On the Gabor frame set for compactly supported continuous functions
Abstract We identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the known obstructions. Easy verifiable sufficient conditi...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2016-03-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-016-1021-4 |
| _version_ | 1818951199759532032 |
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| author | Ole Christensen Hong Oh Kim Rae Young Kim |
| author_facet | Ole Christensen Hong Oh Kim Rae Young Kim |
| author_sort | Ole Christensen |
| collection | DOAJ |
| description | Abstract We identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the known obstructions. Easy verifiable sufficient conditions for a function to belong to the class are derived, and it is shown that the B-splines B N $B_{N}$ , N ≥ 2 $N\ge2$ , and certain ‘continuous and truncated’ versions of several classical functions (e.g., the Gaussian and the two-sided exponential function) belong to the class. The sufficient conditions for the frame property guarantees the existence of a dual window with a prescribed size of the support. |
| first_indexed | 2024-12-20T09:30:41Z |
| format | Article |
| id | doaj.art-395ba5a769bb47e4a32db62e09d8ea73 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-20T09:30:41Z |
| publishDate | 2016-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-395ba5a769bb47e4a32db62e09d8ea732022-12-21T19:45:05ZengSpringerOpenJournal of Inequalities and Applications1029-242X2016-03-012016111710.1186/s13660-016-1021-4On the Gabor frame set for compactly supported continuous functionsOle Christensen0Hong Oh Kim1Rae Young Kim2Department of Applied Mathematics and Computer Science, Technical University of DenmarkDivision of General Studies, UNISTDepartment of Mathematics, Yeungnam UniversityAbstract We identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the known obstructions. Easy verifiable sufficient conditions for a function to belong to the class are derived, and it is shown that the B-splines B N $B_{N}$ , N ≥ 2 $N\ge2$ , and certain ‘continuous and truncated’ versions of several classical functions (e.g., the Gaussian and the two-sided exponential function) belong to the class. The sufficient conditions for the frame property guarantees the existence of a dual window with a prescribed size of the support.http://link.springer.com/article/10.1186/s13660-016-1021-4Gabor framesframe setB-splines |
| spellingShingle | Ole Christensen Hong Oh Kim Rae Young Kim On the Gabor frame set for compactly supported continuous functions Journal of Inequalities and Applications Gabor frames frame set B-splines |
| title | On the Gabor frame set for compactly supported continuous functions |
| title_full | On the Gabor frame set for compactly supported continuous functions |
| title_fullStr | On the Gabor frame set for compactly supported continuous functions |
| title_full_unstemmed | On the Gabor frame set for compactly supported continuous functions |
| title_short | On the Gabor frame set for compactly supported continuous functions |
| title_sort | on the gabor frame set for compactly supported continuous functions |
| topic | Gabor frames frame set B-splines |
| url | http://link.springer.com/article/10.1186/s13660-016-1021-4 |
| work_keys_str_mv | AT olechristensen onthegaborframesetforcompactlysupportedcontinuousfunctions AT hongohkim onthegaborframesetforcompactlysupportedcontinuousfunctions AT raeyoungkim onthegaborframesetforcompactlysupportedcontinuousfunctions |