Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Sören Häuser
Technische Universität, Kaiserslautern
Shearlet coorbit spaces: traces and embeddings
In recent years it has turned out that shearlets have the potential to retrieve directional information so
that they became interesting for many applications. Moreover the shearlet transform has the outstanding
property to stem from a square integrable group representation. This remarkable fact provides the opportunity
to design associated canonical smoothness spaces, so-called shearlet coorbit spaces by applying
the general coorbit theory derived by Feichtinger and Gröchenig. However, once these abstract smoothness
space are established some natural questions arise. Of course one would like to know how these
spaces look like and how they are related to other known classical smoothness spaces such as Besov
or Triebel-Lizorkin spaces. Moreover, one would like to understand the structure of these new spaces.
That is, it would be desirable to know how these new scales of shearlet coorbit spaces behave under
embeddings and trace operations.
In this talk we examine structural properties of shearlet coorbit spaces in higher dimensions. We prove
embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone in
three dimensions into Besov spaces. The results are based on general atomic decompositions of Besov
spaces. Furthermore, we establish trace results for these subspaces with respect to the coordinate
planes. It turns out that in many cases these traces are contained in lower dimensional shearlet coorbit
spaces. To prove these results we apply the concept of coorbit molecules recently developed by Gröchenig
and Piotrowski.
Joint work with S. Dahlke (Philipps-Universität Marburg), G. Steidl (Technische Universität Kaiserslautern)
and G. Teschke (Hochschule Neubrandenburg).
Lutz Kämmerer
Technische Universität Chemnitz
Stable interpolation of hyperbolic cross trigonometric polynomials
A straightforward discretisation of high-dimensional problems often leads to an exponential growth in the
number of degrees of freedom. So, computational costs of even efficient algorithms like the fast Fourier
transform increase similar.
Trigonometric polynomials with frequencies only supported by hyperbolic crosses allow for a good approximation
of functions of appropriate smoothness and decrease the number of used Fourier coefficients
strongly. As a matter of course, an important issue is the customisation of efficient algorithms to these
thinner discretisations.
Sparse grids are the natural discretisations in the spatial domain. The corresponding Fourier transform
suffers from stability problems. For that reason we consider sets produced by multiples of a so-called
generating vector as spatial discretisations. Therewith an easy and fast evaluation of trigonometric polynomials
at the grid nodes is guaranteed. Some additional assumptions ensure even stability and besides
the fast reconstruction of trigonometric polynomials from the function values. We discuss necessary and
sufficient conditions allowing for the stable reconstruction.
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