An algebraic perspective on multivariate tight wavelet frames
Recent advances in real algebraic geometry and in the theory of polynomial optimization are applied to answer some open questions in the theory of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary E...
| Main Authors: | , , , |
|---|---|
| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2013
|
| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/98752 http://hdl.handle.net/10220/17339 |
| _version_ | 1811677091190538240 |
|---|---|
| author | Stöckler, Joachim. Charina, Maria. Putinar, Mihai. Scheiderer, Claus. |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Stöckler, Joachim. Charina, Maria. Putinar, Mihai. Scheiderer, Claus. |
| author_sort | Stöckler, Joachim. |
| collection | NTU |
| description | Recent advances in real algebraic geometry and in the theory of polynomial optimization are applied to answer some open questions in the theory of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) are given in terms of Hermitian sums of squares of certain nonnegative Laurent polynomials and in terms of semidefinite programming. These formulations merge recent advances in real algebraic geometry and wavelet frame theory and lead to an affirmative answer to the long-standing open question of the existence of tight wavelet frames in dimension d=2. They also provide, for every d, efficient numerical methods for checking the existence of tight wavelet frames and for their construction. A class of counterexamples in dimension d=3 show that, in general, the so-called sub-QMF condition is not sufficient for the existence of tight wavelet frames. Stronger sufficient conditions for determining the existence of tight wavelet frames in dimension d≥3 are derived. The results are illustrated on several examples. |
| first_indexed | 2024-10-01T02:31:51Z |
| format | Journal Article |
| id | ntu-10356/98752 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T02:31:51Z |
| publishDate | 2013 |
| record_format | dspace |
| spelling | ntu-10356/987522020-03-07T12:37:16Z An algebraic perspective on multivariate tight wavelet frames Stöckler, Joachim. Charina, Maria. Putinar, Mihai. Scheiderer, Claus. School of Physical and Mathematical Sciences DRNTU::Science::Mathematics Recent advances in real algebraic geometry and in the theory of polynomial optimization are applied to answer some open questions in the theory of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) are given in terms of Hermitian sums of squares of certain nonnegative Laurent polynomials and in terms of semidefinite programming. These formulations merge recent advances in real algebraic geometry and wavelet frame theory and lead to an affirmative answer to the long-standing open question of the existence of tight wavelet frames in dimension d=2. They also provide, for every d, efficient numerical methods for checking the existence of tight wavelet frames and for their construction. A class of counterexamples in dimension d=3 show that, in general, the so-called sub-QMF condition is not sufficient for the existence of tight wavelet frames. Stronger sufficient conditions for determining the existence of tight wavelet frames in dimension d≥3 are derived. The results are illustrated on several examples. 2013-11-06T05:25:55Z 2019-12-06T19:59:19Z 2013-11-06T05:25:55Z 2019-12-06T19:59:19Z 2013 2013 Journal Article Charina, M., Putinar, M., Scheiderer, C., & Stöckler, J. (2013). An algebraic perspective on multivariate tight wavelet frames. Constructive Approximation, 38(2), 253-276. https://hdl.handle.net/10356/98752 http://hdl.handle.net/10220/17339 10.1007/s00365-013-9191-5 en Constructive approximation |
| spellingShingle | DRNTU::Science::Mathematics Stöckler, Joachim. Charina, Maria. Putinar, Mihai. Scheiderer, Claus. An algebraic perspective on multivariate tight wavelet frames |
| title | An algebraic perspective on multivariate tight wavelet frames |
| title_full | An algebraic perspective on multivariate tight wavelet frames |
| title_fullStr | An algebraic perspective on multivariate tight wavelet frames |
| title_full_unstemmed | An algebraic perspective on multivariate tight wavelet frames |
| title_short | An algebraic perspective on multivariate tight wavelet frames |
| title_sort | algebraic perspective on multivariate tight wavelet frames |
| topic | DRNTU::Science::Mathematics |
| url | https://hdl.handle.net/10356/98752 http://hdl.handle.net/10220/17339 |
| work_keys_str_mv | AT stocklerjoachim analgebraicperspectiveonmultivariatetightwaveletframes AT charinamaria analgebraicperspectiveonmultivariatetightwaveletframes AT putinarmihai analgebraicperspectiveonmultivariatetightwaveletframes AT scheidererclaus analgebraicperspectiveonmultivariatetightwaveletframes AT stocklerjoachim algebraicperspectiveonmultivariatetightwaveletframes AT charinamaria algebraicperspectiveonmultivariatetightwaveletframes AT putinarmihai algebraicperspectiveonmultivariatetightwaveletframes AT scheidererclaus algebraicperspectiveonmultivariatetightwaveletframes |