an inversion formula. As an example, we study these results on the continuous affine group

Reproducing subgroups of Sp(2,R)Sp(2,\mathbb{R}). Part I: algebraic classification

We classify the connected Lie subgroups of the symplectic group $Sp(2,\mathbb{R})$ whose elements are matrices in block lower triangular form. The classification is up to conjugation within $Sp(2,\mathbb{R})$. Their study is motivated by the need of a unified approach to continuous 2D signal analyses, as those provided by wavelets and shearlets.Comment: 26 page

The transmission problem on a three-dimensional wedge

We consider the transmission problem for the Laplace equation on an infinite three-dimensional wedge, determining the complex parameters for which the problem is well-posed, and characterizing the infinite multiplicity nature of the spectrum. This is carried out in two formulations leading to rather different spectral pictures. One formulation is in terms of square integrable boundary data, the other is in terms of finite energy solutions. We use the layer potential method, which requires the harmonic analysis of a non-commutative non-unimodular group associated with the wedge