3,166 research outputs found
Function Spaces on Singular Manifolds
It is shown that most of the well-known basic results for Sobolev-Slobodeckii
and Bessel potential spaces, known to hold on bounded smooth domains in
, continue to be valid on a wide class of Riemannian manifolds
with singularities and boundary, provided suitable weights, which reflect the
nature of the singularities, are introduced. These results are of importance
for the study of partial differential equations on piece-wise smooth domains.Comment: 37 pages, 1 figure, final version, augmented by additional
references; to appear in Math. Nachrichte
Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type
Coorbit space theory is an abstract approach to function spaces and their
atomic decompositions. The original theory developed by Feichtinger and
Gr{\"o}chenig in the late 1980ies heavily uses integrable representations of
locally compact groups. Their theory covers, in particular, homogeneous
Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces, and the
recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces
cannot be covered by their group theoretical approach. Later it was recognized
by Fornasier and the first named author that one may replace coherent states
related to the group representation by more general abstract continuous frames.
In the first part of the present paper we significantly extend this abstract
generalized coorbit space theory to treat a wider variety of coorbit spaces. A
unified approach towards atomic decompositions and Banach frames with new
results for general coorbit spaces is presented. In the second part we apply
the abstract setting to a specific framework and study coorbits of what we call
Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel
spaces of various types of interest as coorbits. We obtain several old and new
wavelet characterizations based on precise smoothness, decay, and vanishing
moment assumptions of the respective wavelet. As main examples we obtain
results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal
spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed
smoothness, and even mixtures of the mentioned ones. Due to the generality of
our approach, there are many more examples of interest where the abstract
coorbit space theory is applicable.Comment: to appear in Journal of Functional Analysi
Collective classification for labeling of places and objects in 2D and 3D range data
In this paper, we present an algorithm to identify types of places and objects from 2D and 3D laser range data obtained in indoor environments. Our approach is a combination of a collective classification method based on associative Markov networks together with an instance-based feature extraction using nearest neighbor. Additionally, we show how to select the best features needed to represent the objects and places, reducing the time needed for the learning and inference steps while maintaining high classification rates. Experimental results in real data demonstrate the effectiveness of our approach in indoor environments
Homogeneous Equations of Algebraic Petri Nets
Algebraic Petri nets are a formalism for modeling distributed systems and algorithms, describing control and data flow by combining Petri nets and algebraic specification. One way to specify correctness of an algebraic Petri net model "N" is to specify a linear equation "E" over the places of "N" based on term substitution, and coefficients from an abelian group "G". Then, "E" is valid in "N" iff "E" is valid in each reachable marking of "N". Due to the expressive power of Algebraic Petri nets, validity is generally undecidable. Stable linear equations form a class of linear equations for which validity is decidable. Place invariants yield a well-understood but incomplete characterization of all stable linear equations. In this paper, we provide a complete characterization of stability for the subclass of homogeneous linear equations, by restricting ourselves to the interpretation of terms over the Herbrand structure without considering further equality axioms. Based thereon, we show that stability is decidable for homogeneous linear equations if "G" is a cyclic group
A note on Hausdorff-Young inequalities in function spaces
The classical Hausdorff-Young inequalities for the Fourier transform acting
between appropriate spaces are cornerstones of Fourier analysis. Here we
extend it to weighted spaces of Besov or Sobolev type where the weight has the
form . This note is not a paper or draft but a
sketchy complement to some earlier results where we dealt with mapping
properties of the Fourier transform
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