Wavelet theory, coorbit spaces and ramifications | Vidéo | Carmin.tv

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Wavelet theory, coorbit spaces and ramifications

By Hans G. Feichtinger

Appears in collections : Special events, 30 Years of Wavelets, 30 years of wavelets / 30 ans des ondelettes, Actions thématiques

Coorbit theory was developed in the late eighties as a unifying principle covering (possible non-)orthogonal frame expansions in the wavelet and in the time-frequency context. Very much in the spirit of « coherent frames » or also reproducing kernels for a Moebius invariant Banach space of analytic functions one can describe a family of function spaces associated with a given integrable and irreducible group representation on a Hilbert space by its generalized wavelet transform, and obtain (among others) atomic decomposition results for the resulting spaces. The theory was flexible enough to cover also more recent examples, such as voice transforms related to the Blaschke group or the spaces (and frames) related to the shearlet transform. As time permits I will talk also on the role of Banach frames and the usefulness of Banach Gelfand triples, especially the one based on the Segal algebra $S_0(G)$, which happens to be a modulation space, in fact the minimal among all time-frequency invariant non-trivial function spaces.

Keywords: wavelet theory - time-frequency analysis - modulation spaces - Banach-Gelfand-triples - Toeplitz operators - atomic decompositions - function spaces - shearlet transform - Blaschke group

Information about the video

Date of recording 24/01/2015
Date of publication 16/03/2015
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.18718203
Cite this video Feichtinger, Hans G. (24/01/2015). Wavelet theory, coorbit spaces and ramifications. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.18718203
URL https://dx.doi.org/10.24350/CIRM.V.18718203

Domain(s)

Bibliography

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[2] Dahlke, S., Steidl, G., & Teschke, G. (2010) The continuous shearlet transform in arbitrary space dimensions. The Journal of Fourier Analysis and Applications, Vol.16(3), 340-364 - http://dx.doi.org/10.1007/s00041-009-9107-8
[3] Daubechies, I., Grossmann, A., & Meyer, Y. (1986). Painless nonorthogonal expansions. Journal of Mathematical Physics, 27(5), 1271-1283 - http://dx.doi.org/10.1063/1.527388
[4] Feichtinger, H.G. (1981). On a new Segal algebra. Monatshefte für Mathematik, 92(4), 269-289 - http://dx.doi.org/10.1007/bf01320058
[5] Feichtinger, H.G., & Gröchenig, K. (1989). Banach spaces related to integrable group representations and their atomic decompositions, I. Journal of Functional Analysis, 86(2), 307-340 - http://dx.doi.org/10.1016/0022-1236(89)90055-4
[6] Feichtinger, H.G. (2003). Modulation spaces of locally compact Abelian groups. In R. Radha, M. Krishna, & S. Thangavelu (Eds.), Proceedings of the International Conference on Wavelets and their Applications (pp. 1-56). New Delhi: Allied Publishers - http://www.univie.ac.at/nuhag-php/bibtex/open_files/120_ModICWA.pdf
[7] Kutyniok, G., & Labate, D. (2009). Resolution of the wavefront set using continuous shearlets. Transactions of the American Mathematical Society, 361(5), 2719-2754 - http://dx.doi.org/10.1090/S0002-9947-08-04700-4
[8] Pap, M. (2010). The voice transform generated by a representation of the Blaschke group on the weighted Bergman spaces. Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Computatorica, 33, 321-342 - http://ac.inf.elte.hu/Vol_033_2010/321.pdf

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