• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

Phase-space methods such as Wigner transforms have long been used for the study of semiclassical limits. Regularized objects, such as Husimi transforms, have often been proposed as an interesting alternative, but have not been widely used, due to the lack of an appropriate framework to study the equations governing them.
We present a new framework, in which exact smoothed Wigner and Husimi equations can be formulated rigorously for a wide class of problems, and the convergence to semiclassical limits can be established. It must be noted that there are several similarities to the Gelfand-Shilov functions of type $S^{b,B}$. Special emphasis is based on the points where this constitutes a strengthening of the existing Wigner-transform-based state of the art. In particular, it is seen that the coarse-scale approach works with less regular potentials.

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

Several deep results in time-frequency analysis, known as Feichtinger-Groechenig theory, are crucial for understanding Gabor frames. We point out a few aspects.

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

Informal discussion, in order to share experience of the speaker on the subject (dating back to his PhD work and habilitation at TU Vienna) with the workshop participants.

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

In the 90's there was wide interest in the use of Wigner and smoothed Wigner transforms in signal processing; several libraries for their computation were created at that time. However, it is fair to say that these transforms are not as widely used today, and one of the reasons is that their computation is "unreasonably" expensive, when compared e.g. to spectrograms. Motivated by a recent scheme for the simulation of caustic development and propagation, we present a new library for the computation of Wigner and smoothed Wigner transforms. The main improvement in comparison to the state of the art is a parallelization of all steps of the process (including output, e.g. plotting) which allows a better behavior for much larger signals.

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

ABSTRACT: The limit Wigner measure of a WKB function satisfies a simple transport equation in phase-space and is well suited for capturing oscillations at scale of order $O(epsilon)$, but it fails, for instance, to provide the correct amplitude on caustics where different scales appear. We define the semi-classical Wigner function of an $N$-dimensional WKB function, as a suitable formal approximation of its scaled Wigner function. The semi-classical Wigner function is an oscillatory integral that provides an $epsilon$-dependent regularization of the limit Wigner measure, it obeys a transport-dispersive evolution law in phase space, and it is well defined even at simple caustics.

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

It is well known that the continuous wavelet transform has the ability to identify the set of singularities of a function or distribution. It was recently shown that certain multidimensional generalizations of the wavelet transform are useful to capture additional information about the geometry of the singularities.
The continuous shearlet transform, recently introduced by the author and his collaborators is a novel approach which combines the power of multiscale analysis with ability to deal with the anisotropy and directional features of multidimensional data. In this talk, we show that the continuous shearlet transform can capture both the location and geometry of singularities. This is useful to identify and characterize the geometrical information of the edges in natural images. Numerical examples of edge analysis and detection will also be presented.

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

The Weyl calculus is an important part within the theory of pseudo-differential operators. In physics, the calculus links classcal mechanics with quantum mechanics in the sense that an observable in classical mechanics corresponds to an operator in quantum mechanics. In general, this operator is the Weyl operator, at least with good approximations.
The composition of two such operators corresponds to a somewhat complicated product between the observables. This product is the so called Weyl product.
In the talk we present some continuity results for this product in background of the theory of modulation spaces. Furthermore, the Weyl product, on the Fourier transform side, is equal to a twisted convolution. Consequently, any continuity result for the Weyl product on modulation spaces gives rise to a similar continuityresult for the twisted convolution on Fourier modulation spaces. We show how these results can be used to get continuity for twisted convolutions on weighted Lebesgue spaces.

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

Consider the solution to the time-dependent Schrödinger equation with initial data f. It is shown by Sjögren and Sjölin in 1989 that there exists f in the Sobolev space $H^s(mathbb{R}^n), ; s=n/2$ such that tangential convergence can not be widened to convergence regions. We show that the corresponding result holds when $-Delta_x$ is replaced by an operator $varphi(D)$, with special conditions on $varphi$. More explicitly we consider the solution to the equation $(varphi(D)+ipartial_t )u =0,$ with the initial condition u(x,0)=f(x). Here $varphi$ should be real-valued and its radial derivatives of first and second order ($varphi' =varphi'_r$ and $varphi'' =varphi''_{rr}$) should be continuous, outside a compact set containing origin. Furthermore, we will require some appropriate conditions on the growth $varphi'$ and $varphi''$. In particular the function $varphi(xi)=|xi|^a$ will satisfy these conditions, for a> 1.

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

We consider a class of pseudodifferential operators defined by smooth symbols which may have almost exponential growth or decay at infinity in phase space. They can be considered as symbol-global type operators, studied by many authors in the context of PDE. In the present lecture, we use time-frequency methods/Gabor frames in the study of spectral asymptotics. Almost exponential growth/decay of symbols is handled by the use of techniques from ultra-distribution theory.

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

Our aim is to emphasize the main known limitations to the use of Wigner measures for Schr"odinger equations.
After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schr"odinger equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the same Wigner measures at some initial time, but different Wigner measures for a larger time. In the case of systems, this difficulty can partially be avoided by considering more refined Wigner measures such as two-scale Wigner measures; however, we give examples of situations where this quadratic approach fails.

This is joint work with Remi Carles, Clotilde ermanian-Kammerer and Hans-Peter Stimming, Comm. Pure and Applied Analysis, (1) 2009

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)

The setting of the Banach Gelfand triple (S0,L2,SO') allows to describe in a clean functional analytic way how a given continuous problem (such as applying a pseudo-differential operator given by its Kohn-Nirenberg symbol that should be applied to a given L2-function, or finding the spreading function of an operator) can be approximated at least in a qualitative way in a constructive way. The setting involves the use of function spaces, Fourier transforms and generalized functions, and aims at the reduction of a continuous problem to a finite-dimensional setting, which in principle is available for numerical implementation, e.g. using MATLAB or any other mathematical software.

• Event: Workshop on "Applied Analysis and Fast Computation in Phase-Space 2008" (2008)