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source: Centre International de Rencontres Mathématiques 2015年8月10日

Jean-Morlet Chair - Research Talks - Feichtinger/Torresani

Semester on 'Computational Time-Frequency and Coorbit Theory'

August 2014 - January 2015

General themes

The central idea for this semester of the Jean Morlet Chair was to promote the connection between mathematical analysis, function spaces, atomic decomposition and application areas within mathematics and applied sciences, which was certainly also in the spirit of Jean Morlet, the inventor of wavelets.

Function spaces have become a central object in analysis in the last decades. Traditionally, Lebesgue spaces and Besov/Sobolev spaces dominated the scene, however modern analysis (real, complex and harmonic) makes use of a large variety of Banach spaces of functions (and distributions) respectively.

These are typically described by certain (Banach frame) expansions, e.g. with respect to wavelets, or Gabor families, or shearlets, and are well suited to describe approximation behavior by finite dimensional subspaces.

As such, they are equally important for considerations concerning the numerical (approximate) realization of linear operators, or their inverse, which ideally should provide the solution with the minimal computational costs to the best possible degree - as measured in one of those function spaces.

The overall goal of the program was to further exploit the unifying view-point that has been brought to the scene with the creation of coorbit theory, which recently has extended deeply into group representation theory, complex analysis and application areas. Modulation spaces are related to Gabor expansions and have turned out to be closely related to slowly varying channels modelling mobile communication, while at the same time being well suited for the description of pseudo-differential operators belonging to the Sjoestrand class.

The overall goal of the program was to bring together people from the different subject areas and promote the unifying perspective of this approach.

http://feichtingertorresani.weebly.com

Hans G. Feichtinger: Mathematical and numerical aspects of frame theory - Part 1 1:30:26

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Motivated by the spectrogram (or short-time Fourier transform) basic principles of linear algebra are explained, preparing for the more general case of Gabor frames in time-frequency analysis. The importance of the singular value decomposition and the four spaces associated with a matrix is pointed out, and based on this the pseudo-inverse (leading later to the dual Gabor frame) and the Loewdin (symmetric) orthogonalization are explained.

Recording during the thematic meeting: "Computational harmonic analysis - with applications to signal and image processing" the October 20, 2014 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

Hans G. Feichtinger: Mathematical and numerical aspects of frame theory - Part 2 1:36:12

Philipp Grohs: Wavelets, shearlets and geometric frames - Part 1 1:35:52

Philipp Grohs: Wavelets, shearlets and geometric frames - Part 2 1:30:06

Nicki Holighaus: Time-frequency frames and applications to audio analysis - Part 1 1:27:48

Matthieu Kowalski: Time-frequency frames and applications to audio analysis - Part 2 1:14:29

Hans Feichtinger: Wavelet theory, coorbit spaces and ramifications 34:20

Kehe Zhu: Products of Toeplitz operators on the Fock space 51:36

Miroslav Englis: Analytic continuation of Toeplitz operators 59:03

Kristian Seip: Hankel and composition operators on spaces of Dirichlet series 1:00:11

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