Frühere Forschungsschwerpunkte
- Construction of Wavelets
- Construction of Interpolating Scaling Functions
The aim of this project is the construction of smooth and localized
refinable functions and wavelets for general scaling matrices. The number
of wavelets and therefore the overall complexity depends on
the determinant of the scaling matrix. Therefore one is
interested in scaling matrices with small determinant. Moreover,
it has turned out that in many applications especially the use
of interpolating scaling functions provides some advantages.
There existed cooperations with
Karl-Heinz Gröchenig,
Peter Maaß
and
Gerd Teschke
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- Dual Systems for General Scalings
For interpolating scaling functions it is easy to derive
a corresponding wavelet basis. This basis can be viewed as
a variant of the well-known hierarchical basis. However,
for stability reasons, this is not always the best choice. To obtain
stable basis for function spaces such as Sobolev or Besov spaces
one needs moreover suitable dual systems.
We cooperated with
Peter Maaß and
Gerd Teschke
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- Wavelet Quadrature Formulas
Any application of wavelets in signal processing or in numerical analysis requires
the computation of integrals of the form
A naive approach using standard quadrature rules would not be successful
since the wavelet
is not necessarily very smooth and, moreover, it might not be known explicitly.
In this context, specific quadrature rules of Gauss type have turned out
to be very powerful.
This was a joint project with
Arne Barinka,
Titus Barsch and
Michael Konik.
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- Applications of Wavelet Methods to the Analysis of Radar Data
The basic radar problem asks to gain information about an object by analyzing
waves reflected from it. This task is relatively easy if
the object can be described as a single point. Additional problems
occur if one wants to observe a reflecting continuum with varying
reflectivity described by a reflectivity density. Then this reflectivity
density cannot be reconstructed by transmitting just one signal.
Therefore we suggest not to transmit just one signal but a family of signals.
Indeed, it can be shown that a complete reconstruction is
possible, provided that the family of outgoing signals forms a frame.
Within this project, we cooperate with
Peter Maaß and
Gerd Teschke
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