Institute for applied mathematics

# Graduate seminar SS2017: Time-Frequency Analysis and Its Applications

Description

In certain problems of PDE, harmonic analysis and stochastics one needs to obtain finer information about the frequency and phase content of a given function $f$, maybe the solution to a PDE or a stochastic process.  One way to make this precise is to introduce a window function" $g$ and define the so-called short time Fourier transform (STFT)  $V_{g}(f)$ with respect to a window function $g$, defined by

$V_{g}(f)(x, \xi) = \int_{\mathbb{R}^{d}} f(t) \overline{g(t- x)} e^{-2 \pi i t \cdot \xi} dt$ .

The STFT basically extracts a local section of the signal.  For $1 \leq p,q \leq \infty$, a non-negative function $m(x, \xi)$ on $\mathbb{R}^{2d}$ and for a window function $g \in \mathcal{S}$;  the modulation space $M_{m}^{p,q}$  is defined by

$M_{m}^{p,q} = \left \{ f \in \mathcal{S}' ~|~ \left(\int_{\mathbb{R}^d} \left( \int_{\mathbb{R}^d} |V_{g}(f)(x, \xi)|^p m(x, \xi)^p dx \right)^{q/p} d \xi \right)^{1/q} < \infty \right\}.$

Modulation spaces capture the behavior of $f$ both on time and frequency domains, thereby giving  fine information about the amplitude and period of oscillations of $f$.  In this graduate seminar, we will be looking at topics in PDE and stochastics where such time-frequency analysis becomes important.  The emphasis will be on understanding main" results and examples.

We will start with a selection of topics (Chapters 2-3 and 10-14) from our main source Foundations of Time-Frequency Analysis"  by Karlheinz Grochenig.  After that, we have the following selection of research papers on applications of time-frequency analysis:

•  Modulation spaces and nonlinear evolution equations. Ruzhansky, Michael, Mitsuru Sugimoto, and Baoxiang Wang.  Evolution equations of hyperbolic and Schrödinger type. Springer Basel, 2012. 267-283.  https://arxiv.org/abs/1203.4651
• Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS.  Benyi, Arpad, Tadahiro Oh, and Oana Pocovnicu.  Excursions in Harmonic Analysis, Volume 4. Springer International Publishing, 2015. 3-25. https://arxiv.org/abs/1405.7326
• On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb{R}^{d}, d \geq 3$.   Benyi, Arpad, Tadahiro Oh, and Oana Pocovnicu. Transactions of the American Mathematical Society, Series B 2.1 (2015): 1-50. https://arxiv.org/abs/1405.7327
• Modulation spaces, Wiener amalgam spaces, and Brownian motions. Benyi, Arpad, and Tadahiro Oh.  Advances in Mathematics 228.5 (2011). https://arxiv.org/abs/1007.1957
• Chapters 10 and also (time-permitting) 11 of  Combescure, Monique, and Didier Robert. Coherent states and applications in mathematical physics. Springer Science and Business Media, 2012.

## News

Prof. Dr. S. Müller erhält den diesjährigen Lehrpreis der Universität Bonn (07.07.17).

Prof. Dr. Michael Ortiz kommt als Bonn Research Chair ans IAM (Pressemitteilung, 29.7.2016).

Prof. Dr. S. Conti erhält den diesjährigen Lehrpreis der Universität Bonn (05.07.2016).

Prof. Dr. Karl-Theodor Sturm, Koordinator des Hausdorff Zentrums für Mathematik an der Universität Bonn, erhält für seine eigene Forschung einen begehrten Advanced Grant des Europäischen Forschungsrats (ERC). (Pressemitteilung 21.04.2016)

Prof. Dr. Stefan Müller has been invited to become full member of the scientific society “Academia Europaea” in November. Only European scientists, who have been recommended by a review board and are confirmed by a vote of the council, can join the society. (06.01.2016)

Contact

Managing Director: Prof. Dr. Anton Bovier
Chief Administrator: Dr. B. Doerffel
geschaeftsfuehrung@iam.uni-bonn.de
Imprint

Mailing address

Institute for Applied Mathematics
University of Bonn
Endenicher Allee 60
D-53115 Bonn / Germany