Elements of postmodern harmonic analysis. (English) Zbl 1329.42031
Gröchenig, Karlheinz (ed.) et al., Operator-related function theory and time-frequency analysis. The Abel symposium 2012, Oslo, Norway, August 20–24, 2012. Cham: Springer (ISBN 978-3-319-08556-2/hbk; 978-3-319-08557-9/ebook). Abel Symposia 9, 77-105 (2015).
The author describes his vision of the so-called “Postmodern Harmonic Analysis” focussing the presentation on the concept of Conceptual Harmonic Analysis (CHA) proposed as a counterpart to abstract and applied resp. computational harmonic analysis. This concept constitutes a more integrative approach based on the emphasis of the connections between the different settings in addition to the analogy already provided by the Abstract Harmonic Analysis (AHA) perspective, pointing out the qualitative aspects. While AHA is helpful for the technical differences between the different settings, it is only establishing the analogy between different groups.
The author suggests to view Harmonic Analysis as a natural subfield of functional analysis, dealing with Banach spaces of functions and distributions, with a variety of group actions on and between them. This constitutes a counterpart to the classical point of view of “Modern Harmonic Analysis”, where the role of \(L^{p}\) spaces, maximal functions, among other concepts, is emphasized.
A very nice historical perspective is presented both for the AHA and CHA with the Fourier transform as a key element. Two paths apparently divergent are considered taking into account the Fast Fourier transform (FFT) as a meeting point in both directions. In other words, does the FFT allow to compute a good approximation of the Fourier transform for “nice” functions? What is the connection between the FFT of a sequence of regular samples of a function and the corresponding samples of its Fourier transform? The distributional setting allows to deal with finite signals as periodic and discrete signals, resp. as discrete periodic measures which will converge to a continuous limit in a natural way.
A naive comparison with number systems is presented. By using an axiomatic request based on the Banach Gelfand triple \((\mathbf{S}_{0}, \mathbf{L}^{2}, \mathbf{S}'_{0})\) (G), where \(\mathbf{S}_{0}(G)\) is the Segal algebra associated with a locally compact abelian group \(G\). Three viewpoints appear in such a way at the lower level typically descriptions can be taken literal, the intermediate level allows the preservation of energy norms and scalar products and, finally, the dual space is large enough to include objects like Dirac measures, Dirac combs or pure frequencies. The use of the Banach Gelfand triple is illustrated through the example of the Fourier transform. Fourier inversion and summability as well as the celebrated Poisson formula are also analyzed in the framework of the distributional setting and constitute a proper setting for the otherwise vague concepts using the classical notion of \(w^*\)-convergence in the dual space \(\mathbf{S}'_{0}(G)\).
As a last oriented message to young researchers, the author suggests to explore new territories where computations or simulations involving Fourier methods on a very intuitive and practical way, without reliable and detailed theoretical analysis behind, but where new mathematics are needed and new problems and concepts must be developed.
The lecture of this kind of contributions constitutes a healthful and stimulating exercise for people working in borderlines of pure mathematics, computation and simulation, and applied sciences.
For the entire collection see [Zbl 1304.00050].
The author suggests to view Harmonic Analysis as a natural subfield of functional analysis, dealing with Banach spaces of functions and distributions, with a variety of group actions on and between them. This constitutes a counterpart to the classical point of view of “Modern Harmonic Analysis”, where the role of \(L^{p}\) spaces, maximal functions, among other concepts, is emphasized.
A very nice historical perspective is presented both for the AHA and CHA with the Fourier transform as a key element. Two paths apparently divergent are considered taking into account the Fast Fourier transform (FFT) as a meeting point in both directions. In other words, does the FFT allow to compute a good approximation of the Fourier transform for “nice” functions? What is the connection between the FFT of a sequence of regular samples of a function and the corresponding samples of its Fourier transform? The distributional setting allows to deal with finite signals as periodic and discrete signals, resp. as discrete periodic measures which will converge to a continuous limit in a natural way.
A naive comparison with number systems is presented. By using an axiomatic request based on the Banach Gelfand triple \((\mathbf{S}_{0}, \mathbf{L}^{2}, \mathbf{S}'_{0})\) (G), where \(\mathbf{S}_{0}(G)\) is the Segal algebra associated with a locally compact abelian group \(G\). Three viewpoints appear in such a way at the lower level typically descriptions can be taken literal, the intermediate level allows the preservation of energy norms and scalar products and, finally, the dual space is large enough to include objects like Dirac measures, Dirac combs or pure frequencies. The use of the Banach Gelfand triple is illustrated through the example of the Fourier transform. Fourier inversion and summability as well as the celebrated Poisson formula are also analyzed in the framework of the distributional setting and constitute a proper setting for the otherwise vague concepts using the classical notion of \(w^*\)-convergence in the dual space \(\mathbf{S}'_{0}(G)\).
As a last oriented message to young researchers, the author suggests to explore new territories where computations or simulations involving Fourier methods on a very intuitive and practical way, without reliable and detailed theoretical analysis behind, but where new mathematics are needed and new problems and concepts must be developed.
The lecture of this kind of contributions constitutes a healthful and stimulating exercise for people working in borderlines of pure mathematics, computation and simulation, and applied sciences.
For the entire collection see [Zbl 1304.00050].
Reviewer: Francisco Marcellán (Leganes)
MSC:
| 42C15 | General harmonic expansions, frames |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
Keywords:
classical Fourier analysis; Fourier transform; fast Fourier transform; Banach Gelfand triple; Segal algebra; Gabor analysis; \(w^*\)-convergenceSoftware:
MatlabReferences:
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