Frames in spaces with finite rate of innovation

Advances in Computational Mathematics

Volumn 28, Issue 4, 2008, Pages 301-329

  Sun, Qiyu ^a  

^a UNIVERSITY OF CENTRAL FLORIDA   (United States)

Author keywords

Banach frame; Frame; Localized frame; Matrix algebra; Refinable function; Signals with finite rate of innovation; Space of homogenous type; Wavelets

Indexed keywords

EID: 41549127940     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-006-9021-4     Document Type: Article

References (61)

1. Aldroubi, A., Gröchenig, K.: Nonuniform sampling and reconstruction in shift-invariant space. SIAM Rev. 43, 585-620 (2001)
2. p. J. Fourier Anal. Appl. 7, 1-21 (2001)
3. Aldroubi, A., Sun, Q., Tang, W.-S.: Convolution, average sampling and a Calderon resolution of the identity for shift-invariant spaces. J. Fourier Anal. Appl. 11, 215-244 (2005)
4. Balan, R., Casazza, P.G., Heil, C., Landau, Z.: Density, overcompleteness and localization of frames I: theorey; II: Gabor system. Preprint (2004)
5. Baskakov, A.G.: Wiener's theorem and asymptotic estimates for elements in inverse matrices. Funktsional Anal. o. Prilozhen. 24, 64-65 (1990)
6. Benedetto, J.J., Li, S.: The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal. 5, 389-427 (1998)
7. n). J. Funct. Anal. 177, 282-309 (2000)
8. Bownik, M.: Anisotropic Hardy spaces and wavelets. Mem. Am. Math. Soc. 781, 22 pp (2003)
9. Butzer, P.L.: A survey of the Whittaker-Shannon sampling theorem and some of its extensions. J. Math. Res. Expo. 3, 185-212 (1983)
10. Casazza, P.: The art of frame theory. Taiwan. J. Math. 4, 129-201 (2000)
11. Casazza, P., Han, D., Larson, D.R.: Frames for Banach spaces. In: The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, Texas, 1999), Contemporary Mathematics, vol. 247, pp. 149-182. (1999)
12. Chen, H.-L., Peng, S.-L.: Localization of dual periodic scaling and wavelet functions. Adv. Comput. Math. 19, 195-210 (2003)
13. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)
14. Christensen, O., Stoeva, D.T.: p-frames in separable Banach spaces. Adv. Comput. Math. 18, 117-126 (2003)
15. Chui, C.K.: An Introduction to Wavelets. Academic, Boston, Massachusetts (1992)
16. Chui, C.K., He, W., Stöckler, J.: Nonstationary tight wavelet frames: I. Bounded intervals; II. Unbounded intervals. Appl. Comput. Harmon. Anal. 17, 141-197 (2004); 18, 25-66 (2005)
17. Chui, C.K., Sun, Q.: Affine frame decompositions and shift-invariant spaces. Appl. Comput. Harmon. Anal. 20, 74-107 (2006)
18. Cohen, A., Dyn, N.: Nonstationary subdivision schemes and multiresolution analysis. SIAM J. Math. Anal. 27, 1745-1769 (1996)
19. Cohen, A., Daubechies, I., Vial, P.: Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1, 54-81 (1993)
20. Coifman, R., Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal. 21, 53-94 (2006)
21. Coifman, R., Weiss, G.: Analyses Harmoniques Noncommutative sur Certains Espaces Homogenes. Springer, Berlin Heidelberg New York (1971)
22. Cramer, R.J.-M., Scholtz, R.A., Win, M.Z.: Evaluation of an ultra wide-band propagation channel. IEEE Trans. Antennas Propag. 50, 561-569 (2002)
23. Daubechies, I.: Ten Lectures on Wavelets. CBMF Conference Series in Applied Mathematics, 61. SIAM, Philadelphia (1992)
24. Fornasier, M., Gröchenig, K.: Intrinsic localization of frames. Constr. Approx. 22, 395-415 (2005)
25. Fornasier, M., Rauhut, H.: Continuous frames, function spaces, and the discretization problem. J. Fourier Anal. Appl. 11, 245-287 (2005)
26. n) using Gabor frames. J. Fourier Anal. Appl. 7, 101-126 (2001)
27. Gröchenig, K.: Describing functions: atomic decompositions versus frames. Monatsh. Math. 112, 1-42 (1991)
28. Gröchenig, K.: Foundation of time-frequency analysis. Birkhäuser, Boston (2001)
29. Gröchenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10, 105-132 (2004)
30. Gröchenig, K.: Time-frequency analysis of Sjöstrand class. Rev. Mat. Iberoam. (To appear)
31. Gröchenig, K., Leinert, M.: Wiener's lemma for twisted convolution and Gabor frames. J. Am. Math. Soc. 17, 1-18 (2003)
32. Gröchenig, K., Leinert, M.: Symmetry of matrix algebras and symbolic calculus for infinite matrices. Trans. Am. Math. Soc. 358, 2695-2711 (2006)
33. Hao, Y., Marziliano, P., Vetterli, M., Blu, T.: Sampling and reconstruction of ECG as a signal with a finite rate of innovation. (Submitted to IEEE Trans. Biomed. Eng.)
34. Higgins, J.R.: Five short stories about the cardinal series. Bull. Am. Math. Soc. 12, 45-89 (1985)
35. Jaffard, S.: Properiétés des matrices bien localisées prés de leur diagonale et quelques applications. Ann. Inst. Henri Poincaré 7, 461-476 (1990)
36. Janssen, A.J.E.M.: Duality and biorthogonality for Weyl-Heisenberg frames. J. Fourier Anal. Appl. 1, 403-436 (1995)
37. Jerri, A.: The Shannon sampling theorem - Its various extensions and applications: A tutorial review. Proc. IEEE 65, 1565-1596 (1977)
38. Jia, R.-Q.: Shift-invariant spaces and linear operator equations. Isr. J. Math. 103, 259-288 (1998)
39. Jia, R.-Q., Micchelli, C.A.: Using the refinement equations for the construction of pre-wavelets. II. Powers of two. In: Curves and Surfaces (Chamonix-Mont-Blanc, 1990), pp. 209-246. Academic, Boston, Massachusetts (1991)
40. Kusuma, J., Maravic, I., Vetterli, M.: Sampling with finite rate of innovation: channel and timing estimatation for UWB and GPS. In: IEEE Conference on Communication 2003. Anchorage, AK
41. Macias, R., Segovia, C.: Lipschitz functions on spaces of homogenous type. Adv. Math. 33, 257-270 (1979)
42. Macias, R., Segovia, C.: A decomposition into atoms of distribution on spaces of homogenous type. Adv. Math. 33, 271-309 (1979)
43. Mallat, S.G.: A Wavelet Tour of Signal Processing. Academic (1999)
44. Maravic, I., Vetterli, M.: Sampling and reconstruction of signals with finite rate of innovation in the presence of noise. IEEE Trans. Signal Process. 53, 2788-2805 (2005)
45. Marziliano, P., Vetterli, M.: Reconstruction of irregular sampled discrete-time band-limited signals with unknown sampling locations. IEEE Trans. Signal Process. 48, 3462-3471 (2000)
46. Marziliano, P., Vetterli, M., Blu, T.: Sampling and exact reconstruction of band-limited signals with shot noise. IEEE Trans. Information Theory. 5, 2230-2233 (2006)
47. Marziliano, P.: Sampling innovations. PhD thesis, Swiss Federal Institute of Technology (April 2001)
48. Meyer, Y.: Ondelettes et Operateurs. Herman, Paris (1990)
49. Nagel, A., Stein, E., Waigner, S.: Ball and metrics defined by vector fields I: Basic properties. Acta Math. 155, 103-147 (1985)
50. Plonka, G.: Periodic spline interpolation with shifted nodes. J. Approx. Theory 76, 1-20 (1994)
51. d). Duke Math. J. 89, 237-282 (1997)
52. Ron, A., Shen, Z.: Generalized shift-invariant systems. Constr. Approx. 22, 1-45 (2005)
53. Schumaker, L.L.: Spline Functions: Basic Theory. Wiley & Sons, New York (1981)
54. Sjöstrand, J.: An algebra of pseudodifferential operators. Math. Res. Lett. 1, 185-192 (1994)
55. Sun, Q.: Stability of the shifts of global supported distributions. J. Math. Anal. Appl. 261, 113-125 (2001)
56. Sun, Q.: Wiener's lemma for infinite matrices with polynomial off-diagonal decay. C. R. Acad. Sci. Paris, Ser I 340, 567-570 (2005)
57. Sun, Q.: Wiener's lemma for infinite matrices. Trans. Am. Math. Soc. (accepted)
58. Sun, Q.: Non-uniform average sampling and reconstruction of signals with finite rate of innovation. SIAM J. Math. Anal. (submitted)
59. Sun, Q., Bi, N., Huang, D.: An Introduction to Multiband Wavelets. Zhejiang University Press, Hangzhou (2000)
60. Vetterli, M., Marziliano, P., Blu, T.: Sampling signals with finite rate of innovation. IEEE Trans. Signal Process. 50, 1417-1428 (2002)
61. Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic, New York (1980)