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THE Banach Gelfand Triple and Its Role in Classical Fourier Analysis and Operator Theory

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THE Banach Gelfand Triple and Its Role in Classical Fourier Analysis and Operator Theory INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Numerical Harmonic Analysis Group THE Banach Gelfand Triple and its role in Classical Fourier Analysis and Operator Theory Hans G. Feichtinger, Univ. Vienna [email protected] www.nuhag.eu TALKS at www.nuhag.eu/talks Internet Presentation f. Southern Federal Unverisity Regional Mathematical Center, 25.06.2020 Supervisors: A.Karapetyants, V. Kravchenko Orig.: Invited Talk, IWOTA 2019, 23.07.2019 THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger, Univ. Vienna [email protected] www.nuhag.eu TALKS at www.nuhag.eu/talks INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. A Zoo of Banach Spaces for Fourier Analysis THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. NEW Abstract (short version), Rostov-on-DonI Psychological Viewpoint: We are coming close to 200 years of the Fourier transform. It is an important and indispensable tool for many areas of mathematica analysis, function spaces, PDE and other areas. It has been crucial in the creation ofnew tools, such asLebesgue integration, Functional Analysis, Hilbert spaces, and more recentlyframes (for Gabor expansions). Analogies: The DFT (realized as FFT) is a simple orthogonal n (better unitary) change of bases, linear mappings on C are just d matrices, but when it comes to continuous variables (FT over R ) only physicists call (δx )x2R a \continuous basis" for (generalized) d functions on R . THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. NEW Abstract (short version), Rostov-on-DonII Technically Speaking: Engineers work with Dirac impulses, Dirac combs, and more recently to some extent the theory of tempered 0 d distributions (S (R )) by Laurent Schwartz is taken into account. \Non-existent" integrals and identities such as Z 1 2πits e ds = δ0(t) −∞ are used to \derive" the Fourier Inversion Theorem and other mysterious manipulation give engineering students the impression that Fourier Analysis is a mystic game with fancy symbols which is only understood correctly by mathematicians. On the other hand they only care about applications, and mathematicians are often not helpful (e.g. because they are viewed as too pedantic). THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. NEW Abstract (short version), Rostov-on-Don III THESIS of my TALK S L2 S0 d The Banach Gelfand Triple ( 0; ; 0)(R ) (which arose in the context of Time-Frequency Analysis) isa simple and useful tool, both for the derivation of mathematically valid theorems AND for teaching relevant concepts to engineers and physicists (and of course mathematicians, interested in applications!). In this context the basic terms of an introductory course on Linear System's Theory can be explained properly: Translation invariant systems viewed as linear operators, which can be described as convolution operator by some impulse response, whose Fourier transform is well defined (and is called transfer function), and S d S0 d there is a kernel theorem: Operators T : 0(R ) ! 0(R ) S0 2d have a \matrix representation" using some σ 2 0(R ). THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Earlier Abstract (for later reading), IWOTA19 talk It is the purpose of this presentation to explain certain aspects of Classical Fourier Analysis from the point of view of distribution theory. The setting of the so-called Banach Gelfand Triple S L2 S0 d ( 0; ; 0)(R ) starts from a particular Segal algebra S d 0(R ); k · kS0 of continuous and Riemann integrable functions. It is Fourier invariant and thus an extended Fourier transform can 0 d be defined for (S ( ); k · kS0 ) the space of so-called mild 0 R 0 p d distributions. Any of the L -spaces contains S0(R ) and is S0 d embedded into 0(R ), for p 2 [1; 1]. We will show how this setting of Banach Gelfand triples resp. rigged Hilbert spaces allows to provide a conceptual appealing approach to most classical parts of Fourier analysis. In contrast to the Schwartz theory of tempered distributions it is expected that the mathematical tools can be also explained in more detail to engineers and physicists. THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. Our Aim: Popularizing Banach Gelfand Triples According to the title I have to first explain the term Banach Gelfand Triples, with the specific emphasis on the BGTr S L2 S0 d S d ( 0; ; 0)(R ), arising from the Segal algebra 0(R ); k · kS0 (as a space of test-functions), alias the modulation spaces 1 d 2 d 1 d (M (R ); k · kM1 ), L (R ); k · k2 and (M (R ); k · kM1 ). Hence I will describe them, provide a selection of different characterizations (there are many of them!) and properties. Finally I will come to the main part, namely applications or use of this (!natural) concept in the framework of classical analysis. THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. The structure of the talk II Compared to other talks I leave out the main application area of this BGTr, namely so-called Gabor Analysis or Time-Frequency Analysis (TFA). Specific aspects of this have been described in my talks in the Gabor Analysis section of this conference. S In fact, the Segal algebra 0(G); k · kS0 , which is well defined on LCA (locally compact groups) has been introduced already 1979, at a winter-school in Vienna, organized by H. Reiter (my advisor). S L2 S0 This makes the BGTr ( 0; ; 0)(G) useful for applications in Abstract Harmonic Analysis. THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. The Life-Cycle of Tools New tools often go through the following development steps: 1 A new tool is developed in order to solve a concrete problem; 2 Once it is observed that the new tool can be applied in other contexts the \trick" becomes a method, and receives a name; 3 Subsequent analysis of the methods specifies the ingredients and conditions of applications; optimization; 4 Exploration of the maximal range of applications; 5 The limitations of the tool are discovered, and so on... 6 Change of view/problems creates the need for new tools! 7 One may find that for various applications a simplified version suffices. THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. The Standard spaces L1; L2; L1 and FL1 THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger 1 2 1 Abbildung: The diagrams showing L ; L ; C0; FL etc. INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. The Universe of Banach Spaces of Tempered Distributions THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Hans G. Feichtinger INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. The Essence: Reducing to the Banach Gelfand Triple THE Banach Gelfand Triple and its rolein Classical Fourier Analysis and Operator Theory Abbildung: Our GOAL:Hans Simplification: G. Feichtinger COMPARISON with Q ⊂ R ⊂ C INTRO The Segal Algebra Key-Players for TF GOALS! Summability Kernels Multipliers Regularization Banach Gelfand Spectral Analysis Shannon Sampling Banach Gelfand Triples Kernel Thm. frametitle: Bibliography for Banach Gelfand TriplesI E. Cordero, H. G. Feichtinger, and F. Luef. Banach Gelfand triples for Gabor analysis. In Pseudo-differential Operators, volume 1949 of Lecture Notes in Mathematics, pages 1{33. Springer, Berlin, 2008. H. G. Feichtinger and M. S. Jakobsen. Distribution theory by Riemann integrals. Mathematical Modelling, Optimization, Analytic and Numerical Solutions, pages 33{76, 2020. H. G. Feichtinger. A sequential approach to mild distributions. Axioms, 9(1):1{25, 2020. H. G. Feichtinger. Classical Fourier Analysis via mild distributions. MESA, Non-linear Studies, 26(4):783{804, 2019. H. G. Feichtinger. Banach Gelfand Triples and some Applications in Harmonic Analysis. In J. Feuto and M. Essoh, editors, Proc. Conf. Harmonic Analysis (Abidjan, May 2018), pages 1{21, 2018.
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