Compactly supported multivariate pairs of dual wavelet frames obtained by convolution

International Journal of Wavelets, Multiresolution and Information Processing

Volumn 6, Issue 2, 2008, Pages 183-208

  Ehler, Martin ^a  

^a PHILIPPS UNIVERSITY MARBURG   (Germany)

Author keywords

Dual frame; Fundamental refinable function; Mixed extension principle; Wavelet frames

Indexed keywords

APPROXIMATION THEORY; FUNCTIONAL ANALYSIS; WAVELET ANALYSIS;

DUAL FRAME; MIXED EXTENSION PRINCIPLE; REFINABLE FUNCTION; WAVELET FRAMES;

WAVELET TRANSFORMS;

EID: 44349117926     PISSN: 02196913     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219691308002306     Document Type: Article

References (37)

1. G. Beylkin, R. Coifman and V. Rokhlin, Frasu wavelet transforms and numerical algorithms, Comm. Pure Appl. Math. 44 (1991) 141-183.
2. C. Cabrelli, C. Heil and U. Molter, Accuracy of several multidimensional refinable distributions, J. Fourier Anal. Appl. 6(5) (2000) 483-502.
3. D. R. Chen, B. Han and S. D. Riemenschneider, Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments, Adv. Comput. Math. 13(2) (2000) 131-165.
4. O. Christensen, An introduction to Frames and Riesz Bases (Birkhäuser, Boston, 2003).
5. C. K. Chui and W. He, Construction of multivariate tight frames via kronecker products, Appl. Comput. Harmon. Anal. 11 (2001) 305-312.
6. C. K. Chui, W. He and J. Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal. 13 (2002) 224-262.
7. A. Coehn and I. Daubechies, Non-separable bi-dimensional wavelet bases, Rev. Mat. Iberoamericana 9 (1993) 51-137.
8. S. Dahlke, K. Gröchenig and P. Maass, A new approach to interpolating scaling functions, Appl. Anal. 72 (1999), 485-500.
9. W. Dahmen, Stability of multiscale transformations, J. Fourier Anal. Appl. 4 (1996) 341-362.
10. W. Dahmen,, Wavelet and multiscale methods for operator equations, Acta Mumer. 6 (1997) 55-228.
11. I. Daubechies, Ten Lectures on Wavelets, Vol. 9, CBMS-NSF Regional Conf. Ser. in Appl. Math., Vol. 61 (SIAM, 1992).
12. I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 1) (2003) 1-46.
13. C. de Boor, R. A. DeVore and A. Ron, On the construction of multivariate (pre)wavelets, Constr. Approx. 9 (1993) 123-166.
14. J. Derado, Multivariate retinable interpolating functions, Appl. Comput. Harmon. Anal. 7 (1999) 165-183.
15. K. Gröchenig and A. Ron, Tight compactly supported wavelet frames of arbitrary high smoothness, Proc. Amer. Math. Soc. 126(4) (1998) 1101-1107.
16. B. Han, Analysis and construction of optimal multivariate biorthogonal wavelets with compact support, SIAM J. Math. Anal. 31 (2) (1999) 274-304
17. B. Han, Symmetry property and construction of wavelets with a general dilation matrix, Linear Algebra Appl. 353 (2002) 207-225.
18. B. Han, Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix, J. Comput. Appl. Math. 155(1) (2003) 43-67.
19. B. Han, Symmetric multivariate orthogonal refinable functions, Appl. Comput. Harmon. Anal. 17 (2004) 277-292.
20. B. Han and R. Jia, Optimal interpolatory subdivision schemes in multidimensional spaces, SIAM J. Numer. Anal. 36(1) (1998) 105-124.
21. B. Han and R. Jia, Quincunx fundamental refinable functions and quincunx biorthogonal wavelets, Math. Comp. 71(237) (2002) 165-196.
22. B. Han and S. D. Riemenschneider, Interpolatory biorthogonal wavelets and cbc algorithm, in Wavelet Analyszs and Applications, eds. D.-G. Deng et al., AMS/IP Studies in Advanced Mathematics, Vol. 25, (Amer. Math. Soc., Providence, RI, 2002), pp. 119-138.
23. W. He and M. J. Lai, Construction of bivariate compactly supported biorthogonal box spline wavelets with arbitrarily high regularities, Appl. Comput. Harm. Anal. 6 (1999) 53-74.
24. W. He and M. J. Lai, Construction of trivariate compactly supported biorthogonal box spline wavelets, J. Approx. Theory 120 (2003) 1-19.
25. H. Ji, S. D. Riemenschneider and Z. Shen, Multivariate compactly supported fundamental mental refinable functions, duals and biorthogonal wavelets, Stud. Appl. Math. 102 (1999) 173-204.
26. R. Jia, Approximation properties of multivariate wavelets, Math. Comp. 67 (1998) 647-665.
27. K. Koch, Multivariate symmetric interpolating scaling vectors with duals, preprint no. 141, DFG priority programme 1114 (2006).
28. M. J. Lai and J. Stöckler, Construction of multivariate compactly supported tight wavelet frames, Appl. Comput. Harmon. Anal. 21 (2006) 324-348.
29. W. Lawton, S. L. Lee and Z. Shen, Stability and orthonormality of multivariate refinable function, SIAM J. Math. Anal. 28(4) (1997) 199-1014.
30. S. Li and H. Ogawa, Pseudoframes for subspaces with applications, J. Fourier Anal. Appl. 10(4) (2004) 409-431.
31. M. Lindemann, Approximation properties of non-separable wavelet bases with isotropic scaling matrices and their relation to besov spaces, Ph.D. thesis, University of Bremen (2005).
32. S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, 1999).
33. S. D. Riemenschneider and Z. Shen, Multidimensional interpolatory subdivision schemes, SIAM J. Numer. Anal. 34 (1997) 2357-2381.
34. d) II: Dual systems, J. Fourier Anal. Appl. 3 (1997) 617-637.
35. d), Math. Comp. 67 (1998) 191-207.
36. I. W. Selesnick and A. F. Abdelnour, Symmetric wavelet tight frames with two generators, Appl. Comput. Harmon. Anal. 17(2) (2004) 211-225.
37. W. Sweldens, The lifting scheme: A custom-design construction of biorthogonal wavelets, Appl. Compat. Harmon. Anal. 3(2) (1996) 186-200.