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H 1 Wavelet theory, coorbit spaces and ramifications

Auteurs : Feichtinger, Hans G. (Auteur de la Conférence)
CIRM (Editeur )

Résumé : 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

Codes MSC :
42C10 - Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42C15 - General harmonic expansions, frames
42C40 - Wavelets and other special systems
43-XX - Abstract harmonic analysis [For other analysis on topological and Lie groups, see 22Exx]
46Exx - Linear function spaces and their duals [See also 30H05, 32A38, 46F05] [For function algebras, see 46J10]

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 16/03/15 Date de captation : 24/01/15 Collection : Special events ; 30 Years of Wavelets Format : quicktime ; audio/x-aac Durée : 00:34:19 Domaine : Analyse & Applications ; Numerical Analysis & Scientific Computing Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-01-24_Feichtinger.mp4 Informations sur la rencontre Nom du congrès : 30 years of wavelets / 30 ans des ondelettesOrganisteurs Congrès : Feichtinger, Hans G. ; Torrésani, BrunoDates : 23/01/15 - 24/01/15 Année de la rencontre : 2015 URL Congrès : http://feichtingertorresani.weebly.com/3...Citation Data DOI : 10.24350/CIRM.V.18718203 Cite this video as: Feichtinger, Hans G. (2015). Wavelet theory, coorbit spaces and ramifications. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18718203 URI : http://dx.doi.org/10.24350/CIRM.V.18718203

Bibliographie

1. [1] Arazy, J., Fisher, S., & Peetre, J. (1985). Möbius invariant function spaces. Journal für die Reine und Angewandte Mathematik, 363, 110-145 - https://eudml.org/doc/152782

2. [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. [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. [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. [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. [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. [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. [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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