A review of old and recent results concerning coherent states will be presented, including semiclassical quantum propagation. We will also show how coherent states allow to construct objects localised in higher dimensional submanifolds of the underlying phase-space, up to the lagrangian case, and the importance, especially for long time evolution, of the freedom that provides continuous representation versus the discrete frame vision.

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

Wigner functions were introduced as a phase space formulation of quantum mechanics, designed especially for the "semiclassical limit" according to the "correspondence principle". In the last 15 years the Wigner measures, as the weak limit of sequences of Wigner functions, have become a mathematical tool on their own for a wide class of "homogenization" problems. We present the key ideas of the concept of Wigner transforms and the problems where they are more or less useful.

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

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

We consider the problem of high-frequency asymptotics for the time-dependent one-dimensional Schrodinger equation with rapidly oscillating initial data. This problem is commonly studied via the WKB method. An alternative method is based on the limit Wigner measure. This approach recovers geometrical optics, but, like the WKB method, it fails at caustics. To remedy this deficiency we employ the semiclassical Wigner function which is a formal asymptotic approximation of the scaled Wigner function but also a regularization of the limit Wigner measure. We obtain Airy-type asymptotics for the semiclassical Wigner function as solutions to the Wigner equation.

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

Modulation spaces play a similar role with respect to Gabor families and within time-frequency analysis as the more classical function spaces (of Besov and Triebel Lizorkin type) with respect to wavelet bases. They can be defined via uniform (as opposed to dyadic) decompositions of the Fourier transform side, and have a natural continous description in terms of the STFT (short-time or gliding window Fourier Transform). On the Fourier transform side they are typical examples of so-called Wiener amalgam spaces, which are a very flexible tool to describe the global behaviour of certain local properties. Especially the convolution relations between Wiener amalgam spaces (decoupling of local and global properties) are a powerful tool.

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

We present some characterizations of classical function spaces, e.g. the Schwartz space of test functions or Gelfand-Shilov spaces, in terms of the short-time Fourier transform. This talk surveys results originally obtained by Feichtinger, Groechenig and Zimmermann.

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

In this talk, I will introduce how to use the frequency-uniform decomposition method to study the cauchy problem of nonlinear Schrodinger equations. I mainly show the global well-posedness of solutions to NLS equations with small rough data in certain modulation spaces.

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

This lecture is dedicated to the jubilee of 25 years since the first technical report on modulation spaces was written. Nowadays, modulation spaces are recognized as the most important spaces of functions/distributions in the growing field of time-frequency analysis and its various applications. In particular, modulation spaces are designed to perform local analysis in time-frequency plane. Our aim is to perform micro-local analysis in the background of modulation spaces. The starting point is to give a reasonable definition of wave-front sets in modulation spaces. This leads to an equivalent notion of wave front sets in Fourier Lebesgue spaces. As applications, we describe (local) products in modulation spaces by the means of the corresponding wave front sets, and show that usual properties for a class of pseudo-differential operators which are valid for classical wave front sets also hold in our framework. The results are the part of ongoing research project with Stevan Pilipovic and Joachim Toft.

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

The theme of this talk is that the theory of charged particles in a uniform magnetic field can be generalized to a large class of operators if one uses an extended a class of Weyl operators which we call "Landau-Weyl operators". The link between standard Weyl calculus and Landau-Weyl calculus is made explicit by the use of an infinite family of intertwining "windowed wavepacket transforms"; this makes possible the use of the theory of modulation spaces to study various regularity properties. Our techniques allow us not only to recover easily the eigenvalues and eigenfunctions of the Hamiltonian operator of a charged particle in a uniform magnetic field, but also to prove global hypoellipticity results, and to study the regularity of the solutions to Schrödinger equations. This is joint work with Franz Luef.

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

We investigate how pseudodifferential operators behave with respect to Gabor frames. If the symbol is taken in a special class of non-smooth symbols that is one of the standard modulation spaces and is known as the Sjöstrand class then the operator is almost diagonalized by time-frequency shifts (phase-space shifts or coherent states) of a single function. In contrast to other almost diagonalization results, the quality of the almost diagonalization characterizes the symbol class. Various modifications and approximation results will be discussed.

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

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

Banach Gelfand triples are an important subset from the modulation spaces. Starting from the Segal algbra S_0(Rd) and its dual one can use the notion of (unitary) Banach Gelfand triple isomorphism to describe e.g. the Fourier transform, or the mapping between operator kernels and their Kohn-Nirenberg symbol or their spreading distribution in a technically not so difficult way, building on standard functional analytic concepts only.

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

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