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Article

A Sequential Approach to Mild Distributions

Faculty of Mathematics, University of Vienna, Vienna 1090, Austria
Axioms 2020, 9(1), 25; https://doi.org/10.3390/axioms9010025
Received: 4 January 2020 / Revised: 5 February 2020 / Accepted: 6 February 2020 / Published: 24 February 2020
(This article belongs to the Special Issue Harmonic Analysis and Applications)
The Banach Gelfand Triple ( S 0 , L 2 , S 0 ) ( R d ) consists of S 0 ( R d ) , · S 0 , a very specific Segal algebra as algebra of test functions, the Hilbert space L 2 ( R d ) , · 2 and the dual space S 0 ( R d ) , whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce S 0 ( R d ) , · S 0 and hence ( S 0 ( R d ) , · S 0 ) , the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short ECmiCS). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra S 0 ( R d ) , · S 0 can be used to establish this natural identification. View Full-Text
Keywords: tempered distribution; sequential approach; mild distribution; Banach Gelfand Triple; Feichtinger’s algebra; Fourier transform; w*-convergence; Short-Time Fourier Transform tempered distribution; sequential approach; mild distribution; Banach Gelfand Triple; Feichtinger’s algebra; Fourier transform; w*-convergence; Short-Time Fourier Transform
MDPI and ACS Style

Feichtinger, H.G. A Sequential Approach to Mild Distributions. Axioms 2020, 9, 25. https://doi.org/10.3390/axioms9010025

AMA Style

Feichtinger HG. A Sequential Approach to Mild Distributions. Axioms. 2020; 9(1):25. https://doi.org/10.3390/axioms9010025

Chicago/Turabian Style

Feichtinger, Hans G. 2020. "A Sequential Approach to Mild Distributions" Axioms 9, no. 1: 25. https://doi.org/10.3390/axioms9010025

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