Interesting Eigenvectors of the Fourier Transform
It is well known that a function can be decomposed uniquely into the sum of an odd and an even function. This notion can be extended to the unique decomposition into the sum of four functions – two of which are even and two odd. These four functions are eigenvectors of the Fourier Transform with fou...
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| Format: | Article |
| Language: | en_US |
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Taylor & Francis
2011
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| Online Access: | http://hdl.handle.net/1721.1/67663 https://orcid.org/0000-0003-3434-391X |
| _version_ | 1826209300503068672 |
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| author | Horn, Berthold Klaus Paul |
| author2 | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory |
| author_facet | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Horn, Berthold Klaus Paul |
| author_sort | Horn, Berthold Klaus Paul |
| collection | MIT |
| description | It is well known that a function can be decomposed uniquely into the sum of an odd and an even function. This notion can be extended to the unique decomposition into the sum of four functions – two of which are even and two odd. These four functions are eigenvectors of the Fourier Transform with four different eigenvalues. That is, the Fourier transformof each of the four components is simply that component multiplied by the corresponding eigenvalue. Some eigenvectors of the discrete Fourier transform of particular interest find application in coding, communication and imaging. Some of the underlying mathematics goes back to the times of Carl Friedrich Gauss. |
| first_indexed | 2024-09-23T14:20:39Z |
| format | Article |
| id | mit-1721.1/67663 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T14:20:39Z |
| publishDate | 2011 |
| publisher | Taylor & Francis |
| record_format | dspace |
| spelling | mit-1721.1/676632022-09-29T08:49:13Z Interesting Eigenvectors of the Fourier Transform Horn, Berthold Klaus Paul Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Horn, Berthold Klaus Paul Horn, Berthold Klaus Paul It is well known that a function can be decomposed uniquely into the sum of an odd and an even function. This notion can be extended to the unique decomposition into the sum of four functions – two of which are even and two odd. These four functions are eigenvectors of the Fourier Transform with four different eigenvalues. That is, the Fourier transformof each of the four components is simply that component multiplied by the corresponding eigenvalue. Some eigenvectors of the discrete Fourier transform of particular interest find application in coding, communication and imaging. Some of the underlying mathematics goes back to the times of Carl Friedrich Gauss. 2011-12-14T14:45:37Z 2011-12-14T14:45:37Z 2010-06 Article http://purl.org/eprint/type/JournalArticle 0035-919X 2154-0098 http://hdl.handle.net/1721.1/67663 Horn, Berthold K.P. “Interesting Eigenvectors of the Fourier Transform.” Transactions of the Royal Society of South Africa 65.2 (2010) : 100-106. https://orcid.org/0000-0003-3434-391X en_US http://dx.doi.org/10.1080/0035919X.2010.510665 Transactions of the Royal Society of South Africa Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Taylor & Francis Prof. Horn |
| spellingShingle | Horn, Berthold Klaus Paul Interesting Eigenvectors of the Fourier Transform |
| title | Interesting Eigenvectors of the Fourier Transform |
| title_full | Interesting Eigenvectors of the Fourier Transform |
| title_fullStr | Interesting Eigenvectors of the Fourier Transform |
| title_full_unstemmed | Interesting Eigenvectors of the Fourier Transform |
| title_short | Interesting Eigenvectors of the Fourier Transform |
| title_sort | interesting eigenvectors of the fourier transform |
| url | http://hdl.handle.net/1721.1/67663 https://orcid.org/0000-0003-3434-391X |
| work_keys_str_mv | AT hornbertholdklauspaul interestingeigenvectorsofthefouriertransform |