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Open AccessArticle

On the Reversibility of Discretization

by Jens V. Fischer 1,2,* and Rudolf L. Stens 2,*
1
German Aerospace Center (DLR), Microwaves and Radar Institute, 82234 Wessling, Germany
2
Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
*
Authors to whom correspondence should be addressed.
Mathematics 2020, 8(4), 619; https://doi.org/10.3390/math8040619
Received: 24 March 2020 / Revised: 10 April 2020 / Accepted: 12 April 2020 / Published: 17 April 2020
“Discretization” usually denotes the operation of mapping continuous functions to infinite or finite sequences of discrete values. It may also mean to map the operation itself from one that operates on functions to one that operates on infinite or finite sequences. Advantageously, these two meanings coincide within the theory of generalized functions. Discretization moreover reduces to a simple multiplication. It is known, however, that multiplications may fail. In our previous studies, we determined conditions such that multiplications hold in the tempered distributions sense and, hence, corresponding discretizations exist. In this study, we determine, vice versa, conditions such that discretizations can be reversed, i.e., functions can be fully restored from their samples. The classical Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem is just one particular case in one of four interwoven symbolic calculation rules deduced below. View Full-Text
Keywords: regularization; localization; truncation; cutoff; finitization; entirization; cyclic dualities; multiplication of distributions; square of the Dirac delta; Whittaker-Kotel’nikov-Shannon regularization; localization; truncation; cutoff; finitization; entirization; cyclic dualities; multiplication of distributions; square of the Dirac delta; Whittaker-Kotel’nikov-Shannon
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MDPI and ACS Style

Fischer, J.V.; Stens, R.L. On the Reversibility of Discretization. Mathematics 2020, 8, 619.

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