Abstract
Starting from any two compactly supported d-refinable function vectors in (L 2(R))r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L 2(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5,12,13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Aldroubi, Portraits of frames, Proc. Amer. Math. Soc. 123 (6) (1995) 1661–1668.
J.J. Benedetto and S.D. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998) 389–427.
C. Cabrelli, C. Heil and U.Molter, Accuracy of lattice translates of several multidimensional refinable functions, J. Approx. Theory 95 (1998) 5–52.
C.K. Chui and W. He, Compactly supported tight frames associated with refinable functions, Appl. Comput. Harmon. Anal. 8 (2000) 293–319.
C.K. Chui, W. He and J. Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal., to appear.
C.K. Chui, W. He, J. Stöckler and Q.Y. Sun, Compactly supported tight affine frames with integer dilations and maximum vanishing moments, Adv. Comput. Math., to appear.
C.K. Chui, X. Shi and J. Stöckler, Affine frames, quasi-frames and their duals, Adv. Comput. Math. 8 (1998) 1–17.
C.K. Chui and Q.Y. Sun, Tight frame oversampling and its equivalence to shift-invariance of frame operators, Proc. Amer. Math. Soc., to appear.
W. Dahmen and C.A. Micchelli, Biorthogonal wavelet expansions, Constr. Approx. 13 (1997) 293–328.
I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990) 961–1005.
I. Daubechies, Ten Lectures on Wavelets, CBMF Conference Series in Applied Mathematics, Vol. 61 (SIAM, Philadelphia, PA, 1992).
I. Daubechies and B. Han, Pairs of dual wavelet frames from any two refinable functions, Constr. Approx., to appear.
I. Daubechies, B. Han, A. Ron and Z.W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal., to appear.
G.C. Donovan, J.S. Geronimo and D.P. Hardin, Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets, SIAM J. Math. Anal. 27 (1996) 1791–1815.
G.C. Donovan, J.S. Geronimo, D.P. Hardin and P. Massopust, Construction of orthogonal wavelets using fractal interpolation function, SIAM J. Math. Anal. 27 (1996) 1158–1192.
B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal. 4 (1997) 380–413.
B. Han, Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, J. Approx. Theory 110 (2001) 18–53.
B. Han, Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix, J. Comput. Appl. Math., to appear.
B. Han and Q. Mo, Tight wavelet frames generated by three symmetric B-spline functions with high vanishing moments, Preprint (2002).
C. Heil, G. Strang and V. Strela, Approximation by translates of refinable functions, Numer. Math. 73 (1996) 75–94.
E. Hernández and G. Weiss, A First Course on Wavelets (CRC Press, Boca Raton, FL, 1996).
R.Q. Jia, S. Riemenschneider and D.X. Zhou, Approximation by multiple refinable functions and multiple functions, Canadian J. Math. 49 (1997) 944–962.
R.Q. Jia and Q.T. Jiang, Approximation power of refinable vectors of functions, in: Proc. of Internat. Conf. on Wavelet Analysis and Applications, to appear.
Q.T. Jiang, Parameterization of masks for tight affine frames with two symmetric/antisymmetric generators, Adv. Comput. Math., to appear.
A. Petukhov, Explicit construction of framelets, Appl. Comput. Harmon. Anal., to appear.
A. Ron and Z.W. Shen, Affine systems in L 2(ℝd ): The analysis of the analysis operator, J. Funct. Anal. 148(2) (1997) 408–447.
A. Ron and Z.W. Shen, Affine systems in L 2(ℝd) II: Dual systems, J. Fourier Anal. Appl. 3 (1997) 617–637.
G. Plonka, Approximation order provided by refinable function vectors, Constr. Approx. 13 (1997) 221–244.
G. Plonka and A. Ron, A new factorization technique of the matrix mask of univariate refinable functions, Numer. Math. 87 (2001) 555–595.
I. Selesnick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2000) 163–181.
Z.W. Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998) 235–250.
Q.Y. Sun, An algorithm for the construction of symmetric and antisymmetric M-band wavelets, in: Wavelet Applications in Signal and Image Processing VIII, eds. A. Aldroubi, A.F. Laine and M. Unser, Proc. SPIE 4119 (2000) 384–394.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Han, B., Mo, Q. Multiwavelet Frames from Refinable Function Vectors. Advances in Computational Mathematics 18, 211–245 (2003). https://doi.org/10.1023/A:1021360312348
Issue Date:
DOI: https://doi.org/10.1023/A:1021360312348