On the Reversibility of Discretization
“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...
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| Format: | Article |
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MDPI AG
2020-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/4/619 |
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| author | Jens V. Fischer Rudolf L. Stens |
| author_facet | Jens V. Fischer Rudolf L. Stens |
| author_sort | Jens V. Fischer |
| collection | DOAJ |
| description | “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. |
| first_indexed | 2024-03-10T20:24:56Z |
| format | Article |
| id | doaj.art-33c1672c0f164ad9a9bb67e8ba10cb06 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T20:24:56Z |
| publishDate | 2020-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-33c1672c0f164ad9a9bb67e8ba10cb062023-11-19T21:52:46ZengMDPI AGMathematics2227-73902020-04-018461910.3390/math8040619On the Reversibility of DiscretizationJens V. Fischer0Rudolf L. Stens1German Aerospace Center (DLR), Microwaves and Radar Institute, 82234 Wessling, GermanyLehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany“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.https://www.mdpi.com/2227-7390/8/4/619regularizationlocalizationtruncationcutofffinitizationentirization |
| spellingShingle | Jens V. Fischer Rudolf L. Stens On the Reversibility of Discretization Mathematics regularization localization truncation cutoff finitization entirization |
| title | On the Reversibility of Discretization |
| title_full | On the Reversibility of Discretization |
| title_fullStr | On the Reversibility of Discretization |
| title_full_unstemmed | On the Reversibility of Discretization |
| title_short | On the Reversibility of Discretization |
| title_sort | on the reversibility of discretization |
| topic | regularization localization truncation cutoff finitization entirization |
| url | https://www.mdpi.com/2227-7390/8/4/619 |
| work_keys_str_mv | AT jensvfischer onthereversibilityofdiscretization AT rudolflstens onthereversibilityofdiscretization |