Abstract
Matrix extension with symmetry is to find a unitary square matrix P of 2π-periodic trigonometric polynomials with symmetry such that the first row of P is a given row vector p of 2π-periodic trigonometric polynomials with symmetry satisfying \(\mathbf {p}\overline{\mathbf{p}}^{T}=1\) . Matrix extension plays a fundamental role in many areas such as electronic engineering, system sciences, wavelet analysis, and applied mathematics. In this paper, we shall solve matrix extension with symmetry by developing a step-by-step simple algorithm to derive a desired square matrix P from a given row vector p of 2π-periodic trigonometric polynomials with complex coefficients and symmetry. As an application of our algorithm for matrix extension with symmetry, for any dilation factor M, we shall present two families of compactly supported symmetric orthonormal complex M-wavelets with arbitrarily high vanishing moments. Wavelets in the first family have the shortest possible supports with respect to their orders of vanishing moments; their existence relies on the establishment of nonnegativity on the real line of certain associated polynomials. Wavelets in the second family have increasing orders of linear-phase moments and vanishing moments, which are desirable properties in numerical algorithms.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bi, N., Han, B., Shen, Z.: Examples of refinable componentwise polynomials. Appl. Comput. Harmon. Anal. 22, 368–373 (2007)
Chui, C.K., Lian, J.A.: Construction of compactly supported symmetric and antisymmetric orthonormal wavelets with scale =3. Appl. Comput. Harmon. Anal. 2, 21–51 (1995)
Cohen, A.: Numerical Analysis of Wavelet Methods. Studies in Mathematics and its Applications, vol. 32. North-Holland, Amsterdam (2003). xviii+366 pp. ISBN:0-444-51124-5
Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods for elliptic operator equations: convergence rates. Math. Comput. 70, 27–75 (2001)
Dahmen, W.: Multiscale and wavelet methods for operator equations. In: Multiscale Problems and Methods in Numerical Simulations. Lecture Notes in Math., vol. 1825, pp. 31–96. Springer, Berlin (2003)
Dahmen, W., Kunoth, A.: Multilevel preconditioning. Numer. Math. 63, 315–344 (1992)
Dahlke, S., Fornasier, M., Raasch, T., Stevenson, R., Werner, M.: Adaptive frame methods for elliptic operator equations: the steepest descent approach. IMA J. Numer. Anal. 27, 717–740 (2007)
Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 674–996 (1988)
Daubechies, I.: Ten Lectures on Wavelets. CBMS Series. SIAM, Philadelphia (1992)
Han, B.: Symmetric orthonormal scaling functions and wavelets with dilation factor 4. Adv. Comput. Math. 8, 221–247 (1998)
Han, B.: Vector cascade algorithms and refinable function vectors in Sobolev spaces. J. Approx. Theory 124, 44–88 (2003)
Han, B.: Symmetric multivariate orthogonal refinable functions. Appl. Comput. Harmon. Anal. 17, 277–292 (2004)
Han, B.: Refinable functions and cascade algorithms in weighted spaces with Hölder continuous masks. SIAM J. Math. Anal. 40, 70–102 (2008)
Han, B.: Symmetric orthonormal complex wavelets with masks of arbitrarily high linear-phase moments and sum rules. Adv. Comput. Math. (2008). doi:10.1007/s10444-008-9102-7. Published online at journal web site
Han, B., Ji, H.: Compactly supported orthonormal complex wavelets with dilation 4 and symmetry. Appl. Comput. Harmon. Anal. 26, 422–431 (2009)
Han, B., Mo, Q.: Splitting a matrix of Laurent polynomials with symmetry and its application to symmetric framelet filter banks. SIAM J. Matrix Anal. Appl. 26, 97–124 (2004)
Heller, P.N.: Rank M wavelets with N vanishing moments. SIAM J. Matrix Anal. Appl. 16, 502–519 (1995)
Ji, H., Shen, Z.: Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions. Adv. Comput. Math. 11, 81–104 (1999)
Jia, R.Q., Micchelli, C.A.: Using the refinement equations for the construction of pre-wavelets. II. Powers of two. In: Curves and Surfaces, pp. 209–246. Academic Press, San Diego (1991)
Lawton, W.: Applications of complex valued wavelet transforms to subband decomposition. IEEE Trans. Signal Proc. 41, 3566–3568 (1993)
Lawton, W., Lee, S.L., Shen, Z.: An algorithm for matrix extension and wavelet construction. Math. Comput. 65, 723–737 (1996)
Lina, J.M., Mayrand, M.: Complex Daubechies wavelets. Appl. Comput. Harmon. Anal. 2, 219–229 (1995)
Petukhov, A.: Construction of symmetric orthogonal bases of wavelets and tight wavelet frames with integer dilation factor. Appl. Comput. Harmon. Anal. 17, 198–210 (2004)
Sun, Q.: M-band scaling functions with minimal support are asymmetric. Appl. Comput. Harmon. Anal. 12, 166–170 (2002)
Vaidyanathan, P.P.: Multirate Systems and Filter Banks. Prentice Hall, Englewood Cliffs (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Stephan Dahlke.
Dedicated to Professor Wolfgang Dahmen on the occasion of his 60th birthday.
Research supported in part by NSERC Canada under Grant RGP 228051.
Rights and permissions
About this article
Cite this article
Han, B. Matrix Extension with Symmetry and Applications to Symmetric Orthonormal Complex M-wavelets. J Fourier Anal Appl 15, 684–705 (2009). https://doi.org/10.1007/s00041-009-9084-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-009-9084-y
Keywords
- Matrix extension with symmetry
- Trigonometric polynomials
- Symmetry
- Orthonormal complex M-wavelets
- General dilation factors
- Linear-phase moments
- Vanishing moments