On the Behavior of the Threshold Operator for Bandlimited Functions
One interesting question is how the good local approximation behavior of the Shannon sampling series for the Paley–Wiener space PW[1 over π] is affected if the samples are disturbed by the non-linear threshold operator. This operator, which is important in many applications, sets all samples whose a...
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| Format: | Article |
| Language: | en_US |
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Springer-Verlag
2014
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| Online Access: | http://hdl.handle.net/1721.1/85974 |
| _version_ | 1811091890582323200 |
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| author | Boche, Holger Monich, Ullrich |
| author2 | Massachusetts Institute of Technology. Research Laboratory of Electronics |
| author_facet | Massachusetts Institute of Technology. Research Laboratory of Electronics Boche, Holger Monich, Ullrich |
| author_sort | Boche, Holger |
| collection | MIT |
| description | One interesting question is how the good local approximation behavior of the Shannon sampling series for the Paley–Wiener space PW[1 over π] is affected if the samples are disturbed by the non-linear threshold operator. This operator, which is important in many applications, sets all samples whose absolute value is smaller than some threshold to zero. In this paper we analyze a generalization of this problem, in which not the Shannon sampling series is disturbed by the threshold operator but a more general system approximation process, were a stable linear time-invariant system is involved. We completely characterize the stable linear time-invariant systems that, for some functions in PW[1 over π], lead to a diverging approximation process as the threshold is decreased to zero. Further, we show that if there exists one such function then the set of functions for which divergence occurs is in fact a residual set. We study the pointwise behavior as well as the behavior of the L[superscript ∞]-norm of the approximation process. It is known that oversampling does not lead to stable approximation processes in the presence of thresholding. An interesting open problem is the characterization of the systems that can be stably approximated with oversampling. |
| first_indexed | 2024-09-23T15:09:38Z |
| format | Article |
| id | mit-1721.1/85974 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T15:09:38Z |
| publishDate | 2014 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/859742022-10-02T00:59:32Z On the Behavior of the Threshold Operator for Bandlimited Functions Boche, Holger Monich, Ullrich Massachusetts Institute of Technology. Research Laboratory of Electronics Monich, Ullrich Monich, Ullrich One interesting question is how the good local approximation behavior of the Shannon sampling series for the Paley–Wiener space PW[1 over π] is affected if the samples are disturbed by the non-linear threshold operator. This operator, which is important in many applications, sets all samples whose absolute value is smaller than some threshold to zero. In this paper we analyze a generalization of this problem, in which not the Shannon sampling series is disturbed by the threshold operator but a more general system approximation process, were a stable linear time-invariant system is involved. We completely characterize the stable linear time-invariant systems that, for some functions in PW[1 over π], lead to a diverging approximation process as the threshold is decreased to zero. Further, we show that if there exists one such function then the set of functions for which divergence occurs is in fact a residual set. We study the pointwise behavior as well as the behavior of the L[superscript ∞]-norm of the approximation process. It is known that oversampling does not lead to stable approximation processes in the presence of thresholding. An interesting open problem is the characterization of the systems that can be stably approximated with oversampling. 2014-03-28T19:44:56Z 2014-03-28T19:44:56Z 2013-01 2012-11 Article http://purl.org/eprint/type/JournalArticle 1069-5869 1531-5851 http://hdl.handle.net/1721.1/85974 Boche, Holger, and Ullrich J. Mönich. “On the Behavior of the Threshold Operator for Bandlimited Functions.” J Fourier Anal Appl 19, no. 1 (February 2013): 1–19. en_US http://dx.doi.org/10.1007/s00041-012-9254-1 Journal of Fourier Analysis and Applications Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Springer-Verlag Monich |
| spellingShingle | Boche, Holger Monich, Ullrich On the Behavior of the Threshold Operator for Bandlimited Functions |
| title | On the Behavior of the Threshold Operator for Bandlimited Functions |
| title_full | On the Behavior of the Threshold Operator for Bandlimited Functions |
| title_fullStr | On the Behavior of the Threshold Operator for Bandlimited Functions |
| title_full_unstemmed | On the Behavior of the Threshold Operator for Bandlimited Functions |
| title_short | On the Behavior of the Threshold Operator for Bandlimited Functions |
| title_sort | on the behavior of the threshold operator for bandlimited functions |
| url | http://hdl.handle.net/1721.1/85974 |
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