Abstract
It is an open problem whether any pair of Bessel sequences with wavelet structure can be extended to a pair of dual frames by adding a pair of singly generated wavelet systems. We consider the particular case where the given wavelet systems are generated by the multiscale setup with trigonometric masks and provide a positive answer under extra assumptions. We also identify a number of conditions that are necessary for the extension to dual (multi-) wavelet frames with any number of generators, and show that they imply that an extension with two pairs of wavelet systems is possible. Along the way we provide examples that demonstrate the extra flexibility in the extension to dual pairs of frames compared with the more popular extensions to tight frames.
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References
Borup, L., Gribonval, R., Nielsen, M.: Bi-framelet systems with few vanishing moments characterize Besov spaces. Appl. Comput. Harmon. Anal. 17, 3–28 (2004)
Bownik, M., Weber, E.: Affine frames, GMRA’s, and the canonical dual. Studia Math. 159, 453–479 (2003)
Charina, M., Putinar, M., Scheiderer, C., Stöckler, J.: A real algebra perspective on multivariate tight wavelet frames. Constr. Approx. 38, 253–276 (2013)
Christensen, O.: An introduction to frames and Riesz bases Birkhäuser (2003)
Christensen, O., Kim, H.O., Kim, R.Y.: Extensions of Bessel sequences to dual pairs of frames. Appl. Comput. Harmon. Anal. 34, 224–233 (2013)
Christensen, O., Kim, H.O., Kim, R.Y.: On Parseval wavelet frames with two or three generators via the unitary extension principle. Can. Math. Bull. 57, 254–263 (2014)
Chui, C.: Wavelets - a tutorial in theory and practice. Academic Press, San Diego (1992)
Chui, C., Shi, X.L.: Inequalities of Littlewood-Paley type for frames and wavelets. SIAM J. Math. Anal. 24, 263–277 (1993)
Chui, C., He, W., Stöckler, J.: Compactly supported tight and sibling frames with maximum vanishing moments. Appl. Comput. Harmon. Anal. 13, 224–262 (2002)
Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)
Daubechies, I., Han, B.: Pairs of dual wavelet frames from any two refinable functions. Constr. Approx. 20, 325–352 (2004)
Daubechies, I., Han, B., Ron, A., Shen, Z.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)
Ehler, M.: On multivariate compactly supported bi-frames. J. Fourier Anal. Appl. 13, 511–532 (2007)
Han, B., Mo, Q.: Splitting a matrix of Laurent polynomials with symmetry and its applications to symmetric framelet filter banks. SIAM J. Matrix Anal. Appl. 26, 97–124 (2004)
Han, B., Shen, Z.: Dual wavelet frames and Riesz bases in Sobolev spaces. Constr. Approx. 29, 369–406 (2009)
Han, D.: Dilations and completions for Gabor systems. J. Fourier Anal. Appl. 15, 201–217 (2009)
Petukhov, A.: Symmetric framelets. Constr. Approx. 19, 309–328 (2003)
Ron, A., Shen, Z.: Affine systems in L 2(\(\mathbb {R}^{d}\)) II: dual systems. J. Fourier Anal. Appl. 3, 617–637 (1997)
Ron, A., Shen, Z.: Compactly supported tight affine spline frames in L 2(R d). Math. Comp. 67, 191–207 (1998)
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Communicated by: Yang Wang
This work was supported by the 2012 Yeungnam University Research Grant.
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Christensen, O., Kim, H.O. & Kim, R.Y. On extensions of wavelet systems to dual pairs of frames. Adv Comput Math 42, 489–503 (2016). https://doi.org/10.1007/s10444-015-9432-1
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DOI: https://doi.org/10.1007/s10444-015-9432-1