arXiv subjects

Hartmut Führ

Publications and source records attributed to Hartmut Führ.

41 records · Page 3Linked to original sources

arXiv2012

A simple and consistent definition of homogeneous Besov spaces on stratified Lie groups

We introduce a general definition of homogeneous Besov spaces on a stratified Lie group $G$, based on a Littlewood-Paley-type decomposition of Schwartz functions with all moments vanishing. We show that under mild and intuitive conditions the spaces thus defined are independent of the decomposition employed. A corollary of this is that previously constructed versions of homogeneous Besov spaces on $G$, relying on the spectral calculus of a sub-Laplacian of the group, are consistent, i.e., independent of the choice of sub-Laplacian. We further prove characterizations of homogeneous Besov spaces using continuous wavelet transforms, with a large variety of analysing wavelets to choose from.

math.FA↗

arXiv2012

Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov space $\dot{B}_{p,q}^s$ in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spaces ${\dot B}_{p,q}^{s}$, with $1 \le p,q < \infty$ and $s \in \mathbb{R}$.

math.FA↗

arXiv2011

Evolution Equations on Gabor Transforms and their Applications

We introduce a systematic approach to the design, implementation and analysis of left-invariant evolution schemes acting on Gabor transform, primarily for applications in signal and image analysis. Within this approach we relate operators on signals to operators on Gabor transforms. In order to obtain a translation and modulation invariant operator on the space of signals, the corresponding operator on the reproducing kernel space of Gabor transforms must be left invariant, i.e. it should commute with the left regular action of the reduced Heisenberg group H_r. By using the left-invariant vector fields on H_r in the generators of our evolution equations on Gabor transforms, we naturally employ the essential group structure on the domain of a Gabor transform. Here we distinguish between two tasks. Firstly, we consider non-linear adaptive left-invariant convection (reassignment) to sharpen Gabor transforms, while maintaining the original signal. Secondly, we consider signal enhancement via left-invariant diffusion on the corresponding Gabor transform. We provide numerical experiments and analytical evidence for our methods and we consider an explicit medical imaging application.

math.AP↗

arXiv2011

Characterizing abelian admissible groups

By definition, admissible matrix groups are those that give rise to a wavelet-type inversion formula. This paper investigates necessary and sufficient admissibility conditions for abelian matrix groups. We start out by deriving a block diagonalization result for commuting real valued matrices. We then reduce the question of deciding admissibility to the subclass of connected and simply connected groups, and derive a general admissibility criterion for exponential solvable matrix groups. For abelian matrix groups with real spectra, this yields an easily checked necessary and sufficient characterization of admissibility. As an application, we sketch a procedure how to check admissibility of a matrix group generated by finitely many commuting matrices with positive spectra.

math.FA↗

arXiv2009

Generalized Calderón conditions and regular orbit spaces

The construction of generalized continuous wavelet transforms on locally compact abelian groups $A$ from quasi-regular representations of a semidirect product group $G = A \rtimes H$ acting on ${\rm L}^2(A)$ requires the existence of a square-integrable function whose Plancherel transform satisfies Calderón-type resolution of the identity. The question then arises under what conditions such square-integrable functions exist. The existing literature on this subject leaves a gap between sufficient and necessary criteria. In this paper, we give a characterization in terms of the natural action of the dilation group $H$ on the character group of $A$. We first prove that a Calderón-type resolution of the identity gives rise to a decomposition of Plancherel measure of $A$ into measures on the dual orbits, and then show that the latter property is equivalent to regularity conditions on the orbit space of the dual action. Thus we obtain, for the first time, sharp necessary and sufficient criteria for the existence of a wavelet inversion formula. As a byproduct and special case of our results we obtain that discrete series subrepresentations of the quasiregular representation correspond precisely to dual orbits with positive Plancherel measure and associated compact stabilizers. Only sufficiency of the conditions was previously known.

math.FA↗