报告人: Prof. Hans G. Feichtinger （奥地利 维也纳大学)
报告时间:
2024年7月22日: 上午10:00-12:00
2024年7月22日: 下午15:00-17:00
2024年7月24日: 下午15:00-17:00
报告地点：章辉楼442
联系人：王保祥 教授
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报告摘要: In recent decades, the range of function spaces used to analyze various operators has significantly expanded. Different types of expansions, such as wavelets, Gabor expansions, and shearlets, now offer highly adaptable spaces that can be finely tuned to specific types of problems. Sometimes, the problems themselves lead to the development of tools—for instance, wavelets were developed to address problems involving Calderon-Zygmund operators. Conversely, in other cases, the tools, like modulation spaces, have proven instrumental in advancing areas such as time-frequency analysis and the theory of pseudo-differential operators. In light of these developments, it is crucial to explore the various principles underlying these constructions, their interrelationships, as well as their shared characteristics and distinctions.
We will start from the traditional theory of Besov spaces, which have been introduced in the spirit of Lipschitz-type spaces derived from the classical L^p-spaces by controlling (divided) difference operators for h to 0. Later on (Peetre and Triebel) it was realized that they can be described using dyadic partitions on the Fourier transform side. This allows among other to show relatively easily that this family is closed under interpolation and duality.
Wiener amalgam spaces started with the work of Norbert Wiener on the Tauberian Theorems (in his book on the Fourier Integral, from 1932). They have been generalized in 1980 (published 1983) by HGFei. In short, they allow to describe the global behaviour of a local norm. This is easy if one starts with a solid BF-space (more or less a Banach function space, or Banach lattice), because than multiplication with the indicator functions of a fundamental domain of a lattice are pointwise multipliers, think of the lattice Z^d in R^d, and the unit cube. Putting a sequence space norm, e.g. a weighted l^q-norm of local L^p-norms gives the classical amalgam spaces. For more general local components one has to resort to the use of so-called BUPUs (typically smooth partitions of unity, such as cubic B-splines on R).
Applying such principles on the Fourier transform side allows to define what is now known as modulation spaces, using BUPUs instead of the dyadic (smooth) decompositions of unity for Besov spaces. One can however derive many similar properties in this way, combining the two first principles. The core part of this series of lectures will be the discussion of methods which provide a wider scope and go for unifying principles, turning these examples into special cases of a (much) wider class of function spaces.
The first approach (which started with the paper with P. Groebner in 1985) was the development of the so-called decomposition spaces. Looking at the construction of Besov or modulation spaces strictly on the Fourier transform side, with a local norm arising from FL^p (image of L^p under the Fourier transform), it is clear that the geometry of the covering of the Fourier domain is the relevant difference. In the Besov case it is defined by dilation (providing dyadic rings at first sight, which can however be decomposed into a fixed number of cubes), while in the modulation space one can use translations. The theory of decomposition spaces shows that the (uniform) finite overlapping condition of the supports of the corresponding BAPUs (bounded admissible partitions of unity) is the critical assumption. There is a notion of equivalence of coverings, and two equivalent coverings define the same spaces and equivalent norms. The typical family of spaces arising in this way are the so-called alpha-modulation spaces, with balls growing slower that the dyadic ones, but faster than the uniform ones (no growth in this case, in fact).
The other joint view-point arises from the observation that the classical description can be replaced by a more group theoretical description. In the case of Besov spaces one can show that an equivalent (characteristic) norm on Besov spaces can be obtained by applying to the continuous wavelet transform (with respect to a suitable wavelet) a corresponding weighted mixed norm condition. Changing simply the order of integration one obtains continuous characterizations of Triebel-Lizorkin spaces. In a similar way modulation spaces (in full generality) can be characterized by the finiteness of again
weighted mixed norm spaces of the SFTT (Short-Time Fourier transform) over phase space.
Coorbit theory, developed jointly with K. Groechenig around 1989, establishes a common approach to these two (and many more, similar) situations. Starting from an so-called ``integrable group representation'' of a given locally compact group on a suitable Hilbert space one can construct a family of ``coorbit spaces'', which again is essentially closed under duality and interpolation. Using again facts concerning Wiener amalgam spaces over locally compact groups one comes up with atomic decompositions, which show essentially that any good enough atom can be used the represent the members of those coorbit spaces via convergent series expansions (unconditionally convergent).
报告人简介：Hans Georg Feichtinger是奥地利维也纳大学数学学院（Faculty of Mathematics, University of Vienna）教授。他目前是国际调和分析领域，尤其是时频分析领域的著名专家。目前担任国际著名杂志The Journal of Fourier Analysis and Applications (傅里叶分析及其应用杂志)的主编。Feichtinger教授目前已在顶尖期刊Transactions of the American Mathematical Society，Annales de l'Institut Henri Poincare C. Analyse Non Lineaire，Applied and Computational Harmonic Analysis，SIAM journal on mathematical analysis，Journal of Functional Analysis等上发表文章多篇。
