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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.

CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université
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.

CIRM - ...

15-XX ; 41-XX ; 42-XX ; 46-XX

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I will give a survey of the operator theory that is currently evolving on Hardy spaces of Dirichlet series. We will consider recent results about multiplicative Hankel operators as introduced and studied by Helson and developments building on the Gordon-Hedenmalm theorem on bounded composition operators on the $H^2$ space of Dirichlet series.

47B35 ; 30B50 ; 30H10

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*Outreach;Mathematics Education and Popularization of Mathematics*

The Jean Morlet Chair is a scientific collaboration between CIRM -CNRS-SMF-, Aix-Marseille Université and the City of Marseille. Two international calls are launched every year to attract innovative researchers in an area of mathematical sciences. Selected candidates who must come from a foreign institution can spend a semester in residence at CIRM, where they run a full program of mathematical events in collaboration with a local project holder. Hans-Georg Feichtinger (University of Vienna) and Bruno Torresani (I2M Marseille) have been in charge of the second semester 2014 which will end in January 2015. The focus is on 'Computational Time-Frequency and Coorbit Theory'. Starting with a Research in Pairs event at the end of August, then three larger events-a School for young scientists, a main Conference and Small group- rather close in dates to enable participants to stay for more than one event, their semester will end on a second Research in Pairs in January 2015 and a celebratory event at the very end of the semester to celebrate 30 years of wavelets.

CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université
The Jean Morlet Chair is a scientific collaboration between CIRM -CNRS-SMF-, Aix-Marseille Université and the City of Marseille. Two international calls are launched every year to attract innovative researchers in an area of mathematical sciences. Selected candidates who must come from a foreign institution can spend a semester in residence at CIRM, where they run a full program of mathematical events in collaboration with a local project ...

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- 373 p.

ISBN 978-3-85403-133-8

**Localisation : **Colloque 1er étage (VIEN)

mathématiques et musique # Diderot Mathematical Forum

00B25 ; 00A65

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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.

CIRM - Chaire Jean-Morlet 2014 - Aix-Marseille Université
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.

CIRM - ...

15-XX ; 41-XX ; 42-XX ; 46-XX

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In several applications in signal processing it has proven useful to decompose a given signal in a multiscale dictionary, for instance to achieve compression by coefficient thresholding or to solve inverse problems. The most popular family of such dictionaries are undoubtedly wavelets which have had a tremendous impact in applied mathematics since Daubechies' construction of orthonormal wavelet bases with compact support in the 1980s. While wavelets are now a well-established tool in numerical signal processing (for instance the JPEG2000 coding standard is based on a wavelet transform) it has been recognized in the past decades that they also possess several shortcomings, in particular with respect to the treatment of multidimensional data where anisotropic structures such as edges in images are typically present. This deficiency of wavelets has given birth to the research area of geometric multiscale analysis where frame constructions which are optimally adapted to anisotropic structures are sought. A milestone in this area has been the construction of curvelet and shearlet frames which are indeed capable of optimally resolving curved singularities in multidimensional data.

In this course we will outline these developments, starting with a short introduction to wavelets and then moving on to more recent constructions of curvelets, shearlets and ridgelets. We will discuss their applicability to diverse problems in signal processing such as compression, denoising, morphological component analysis, or the solution of transport PDEs. Implementation aspects will also be covered. (Slides in attachment).
In several applications in signal processing it has proven useful to decompose a given signal in a multiscale dictionary, for instance to achieve compression by coefficient thresholding or to solve inverse problems. The most popular family of such dictionaries are undoubtedly wavelets which have had a tremendous impact in applied mathematics since Daubechies' construction of orthonormal wavelet bases with compact support in the 1980s. While ...

42C15 ; 42C40

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In several applications in signal processing it has proven useful to decompose a given signal in a multiscale dictionary, for instance to achieve compression by coefficient thresholding or to solve inverse problems. The most popular family of such dictionaries are undoubtedly wavelets which have had a tremendous impact in applied mathematics since Daubechies' construction of orthonormal wavelet bases with compact support in the 1980s. While wavelets are now a well-established tool in numerical signal processing (for instance the JPEG2000 coding standard is based on a wavelet transform) it has been recognized in the past decades that they also possess several shortcomings, in particular with respect to the treatment of multidimensional data where anisotropic structures such as edges in images are typically present. This deficiency of wavelets has given birth to the research area of geometric multiscale analysis where frame constructions which are optimally adapted to anisotropic structures are sought. A milestone in this area has been the construction of curvelet and shearlet frames which are indeed capable of optimally resolving curved singularities in multidimensional data.

In this course we will outline these developments, starting with a short introduction to wavelets and then moving on to more recent constructions of curvelets, shearlets and ridgelets. We will discuss their applicability to diverse problems in signal processing such as compression, denoising, morphological component analysis, or the solution of transport PDEs. Implementation aspects will also be covered. (Slides in attachment).
In several applications in signal processing it has proven useful to decompose a given signal in a multiscale dictionary, for instance to achieve compression by coefficient thresholding or to solve inverse problems. The most popular family of such dictionaries are undoubtedly wavelets which have had a tremendous impact in applied mathematics since Daubechies' construction of orthonormal wavelet bases with compact support in the 1980s. While ...

42C15 ; 42C40

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Time-frequency (or Gabor) frames are constructed from time- and frequency shifts of one (or several) basic analysis window and thus carry a very particular structure. On the other hand, due to their close relation to standard signal processing tools such as the short-time Fourier transform, but also local cosine bases or lapped transforms, in the past years time-frequency frames have increasingly been applied to solve problems in audio signal processing.

In this course, we will introduce the basic concepts of time-frequency frames, keeping their connection to audio applications as a guide-line. We will show how standard mathematical tools such as the Walnut representations can be used to obtain convenient reconstruction methods and also generalizations such the non-stationary Gabor transform. Applications such as the realization of an invertible constant-Q transform will be presented. Finally, we will introduce the basic notions of transform domain modelling, in particular those based on sparsity and structured sparsity, and their applications to denoising, multilayer decomposition and declipping. (Slides in attachment).
Time-frequency (or Gabor) frames are constructed from time- and frequency shifts of one (or several) basic analysis window and thus carry a very particular structure. On the other hand, due to their close relation to standard signal processing tools such as the short-time Fourier transform, but also local cosine bases or lapped transforms, in the past years time-frequency frames have increasingly been applied to solve problems in audio signal ...

42C15

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Time-frequency (or Gabor) frames are constructed from time- and frequency shifts of one (or several) basic analysis window and thus carry a very particular structure. On the other hand, due to their close relation to standard signal processing tools such as the short-time Fourier transform, but also local cosine bases or lapped transforms, in the past years time-frequency frames have increasingly been applied to solve problems in audio signal processing.

In this course, we will introduce the basic concepts of time-frequency frames, keeping their connection to audio applications as a guide-line. We will show how standard mathematical tools such as the Walnut representations can be used to obtain convenient reconstruction methods and also generalizations such the non-stationary Gabor transform. Applications such as the realization of an invertible constant-Q transform will be presented. Finally, we will introduce the basic notions of transform domain modelling, in particular those based on sparsity and structured sparsity, and their applications to denoising, multilayer decomposition and declipping. (Slides in attachment).
Time-frequency (or Gabor) frames are constructed from time- and frequency shifts of one (or several) basic analysis window and thus carry a very particular structure. On the other hand, due to their close relation to standard signal processing tools such as the short-time Fourier transform, but also local cosine bases or lapped transforms, in the past years time-frequency frames have increasingly been applied to solve problems in audio signal ...

94A12

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Generalizing results of Rossi and Vergne for the holomorphic discrete series on symmetric domains, on the one hand, and of Chailuek and Hall for Toeplitz operators on the ball, on the other hand, we establish existence of analytic continuation of weighted Bergman spaces, in the weight (Wallach) parameter, as well as of the associated Toeplitz operators (with sufficiently nice symbols), on any smoothly bounded strictly pseudoconvex domain. Still further extension to Sobolev spaces of holomorphic functions is likewise treated.
Generalizing results of Rossi and Vergne for the holomorphic discrete series on symmetric domains, on the one hand, and of Chailuek and Hall for Toeplitz operators on the ball, on the other hand, we establish existence of analytic continuation of weighted Bergman spaces, in the weight (Wallach) parameter, as well as of the associated Toeplitz operators (with sufficiently nice symbols), on any smoothly bounded strictly pseudoconvex domain. Still ...

47B35 ; 30H20

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Let $f$ and $g$ be functions, not identically zero, in the Fock space $F^2$ of $C^n$. We show that the product $T_fT_\bar{g}$ of Toeplitz operators on $F^2$ is bounded if and only if $f= e^p$ and $g= ce^{-p}$, where $c$ is a nonzero constant and $p$ is a linear polynomial.

47B35 ; 30H20

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*Special events;30 Years of Wavelets*

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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*Special events;30 Years of Wavelets;Mathematics in Science and Technology*

In this conference, I start by presenting the first applications and developments of wavelet methods made in Marseille in 1985 in the framework of sounds and music. A description of the earliest wavelet transform implementation using the SYTER processor is given followed by a discussion related to the first signal analysis investigations. Sound examples of the initial sound transformations obtained by altering the wavelet representation are further presented. Then methods aiming at estimating sound synthesis parameters such as amplitude and frequency modulation laws are described. Finally, new challenges brought by these early works are presented, focusing on the relationship between low-level synthesis parameters and sound perception and cognition. An example of the use of the wavelet transforms to estimate sound invariants related to the evocation of the "object" and the "action" is presented.

Keywords : sound and music - first wavelet applications - signal analysis - sound synthesis - fast wavelet algorithms - instantaneous frequency estimation - sound invariants
In this conference, I start by presenting the first applications and developments of wavelet methods made in Marseille in 1985 in the framework of sounds and music. A description of the earliest wavelet transform implementation using the SYTER processor is given followed by a discussion related to the first signal analysis investigations. Sound examples of the initial sound transformations obtained by altering the wavelet representation are ...

00A65 ; 42C40 ; 65T60 ; 94A12 ; 97M10 ; 97M80

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*Special events;30 Years of Wavelets*

In my talk I am presenting a link between time-frequency analysis and noncommutative geometry. In particular, a connection between the Moyal plane, noncommutative tori and time-frequency analysis. After a brief description of a dictionary between these two areas I am going to explain some consequences for time-frequency analysis and noncommutative geometry such as the construction of projections in the mentioned operator algebras and Gabor frames.

Keywords: modulation spaces - Banach-Gelfand triples - noncommutative tori - Moyal plane - noncommutative geometry - deformation quantization
In my talk I am presenting a link between time-frequency analysis and noncommutative geometry. In particular, a connection between the Moyal plane, noncommutative tori and time-frequency analysis. After a brief description of a dictionary between these two areas I am going to explain some consequences for time-frequency analysis and noncommutative geometry such as the construction of projections in the mentioned operator algebras and Gabor ...

46Fxx ; 46Kxx ; 46S60 ; 81S05 ; 81S10 ; 81S30

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*Special events;30 Years of Wavelets*

The introduction of wavelets in the mid 80's has significantly reshaped some areas of the scientific landscape by establishing bridges between previously disconnected domains, and eventually leading to a new paradigm. This generally accepted-yet loose-claim can be given a more precise form by exploiting bibliometric databases such as the ISI Web of Science. Preliminary results in this direction will be reported here, based on multiple entries where authors, references, keywords and disciplines are used as nodes of a network in which the links correspond to their co-appearance in the same paper. While the evolution in time of such an " heterogeneous net " gives a quantified perspective on the birth and growth of wavelets as a well-identified scientific field of its own, it also raises many interpretation issues (related, e.g., to automation vs. expertise) whose implications go beyond this peculiar case study.

Keywords : wavelets - history - bibliometry - network, paradigm
The introduction of wavelets in the mid 80's has significantly reshaped some areas of the scientific landscape by establishing bridges between previously disconnected domains, and eventually leading to a new paradigm. This generally accepted-yet loose-claim can be given a more precise form by exploiting bibliometric databases such as the ISI Web of Science. Preliminary results in this direction will be reported here, based on multiple entries ...

01-XX ; 42-XX ; 68-XX ; 94-XX

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*Special events;30 Years of Wavelets*

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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*Special events;30 Years of Wavelets*

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
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 ...

43-XX ; 46Exx ; 42C40 ; 42C15 ; 42C10

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*Special events;30 Years of Wavelets*

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

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