## Contributed Talk## Kernel-based frames in Clifford analysisSwanhild BernsteinTU Bergakademie Freiberg, Germany AbstractWavelets and frames are important tools in mathematical theory and applications.
Especially Gabor frames [3] are important due to their interplay of time shifts and
frequency shifts. These connections are based in the Fourier transform. Unfortunately, there
is no general Fourier transform for groups and homogeneous spaces that has similar properties
concerning time and frequency shifts. On the other hand time shifts can usually be described
by group actions and the frequency shifts will be realized by modulations which are given by a
certain convolution kernel. The considered kernels behave similar to the Fourier kernel.
Following the ideas in [4] we can build continuous frames in L2 using reproducing approximate
identities which replace the Fourier kernel in view of modulations, i.e. frequency shifts, and time
shifts realized by convolution with the reproducing approximate identity k: {T_g k_h : g in G; h in R+},
where G is a Lie goup or homogeneous space and T_g the translation induced by an group action
and R+ := {t in R; t > 0}: These ideas has been already used to construct wavelets [2] and [1].
Here, we will use them for frames to get discrete versions and specifically Gabor-type frames.
We will discretize the continuous frame to {T_alpha_g k_beta_h : alpha, beta in R+}
and fixed elements g in G and h in R+. We will give examples how to construct frames in L2 for
groups and homogeneous spaces using Clifford-valued kernels. ISSN 1611 - 4086 | © IKM 2015 |