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On Inverses of the Dirac Comb

1
German Aerospace Center (DLR), Microwaves and Radar Institute, 82234 Wessling, Germany
2
Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
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Author to whom correspondence should be addressed.
Mathematics 2019, 7(12), 1196; https://doi.org/10.3390/math7121196
Received: 8 November 2019 / Revised: 30 November 2019 / Accepted: 2 December 2019 / Published: 6 December 2019
We determine tempered distributions which convolved with a Dirac comb yield unity and tempered distributions, which multiplied with a Dirac comb, yield a Dirac delta. Solutions of these equations have numerous applications. They allow the reversal of discretizations and periodizations applied to tempered distributions. One of the difficulties is the fact that Dirac combs cannot be multiplied or convolved with arbitrary functions or distributions. We use a theorem of Laurent Schwartz to overcome this difficulty and variants of Lighthill’s unitary functions to solve these equations. The theorem we prove states that double-sided (time/frequency) smooth partitions of unity are required to neutralize discretizations and periodizations on tempered distributions. View Full-Text
Keywords: tempered distribution; partition of unity; unitary function; Poisson summation formula; Heisenberg uncertainty principle; Paley-Wiener function; Whittaker-Kotel’nikov-Shannon (WKS) tempered distribution; partition of unity; unitary function; Poisson summation formula; Heisenberg uncertainty principle; Paley-Wiener function; Whittaker-Kotel’nikov-Shannon (WKS)
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MDPI and ACS Style

Fischer, J.V.; Stens, R.L. On Inverses of the Dirac Comb. Mathematics 2019, 7, 1196.

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