Self-similarity and multiwavelets in higher dimensions

Memoirs of the American Mathematical Society

Volumn 170, Issue 807, 2004, Pages

  Cabrelli, Carlos A ^a   Heil, Christopher ^a   Molter, Ursula M ^a  

^a AMERICAN MATHEMATICAL SOCIETY   (United States)

Author keywords

Cascade Algorithm; Dilation matrix; Joint spectral radius; Multiresolution analysis; Multiwavelets; Nonseparable wavelets; Refinable functions; Refinement equations; Scaling functions; Self similarity; Wavelets

Indexed keywords

EID: 2642532926     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: 10.1090/memo/0807     Document Type: Article

References (79)

1. 2 for the sparse representation of integral operators, SIAM J. Math. Anal. 24 (1993), 246-262.
2. [AK97] R. Ashino and M. Kametani, A lemma on matrices and a construction of multiwavelets, Math. Japan 45 (1997), 267-287.
3. 2) of arbitrarily high regularity, Rev. Mat. Iberoamericana 15 (1999), 37-58.
4. [Aya99a] A. Ayache, New filter banks that may generate nonseparable, compactly supported, orthonormal wavelet bases of arbitrarily high regularity, preprint (1999).
5. n, Proc. Amer. Math. Soc. 112 (1991), 549-562.
6. [BW92] M. A. Berger and Y. Wang, Bounded semi-groups of matrices, Linear Algebra Appl. 166 (1992), 21-27.
7. 2, SIAM J. Math. Anal. 30 (1999), 678-697.
8. [BL01] J. J. Benedetto and M. Leon, The construction of single wavelets in D-dimensions, J. Geom. Anal. 11 (2001), 1-15.
9. d), Trans. Amer. Math. Soc. 341 (1994), 787-806.
10. d), J. Funct. Anal. 119 (1994), 37-78.
11. [BZ00] M. Bröker and X. Zhou, Characterization of continuous, four-coefficient scaling functions via matrix spectral radius, SIAM J. Matrix Anal. Appl. 22 (2000), 242-257.
12. [CHM98] C. Cabrelli, C. Heil, and U. Molter, Accuracy of lattice translates of several multidimensional refinable functions, J. Approx. Theory 95 (1998), 5-52.
13. [CHM99] C. A. Cabrelli, C. Heil, and U. M. Molter, Polynomial reproduction by refinable functions, Advances in Wavelets (Hong Kong, 1997), K.-S. Lau, ed., Springer-Verlag, Singapore, 1999, pp. 121-161.
14. [CHM00] C. Cabrelli, C. Heil, and U. Molter, Accuracy of several multidimensional refinable distributions, J. Fourier Anal. Appl. 6 (2000), 483-502.
15. [CM94] C. A. Cabrelli and U. M. Molter, Generalization of a functional equation and its application to the construction of wavelets, Preprint No. 78, Departamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, 1994.
16. [CM99] C. A. Cabrelli and U. M. Molter, Generalized self-similarity, J. Math. Anal. Appl. 230 (1999), 251-260.
17. n, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 1-10.
18. s), Proc. Amer. Math. Soc. 125 (1997), 2883-2889.
19. [Coh90] A. Cohen, Ondelettes, analyses multirésolutions et filtres miroirs en quadrature, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 439-459.
20. [CD93] A. Cohen and I. Daubechies, Non-separable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9 (1993), 51-137.
21. [CDP97] A. Cohen, I. Daubechies, and G. Plonka, Regularity of refinable function vectors, J. Fourier Anal. Appl. 3 (1997), 295-324.
22. [CGV99] A. Cohen, K. Gröchenig, and L. F. Villemoes, Regularity of multivariate refinable functions, Constr. Approx. 15 (1999), 241-255.
23. [CH92] D. Colella and C. Heil, The characterization of continuous, four-coefficient scaling functions and wavelets, IEEE Trans. Inf. Th. 38 (1992), 876-881.
24. [CH94] D. Colella and C. Heil, Characterizations of scaling functions: Continuous solutions, SIAM J. Matrix Anal. Appl. 15 (1994), 496-518.
25. [DM97] S. Dahlke and P. Maass, Interpolating refinable functions and wavelets for general scaling matrices, Numer. Funct. Anal. Optim. 18 (1997), 521-539.
26. n, J. Fourier Anal. Appl. 3 (1997), 451-456.
27. [Dau92] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.
28. [DL92] I. Daubechies and J. C. Lagarias, Two-scale difference equations: II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992), 1031-1079.
29. [Der99] J. Derado, Multivariate refinable interpolating functions, Appl. Comput. Harmon. Anal. 7 (1999), 165-183.
30. [DGH95] G. C. Donovan, J. S. Geronimo, and D. P. Hardin, A class of orthogonal multiresolution analyses in 2D, Mathematical Methods for Curves and Surfaces, M. Dæhlen, T. Lyche, and L. L. Schumaker, eds., Vanderbilt University Press, Nashville, 1995, pp. 99-110.
31. [DGHM96] G. Donovan, J. S. Geronimo, D. P. Hardin, and P. R. Massopust, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal. 47 (1996), 1158-1192.
32. [Eir92] T. Eirola, Sobolev characterization of solutions of dilation equations, SIAM J. Math. Anal. 23 (1992). 1015-1030.
33. [GHM94] J. S. Geronimo, D. P. Hardin, and P. R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Th. 78 (1994), 373-401.
34. [GL94] T. N. T. Goodman, S. L. Lee, and W.-S. Tang, Wavelets of multiplicity r, Trans. Amer. Math. Soc. 342 (1994), 307-324.
35. [GLT93] T. N. T. Goodman, S. L. Lee, and W.-S. Tang, Wavelets in wandering subspaces, Trans. Amer. Math. Soc. 338 (1993), 639-654.
36. [Gri96] G. Gripenberg, Computing the joint spectral radius, Linear Algebra Appl. 234 (1996), 43-60.
37. [Grö87] K. Gröchenig, Analyse multiéchelles et bases d'ondelettes, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 13-15.
38. n, IEEE Trans. Inform. Theory 38 (1992), 556-568.
39. [GR98] K. Gröchenig and A. Ron, Tight compactly supported wavelet frames of arbitrarily high smoothness, Proc. Amer. Math. Soc. 126 (1998), 1101-1107.
40. [Han97] B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal. 4 (1997), 380-413.
41. [HL97] W. He and M.-J. Lai, Examples of bivariate nonseparable compactly supported orthonormal continuous wavelets, Wavelet Applications in Signal and Image Processing IV, Proc. SPIE Vol. 2825, M. A. Unser et al., eds., SPIE, Bellingham, WA, 1997, pp. 303-314.
42. [HL98] W. He and M.-J. Lai, Construction of bivariate compactly supported orthonormal multiwavelets with arbitrarily high regularity, preprint, U. Georgia (1998).
43. [HL99] W. He and M.-J. Lai, Construction of bivariate compactly supported biorthogonal box spline wavelets with arbitrarily high regularities, Appl. Comput. Harmon. Anal. 6 (1999), 53-74.
44. [Hei94] C. Heil, Some stability properties of wavelets and scaling functions, Wavelets and Their Applications, J. S. Byrnes et al., eds., Kluwer Academic Publishers, Dordrecht, 1994, pp. 19-38.
45. [HC93] C. Heil and D. Colella, Dilation equations and the smoothness of compactly supported wavelets, Wavelets: Mathematics and Applications, J. J. Benedetto and M. W. Frazier, eds., CRC Press, Boca Raton, FL, 1993, pp. 161-200.
46. [HS95] C. Heil and G. Strang, Continuity of the joint spectral radius, Linear Algebra for Signal Processing, A. Bojanczyk and G. Cybenko, eds., IMA Vol. Math. Appl. 69, Springer-Verlag, New York, 1995, pp. 51-61.
47. [HSS96] C. Heil, G. Strang, and V. Strela, Approximation by translates of refinable functions, Numerische Math. 73 (1996), 75-94.
48. [Hut81] J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747.
49. [JRS99] H. Ji, S. D. Riemenschneider, and Z. Shen, Multivariate compactly supported fundamental refinable functions, duals and biorthogonal wavelets, Stud. Appl. Math. 102 (1999), 173-204.
50. p spaces, Adv. Comput. Math. 3 (1995), 309-341.
51. [Jia99] R.-Q. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. 351 (1999), 4089-4112.
52. [Jng99] Q. Jiang, Multivariate matrix refinable functions with arbitrary matrix dilation, Trans. Amer. Math. Soc. 351 (1999), 2407-2438.
53. n), CRM Proc. Lecture Notes 18, Amer. Math. Soc., Providence, RI, 1999, pp. 135-151.
54. n, IEEE Trans. Inform. Theory 38 (1992), 533-555.
55. [KoV95] J. Kovačević and M. Vetterli, Nonseparable two- and three-dimensional wavelets, IEEE Trans. Signal Proc. 43 (1995), 1269-1273.
56. [KS00] J. Kovačević and W. Sweldens, Wavelet families of increasing order in arbitrary dimensions, IEEE Trans. Image Proc. 9 (2000¡ 480-396.
57. 2, J. Fourier Anal. Appl. 2 (1995), 1-14.
58. [LagW95b] J. C. Lagarias and Y. Wang, The finiteness conjecture for the generalized spectral radius of a set of matrices, Linear Algebra Appl. 214 (1995), 17-42.
59. n) and algebraic number theory, J. Number Theory 57 (1996), 181-197.
60. n. II. Lattice tilings, J. Fourier Anal. Appl. 3 (1997), 83-102.
61. n) and algebraic number theory, J. Number Theory 76 (1999), 330-336.
62. 2-scaling functions, Asian J. Math. 1 (1997), 272-292.
63. p- solutions for the two-scale dilation equation, SIAM J. Math. Anal. 26 (1995), 1018-1046.
64. [Mae95] M. Maesumi, Optimum ball for joint spectral radius: an example from four-coefficient MRA, Approximation Theory VIII, Vol. 2, C. K. Chui and L. L. Schumaker, eds., World Scientific, Singapore, 1995, pp. 267-274.
65. [Mey92] Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1992.
66. [MP89] C. A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989), 841-870.
67. [MS97] C. A. Micchelli and T. Sauer, Regularity of multiwavelets, Adv. Comput. Math. 7 (1997), 455-545.
68. 2(R) generated by refinable function vectors, Constr. Approx. 13 (1997), 221-244.
69. [Pot97] A. Potiopa, A problem of Lagarias and Wang, Master's Thesis, Siedlce University, Siedlce, Poland, 1997 (Polish).
70. [RS60] G. C. Rota and G. Strang, A note on the joint spectral radius, Kon. Nederl. Akad. Wet. Proc. A 63 (1960), 379-381.
71. [Rue02] A. M. C. Ruedin, Construction of nonseparable multiwavelets for nonlinear image compression, EURASIP J. Appl. Signal Process. 2002 (2002), 73-79.
72. [Sun91] Q. Sun, Two-scale difference equations: Local and global linear independence, unpublished manuscript (1991).
73. [Vil92] L. F. Villemoes, Energy moments in time and frequency for two-scale difference equation solutions and wavelets, SIAM J. Math. Anal. 23 (1992), 1519-1543.
74. [Vil94a] L. F. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal. 25 (1994), 1433-1460.
75. [Vil94b] L. F. Villemoes, Continuity of nonseparable quincunx wavelets, Appl. Comput. Harmon. Anal. 1 (1994), 180-187.
76. [Wan95] Y. Wang, Two-scale dilation equations and the cascade algorithm, Random Comput. Dynam. 3 (1995), 289-307.
77. [Wan96] Y. Wang, Two-scale dilation equations and the mean spectral radius, Random Comput. Dynam. 4 (1996), 49-72.
78. [Woj97] P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge University Press, Cambridge, 1997.
79. [Zho98] D.-X. Zhou, The p-norm joint spectral radius for even integers, Methods Appl. Anal. 5 (1998), 39-54.