Contributed Talk

Kernel-based frames in Clifford analysis

Swanhild Bernstein
TU Bergakademie Freiberg, Germany


Wavelets 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.

[1] Bernstein, S., A Lie group approach to diffusive wavelets, EURASIP conf. proc., 301-304, (2013).
[2] Bernstein, S. and Ebert, S. Kernel based Wavelets on S3, J. Concr. Appl. Mathematics, 8(1), 110-124, (2010).
[3] Gröchenig, K.-H.,foundations of Time-Frequency Analysis, Birkhäuser, Boston, (2001).
[4] Freeden, W., Gervens, T. and Schreiner, M., Constructive Approximation on the Sphere, Oxford University Press Inc., Oxford, (1998).

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