Continuous (semi-)frames revisited | Vidéo | Carmin.tv

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Continuous (semi-)frames revisited

By Jean-Pierre Antoine

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

We start by recalling the essential features of frames, both discrete and continuous, with some emphasis on the notion of frame duality. Then we turn to generalizations, namely upper and lower semi-frames, and their duality. Next we consider arbitrary measurable maps and examine the standard operators, analysis, synthesis and frame operators, and study their properties. Finally we analyze the recent notion of reproducing pairs. In view of their duality structure, we introduce two natural partial inner product spaces and formulate a number of open questions.

Keywords: continuous frames - semi-frames - frame duality - reproducing pairs - partial inner product spaces

Information about the video

Date of recording 23/01/2015
Date of publication 11/03/2015
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.18715703
Cite this video Antoine, Jean-Pierre (23/01/2015). Continuous (semi-)frames revisited. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.18715703
URL https://dx.doi.org/10.24350/CIRM.V.18715703

Domain(s)

Bibliography

[1] Ali, S.T., Antoine, J-P., & Gazeau, J-P. (1991). Square integrability of group representations on homogeneous spaces. I: reproducing triples and frames. Annales de l'Institut Henri Poincaré. Physique Théorique, 55(4), 829-855 - https://eudml.org/doc/76555
[2] Ali, S.T., Antoine, J-P., & Gazeau, J-P. (1993). Continuous frames in Hilbert space. Annals of Physics, 222(1), 1-37 - http://dx.doi.org/10.1006/aphy.1993.1016
[3] Antoine, J-P., & Balazs, P. (2011). Frames and semi-frames. Journal of Physics A: Mathematical and Theoretical, 44(20), 205201; corrigendum ibid. 44(47), 479501 - http://dx.doi.org/10.1088/1751-8113/44/20/205201
[4] Antoine, J-P., & Balazs, P. (2012). Frames, semi-frames, and Hilbert scales. Numerical Functional Analysis and Optimization, 33(7-9), 736-769 - http://dx.doi.org/10.1080/01630563.2012.682128
[5] Antoine, J-P., & Trapani, C. (2009). Partial Inner Product Spaces: theory and Applications. Berlin, Heidelberg: Springer. (Lecture Notes in Mathematics, 1986) - http://dx.doi.org/10.1007/978-3-642-05136-4
[6] Speckbacher, M., & Balazs, P. (2014). The continuous non stationary Gabor transform on LCA groups with applications to representations of the affine Weyl-Heisenberg group. <arXiv:1404.6830> - http://arxiv.org/abs/1407.6830v1

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