Choosing Function Spaces in Harmonic Analysis

Abstract

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 result s. 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.