Choosing function spaces in harmonic analysis. (English) Zbl 1377.42028
Balan, Radu (ed.) et al., Excursions in harmonic analysis, Volume 4. The February Fourier talks at the Norbert Wiener Center, College Park, MD, USA, 2002–2013. Cham: Birkhäuser/Springer (ISBN 978-3-319-20187-0/hbk; 978-3-319-20188-7/ebook). Applied and Numerical Harmonic Analysis, 65-101 (2015).
Summary: Without doubt function spaces play a crucial role in harmonic analysis. Moreover many function spaces arose from questions in Fourier analysis. Here we would like to draw the attention to the question: “Which function spaces are useful for which problem?” Looking into the books on Fourier analysis one may come to the conclusion that it has almost become a dogma that one has to study Lebesgue integration and \(\boldsymbol L^p\)-spaces properly in order to have a chance to understand the Fourier transform. For the study of PDEs one has to resort to Sobolev spaces, or the Schwartz theory of tempered distribution, where suddenly Lebesgue spaces play a minor role. Finally, numerical applications make use of the FFT (fast Fourier transform), which has a vast range of applications in signal processing, but in the corresponding engineering books neither Lebesgue nor Schwartz theory plays a significant role. “Strange objects” like a Dirac distribution or Dirac combs (used to prove sampling theorems) are often used in a mysterious way, divergent integrals are giving magically useful results. Cautious authors provide some hint to the fact that “mathematicians know how to give those objects a correct meaning”. More recently, other function systems such as wavelets and Gabor expansions have come into the picture as well as the theory of spline-type spaces and irregular sampling have gained importance. In this context the classical function spaces such as \(\boldsymbol L^p\)-spaces or even Sobolev and Besov spaces are not really helpful and do not allow to derive good results. Instead, Wiener amalgam spaces and modulation spaces are playing a major role there. It is the purpose of this chapter to initiate a discussion about the “information content” of function spaces and their “usefulness”. In fact, even the discussion of the meaning of such words may be a stimulating challenge for the community and worth the effort. When I illustrate this circle of problems in the context of time-frequency analysis, but also with respect to potential usefulness for the teaching of the subject to engineers, I do not mean to specifically promote my favorite spaces, but rather show – in a context very familiar to me – how I want to understand the question. Of course such a description is subjective, while, on the other hand, it provides a kind of experience report, indicating that I personally found those spaces useful for many of the things I have been doing in the last decades. It also favors obviously less popular spaces over the well-known and frequently used ones.
For the entire collection see [Zbl 1357.42001].
For the entire collection see [Zbl 1357.42001].
MSC:
| 42B35 | Function spaces arising in harmonic analysis |
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