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Multi angle  High dimensional learning from images to physics
Mallat, Stéphane (Auteur de la Conférence) | CIRM (Editeur )

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Multi angle  From function spaces to stylometry: the wavelet bridge
Jaffard, Stéphane (Auteur de la Conférence) | CIRM (Editeur )

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Multi angle  Continuous and discrete uncertainty principles
Torrésani, Bruno (Auteur de la Conférence) | CIRM (Editeur )

Uncertainty principles go back to the early years of quantum mechanics. Originally introduced to describe the impossibility for a function to be sharply localized in both the direct and Fourier spaces, localization being measured by variance, it has been generalized to many other situations, including different representation spaces and different localization measures.
In this talk we first review classical results on variance uncertainty inequalities (in particular Heisenberg, Robertson and Breitenberger inequalities). We then focus on discrete (and in particular finite-dimensional) situations, where variance has to be replaced with more suitable localization measures. We then present recent results on support and entropic inequalities, describing joint localization properties of vector expansions with respect to two frames.

Keywords: uncertainty principle - variance of a function - Heisenberg inequality - support inequalities - entropic inequalities
Uncertainty principles go back to the early years of quantum mechanics. Originally introduced to describe the impossibility for a function to be sharply localized in both the direct and Fourier spaces, localization being measured by variance, it has been generalized to many other situations, including different representation spaces and different localization measures.
In this talk we first review classical results on variance uncertainty ...

94A12 ; 94A17 ; 26D20 ; 42C40

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Multi angle  Continuous (semi-)frames revisited
Antoine, Jean-Pierre (Auteur de la Conférence) | CIRM (Editeur )

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
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 ...

42C15 ; 42C40 ; 46C50 ; 65T60

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Multi angle  Compressive sensing with time-frequency structured random matrices
Rauhut, Holger (Auteur de la Conférence) | CIRM (Editeur )

One of the important "products" of wavelet theory consists in the insight that it is often beneficial to consider sparsity in signal processing applications. In fact, wavelet compression relies on the fact that wavelet expansions of real-world signals and images are usually sparse. Compressive sensing builds on sparsity and tells us that sparse signals (expansions) can be recovered from incomplete linear measurements (samples) efficiently. This finding triggered an enormous research activity in recent years both in signal processing applications as well as their mathematical foundations. The present talk discusses connections of compressive sensing and time-frequency analysis (the sister of wavelet theory). In particular, we give on overview on recent results on compressive sensing with time-frequency structured random matrices.

Keywords: compressive sensing - time-frequency analysis - wavelets - sparsity - random matrices - $\ell_1$-minimization - radar - wireless communications
One of the important "products" of wavelet theory consists in the insight that it is often beneficial to consider sparsity in signal processing applications. In fact, wavelet compression relies on the fact that wavelet expansions of real-world signals and images are usually sparse. Compressive sensing builds on sparsity and tells us that sparse signals (expansions) can be recovered from incomplete linear measurements (samples) efficiently. This ...

94A20 ; 94A08 ; 42C40 ; 60B20 ; 90C25

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Multi angle  A test for local white noise (and the absence of aliasing) in locally stationary wavelet time series
Nason, Guy (Auteur de la Conférence) | CIRM (Editeur )

This talk develops a new test for local white noise which also doubles as a test for the lack of aliasing in a locally stationary wavelet process. We compare and contrast our new test with the aliasing test for stationary time series due to Hinich and co-authors. We show that the test is robust to mismatch of analysis and synthesis wavelet. We demonstrate the effectiveness of the test on some simulated examples and on an example from wind energy.

42C40 ; 60G10 ; 62M10 ; 62M15

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Multi angle  30 years of $T(b)$ theorems
Auscher, Pascal (Auteur de la Conférence) | CIRM (Editeur )

The $T(b)$ theorem proved 30 years ago by David, Journé and Semmes, following a first result of McIntosh and Meyer, has proved to be a powerful and versatile tool for a number of applications. We will discuss history and main applications including recent ones.

42B20 ; 42B25 ; 42C40

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9278018146910962909" class='linkstyle1'>97M10 ; 97M80

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Multi angle  Somes perspectives of computational harmonic analysis in numerics
Grohs, Philipp (Auteur de la Conférence) | CIRM (Editeur )

Wavelets are standard tool in signal- and image processing. It has taken a long time until wavelet methods have been accepted in numerical analysis as useful tools for the numerical discretization of certain PDEs. In the signal- and image processing community several new frame constructions have been introduced in recent years (curvelets, shearlets, ridgelets, ...). Question: Can they be used also in numerical analysis? This talk: Small first step. Wavelets are standard tool in signal- and image processing. It has taken a long time until wavelet methods have been accepted in numerical analysis as useful tools for the numerical discretization of certain PDEs. In the signal- and image processing community several new frame constructions have been introduced in recent years (curvelets, shearlets, ridgelets, ...). Question: Can they be used also in numerical analysis? This talk: Small first ...

42C15 ; 42C40 ; 65Txx