Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for L2(ℝd)

Applied and Computational Harmonic Analysis

Volumn 17, Issue 2 SPEC. ISS., 2004, Pages 119-140

  Aldroubi, Akram ^a   Cabrelli, Carlos ^b,c   Molter, Ursula M ^b,c  

^a VANDERBILT UNIVERSITY   (United States)

^b UNIVERSIDAD DE BUENOS AIRES   (Argentina)

^c Facultad de Ciencias Naturales e Instituto M Lillo   (Argentina)

Author keywords

Frames; Irregular sampling; Wavelet sets; Wavelets

Indexed keywords

ATOMS; CRYSTAL LATTICES; DECOMPOSITION; DIGITAL STORAGE; HARMONIC ANALYSIS; IMAGE ANALYSIS; IMAGE COMPRESSION; SAMPLING; SPEECH CODING;

CONVERGENCE; FRAMES; IRREGULAR SAMPLING; WAVELET SETS;

WAVELET TRANSFORMS;

EID: 4344631998     PISSN: 10635203     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.acha.2004.03.005     Document Type: Article

References (48)

1. R. Ashino, S.J. Desjardins, C. Heil, M. Nagase, R. Vaillancourt, Smooth tight frame wavelets and image analysis in Fourier space, Comput. Math. Appl. 45 (2003) 1551-1579.
2. R. Ashino, C. Heil, M. Nagase, R. Vaillancourt, Microlocal filtering with multiwavelets, Comput. Math. Appl. 41 (2001) 111-133.
3. n, J. Fourier Anal. Appl. 5 (1999) 563-573.
4. R. Balan, Stability theorems for Fourier frames and wavelet Riesz bases, J. Fourier Anal. Appl. 3 (5) (1997) 499-504, dedicated to the memory of Richard J. Duffin.
5. R. Balan, P.G. Casazza, C. Heil, Z. Landau, Excesses of Gabor frames, Appl. Comput. Harmon. Anal. 14 (2003) 87-106.
6. d, AMS Contemp. Math. 247 (1999) 43-74.
7. J.J. Benedetto, M. Leon, The construction of single wavelets in d-dimensions, J. Geom. Anal. (2001) 1-15.
8. J.J. Benedetto, S. Sumetkijakan, Tight frames and geometric properties of wavelet sets, Preprint, 2003.
9. J.J. Benedetto, H.-C. Wu, Non-uniform sampling and spiral MRI reconstruction, in: Proceedings of SampTA 1999, Trondheim, Norway, 1999.
10. A. Beurling, Local harmonic analysis with some applications to differential operators, in: Proc. Ann. Science Conf., Belfer Grad. School of Science, 1966, pp. 109-125.
11. O. Christensen, Moment problems and stability results for frames with applications to irregular sampling and Gabor frames, Appl. Comput. Harmon. Anal. 3(1) (1996) 82-86.
12. O. Christensen, Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl. 3 (1997) 543-557.
13. O. Christensen, B. Deng, C. Heil, Density of Gabor frames, Appl. Comput. Harmon, Anal. 7 (3) (1999) 292-304.
14. O. Christensen, S. Favier, F. Zó, Irregular wavelet frames and Gabor frames, Approx. Theor. Appl. 17 (3) (2001) 90-101.
15. O. Christensen, C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr. 185 (1997) 33-47.
16. C.K. Chui, W. He, J. Stöckler, Compactly supported tight and sibling frames with maximal vanishing moments, Appl. Comput. Harmon. Anal. 13 (3) (2002) 224-262.
17. C.K. Chui, W. He, J. Stöckler, Q. Sun, Compactly supported tight affine frames with integer dilations and maximum vanishing moments, Adv. Comput. Math. 18 (2) (2003) 159-187, special issue on frames.
18. C.K. Chui, X. Shi, Orthonormal wavelets and tight frames with arbitrary real dilations, Appl. Comput. Harmon. Anal. 17 (2000) 243-264.
19. n, J. Fourier Anal. Appl. 3 (4) (1997) 451-456.
20. n, in: Wavelets, Multiwavelets, and Their Applications, vol. II, San Diego, CA, 1997, American Mathematical Society, Providence, RI, 1998, pp. 15-40.
21. I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (5) (1986) 1271-1283.
22. S.J. Favier, R.A. Zalik, On the stability of frames and Riesz bases, Appl. Comput. Harmon. Anal. 2 (2) (1995) 160-173.
23. H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decomposition, J. Funct. Anal. (1989) 307-340.
24. H.G. Feichtinger, Banach spaces of distributions defined by decomposition methods II, Math. Nachr. 132 (1987) 207-237.
25. H.G. Feichtinger, P. Gröbner, Banach spaces of distributions defined by decomposition method I, Math. Nachr. 123 (1985) 97-120.
26. H.G. Feichtinger, T. Werther, Atomic systems for subspaces, in: L. Zayed (Ed.), Proceedings SampTA 2001, Orlando, FL, 2001, pp. 163-165.
27. K. Gröchenig, Describing functions: Atomic decompositions versus frames, Monatsh. Math. 112 (1) (1991) 1-42.
28. K. Gröchenig, Irregular sampling of wavelet and short-time Fourier transforms, Constr. Approx. 9 (2-3) (1993) 283-297.
29. D. Han, D. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000).
30. C. Heil, G. Kutyniok, Density of weighted wavelet frames, J. Geom. Anal. 13 (2003) 479-493.
31. C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989) 628-666.
32. E. Hernández, D. Labate, G. Weiss, A unified characterization of reproducing systems generated by a finite family II, J. Geom. Anal. 12 (4) (2002) 615-662.
33. E. Hernández, D. Labate, G. Weiss, E. Wilson, Oversampling quasi affine frames and wave packets, 2003, in preparation.
34. R. Horn, C. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991.
35. D. Labate, A unified characterization of reproducing systems generated by a finite family, J. Geom. Anal. 12 (3) (2002) 469-491.
36. S. Li, H. Ogawa, A theory of pseudoframes for subspaces with applications, Technical report, 1998.
37. n and frames, Preprint, 2003.
38. G. Ólafsson, D. Speegle, Groups, wavelets and wavelet sets, Preprint, 2003.
39. P.A. Olsen, K. Seip, A note on irregular discrete wavelet transforms, IEEE Trans. Inform. Theor. Part 2 38 (2) (1992) 861-863.
40. M. Papadakis, G. Gogoshin, I.A. Kakadiarjs, D.J. Kouri, D.K. Hoffman, Non-separable radial frame multiresolution analysis in multidimensions, Numer. Funct. Anal. Optimiz., in press.
41. M. Papadakis, G. Gogoshin, I.A. Kakadiaris, D.J. Kouri, D.K. Hoffman, Non-separable radial frame multiresolution analysis in multidimensions and isotropic fast wavelet algorithms, SPIE-Wavelet X, in press.
42. J. Ramanathan, T. Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal. 2 (2) (1995) 148-153.
43. W. Sun, X. Zhou, Irregular wavelet frames, Sci. China Ser. A 43 (2) (2000) 122-127.
44. W. Sun, X. Zhou, On the stability of Gabor frames, Adv. Appl. Math. 26 (3) (2001) 181-191.
45. W. Sun, X. Zhou, Irregular wavelet/Gabor frames, Appl. Comput. Harmon. Anal. 13 (2002) 63-76.
46. W. Sun, X. Zhou, Irregular Gabor frames and their stability, Proc. Amer. Math. Soc. 131 (9) (2003) 2883-2893.
47. W. Sun, X. Zhou, Density of irregular wavelet frames, Proc. Amer. Math. Soc., 2003, in press.
48. Y. Wang, Wavelets, tiling, and spectral sets, Duke Math. J. 114 (1) (2002) 43-57.