Thoughts on numerical and conceptual harmonic analysis

If one compares the literature on abstract Harmonic Analysis or publications with the word “Fourier Analysis” in the title with books describing the foundations of digital signal processing or systems theory one may easily get the impression that they describe two different worlds having little in common except for some vocabulary, indicating their joint roots. It is the purpose of this article to shed some light on the existing and sometimes buried interconnections between these two branches of Fourier Analysis. We suggest to take a broader perspective on Harmonic Analysis, re-emphasizing how they are tied together in many ways and how numerical experiments may help to understand concepts of Harmonic Analysis. Based on our experience we firmly believe that Fourier Analysis provides opportunities to relate relevant mathematical theorems to valuable algorithms, which also have to be properly implemented (typically making use of the FFT, the Fast Fourier Transform). Thus this note tries to bridge the gap between pure mathematics and real-world engineering applications. In a previous paper (H. G. Feichtinger, Elements of Postmodern Harmonic Analysis, pages 1 – 27, Springer, 2015) the author has already outlined some ideas in this direction, by introducing the idea of Conceptual Harmonic Analysis, as a link between the two worlds. The best way to describe the connections between the two worlds is by means of generalized functions (often called distributions). We will indicate that the Banach Gelfand triple (S,L,S′)(ℝd), which is based on a particular Banach algebra of continuous functions, namely the Segal algebra (S(ℝd),∥⋅∥S0), provides convenient vehicle to express many otherwise “soft transition” in a distributional sense. In the context of the space S(ℝd) of test functions approximation by finite sequences makes perfect sense and one can justify many transitions between the two worlds (continuous versus finite signals) in a clear mathematical context (see H. G. Feichtinger and N. Kaiblinger, Quasi-interpolation in the Fourier algebra, J. Approx. Theory, 144(1):103–118, 2007).The current text also contains various considerations of a general nature, which are supposed to stimulate discussions and reflection of the work done in the field of abstract and applied Harmonic Analysis in general.