Density, overcompleteness, and localization of frames. I. Theory

Journal of Fourier Analysis and Applications

Volumn 12, Issue 2, 2006, Pages 105-143

  Balan, Radu ^a   Casazza, Peter G ^b   Heil, Christopher ^c   Landau, Zeph ^d  

^a SIEMENS CORPORATE RESEARCH   (United States)

^b UNIVERSITY OF MISSOURI   (United States)

^c GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^d CITY COLLEGE OF NEW YORK   (United States)

Author keywords

Density; Excess; Frames; Gabor systems; Modulation spaces; Overcompleteness; Riesz bases; Wavelets; Weyl Heisenberg systems

Indexed keywords

EID: 33646490258     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00041-006-6022-0     Document Type: Article

References (59)

1. Aldroubi, A. and Gröchenig, K. (2001). Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev. 43, 585-620.
2. Baggett, L. (1990). Processing a radar signal and representations of the discrete Heisenberg group. Colloq. Math. 60/61, 195-203.
3. Balan, R. (1997). Stability theorems for Fourier frames and wavelet Riesz bases, J. Fourier Anal. Appl. 3(5), 499-504.
4. Balan, R., Casazza, P. G., Heil. C., and Landau, Z. (2003). Deficits and excesses of frames, Adv. Comput. Math. 18, 93-116.
5. Balan, R., Casazza, P. G., Heil, C., and Landau, Z. Density, overcompleteness, and localization of frames. II. Gabor frames, J. Fourier Anal. Appl. to appear.
6. Balan, R., Casazza, P. G., Heil, C., and Landau, Z. (2005). Density, overcompleteness, and localization of frames, research announcement.
7. Baskakov, A.G. (1990). Wiener's theorem and asymptotic estimates for elements of inverse matrices, Funktional. Anal. i Prilozhen. 24(1990), 64-65;
8. translation in Fund. Anal. Appl. 24, 222-224.
9. Baskakov, A.G. (1997). Estimates for the entries of inverse matrices and the spectral analysis of linear operators, Izv. Ross. Akad. Nauk Ser. Mat. 61(1997), 3-26;
10. translation in Izv. Math. 61, 1113-1135.
11. Benedetto, J. J., Powell, A., and Yilmaz, O. Sigma-Delta quantization and finite frames, IEEE Trans, Inform. Theory to appear.
12. Bodmann, B. G. and Paulsen, V. I. (2005). Frames, graphs and erasures, Linear Algebra Appl. 404, 118-146.
13. Bölcskei, H., Hlawatsch, F., and Feichtinger, H.G. Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Process. 46, 3256-3268.
14. Bownik, M. and Rzeszotnik, Z. (2003). The spectral function of shift-invariant spaces, Michigan Math. J. 51, 387-414.
15. Bownik, M. and Speegle, D. Feichtinger conjecture for wavelet frames, Gabor frames and frames of translates. Canad. J. Math. to appear.
16. Candès, E.J. and Donoho, D. L. (2005). Continuous curvet transform: II. Discretization and frames. Appl. Comput. Hannon. Anal. 19, 198-222.
17. Casazza, P. G. (2000). The art of frame theory, Taiwanese J. Math. 4, 129-201.
18. Casazza, P. G., Christensen, O., Lindner, A., and Vershynin, R. (2005). Frames and the Feichtinger conjecture. Proc. Amer. Math. Soc. 133, 1025-1033.
19. Casazza, P. G. and Tremain, J. C. (2006). The Kadison-Singer problem in mathematics and engineering. Proc. Natl. Acad. Sci. USA 103, 2032-2039.
20. Christensen, O. (2003). An Introduction to Frames and Riesz Bases, Birkhäuser, Boston.
21. Christensen, O., Deng, B., and Heil, C. (1999). Density of abor frames. Appl. Comput. Harmon. Anal, 7, 292-304.
22. Daubechies, I. (1990). The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory 39, 961-1005.
23. Daubechies, I. (1992). Ten Lectures on Wavelets, SIAM, Philadelphia.
24. Daubechies, I., Landau, H., and Landau, Z. (1995). Gabor time-frequency lattices and the Wexler-Raz identity, J. Fourier Anal. Appl. 1, 437-478.
25. 1 minimization, Proc. Natl. Acad. Sci. USA 100, 2197-2202.
26. Eldar, Y. C. and Forney, G. D. (2002). Optimal tight frames and quantum measurement, IEEE Trans. Inform. Theory 48, 599-610.
27. Fornasier, M. (2003). Constructive methods for numerical applications in signal processing and homogenization problems, PhD Thesis, U. Padua.
28. Gohberg, I., Kaashoek, M. A., and Woerdeman, H. J. (1989). The band method for positive and contractive extension problems, J. Operator Theory 22, 109-155.
29. Goyal, V. K., Kovačević, J., and Keiner, J. A. (2001). Quantized frame extensions with erasures, Appl. Comput. Harmon. Anal. 10, 203-233.
30. N: Analysis, synthesis, and algorithms, IEEE Trans. Inform. Theory 44, 16-31.
31. Gribonval, R. and Nielsen, M. (2004). Nonlinear approximation with dictionaries, I. Direct estimates, J. Fourier Anal. Appl. 10(1), 51-71.
32. Gröchenig, K. (2001). Foundations of Time-Frequency Analysis, Birkhäuser, Boston.
33. Gröchenig, K. (2003). Localized frames are finite unions of Riesz sequences, Adv. Comput. Math. 18, 149-157.
34. Groehenig, K. (2004). Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl. 10, 105-132.
35. Gröehenig, K. and Fornasier, M. (2005). Intrinsic localization of frames, Constr. Approx. 22, 395-415.
36. Gröehenig, K. and Heil. C. (1999). Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory 34, 439-457.
37. Gröehenig, K. and Razafinjatovo, H. (1996). On Landau's necessary density conditions for sampling and interpolation of band-limited functions. J. London Math. Soc. (2) 54, 557-565.
38. Han, D. and Larson, D. R. (2000). Frames, bases and group representations Mem. Amer. Math, Soc. 147(697).
39. Heil. C. E. and Walnut, D. F. (1989), Continuous and discrete wavelet transforms. SIAM Review 31, 628-666.
40. Heil, C. and Kutyniok, G. (2003). Density of weighted wavelet frames, J. Geom. Anal. 13, 479-493.
41. Hindman, N. and Strauss, D. (1998). Algebra in the Stone-Čech Compactification, de Gruyter Expositions in Mathematics, 27, Walter de Gruyter and Co., Berlin.
42. Holmes, R. B. and Paulsen, V. I. (2004). Optimal frames for erasures. Linear Algebra Appl. 377, 31-51.
43. Jaffard, S. (1990). Propriétés des matrices "bien localisées" près de leur diagonale et quelques applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 7, 461-476.
44. Janssen, A. J. E. M. (1994). Signal analytic proofs of two basic results on lattice expansions. Appl. Comput. Harmon. Anal. 1, 350-354.
45. Janssen, A. J. E. M. (1995). Duality and biorthogonality for Weyl-Heisenberg frames, J. Fourier Anal. Appl. 1, 403-436.
46. Kurbatov, V. G. (1990). Algebras of difference and integral operators. Funktsional. Anal. i Prilozhen. 24, 87-88;
47. translation in Funct. Anal. Appl. 24, (1990), 156-158.
48. p estimates on the bilinear Hubert transform for 2 < p < ∞, Ann. of Math. (2) 146, 693-724.
49. Landau, H. (1993). On the density of phase-space expansions, IEEE Trans. Inform. Theory 39, 1152-1156.
50. Margrave, G. F., Gibson, P. C., Grossman, J. P., Henley, D. C., Iliescu, V., and Lamoureux, M. P. (2005). The Gabor transform, pseudodifferential operators, and seismic deconvolution, Integrated Computer-Aided Engineering 12, 43-55.
51. Ramanathan, J. and Steger, T. (1995). Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal. 2, 148-153.
52. Rieffel, M. (1981). Von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257. 403-418.
53. Riesz, F. and Sz, B. (1990). Functional Analysis, Dover Publications, New York.
54. d). Duke Math. J. 89, 237-282.
55. Sjöstrand, J. (1995). Wiener type algebras of pseudodifferential perators, Séminaire sur les Equations aux Dérivées Partielles. 1994-1995, Exp. No. IV. École Polytech., Palaiseau.
56. Strohmer, T. and Beaver, S. (2003). Optimal OFDM system design for time-frequency dispersive channels. IEEE Trans. Communications 51, 1111-1122.
57. Strohmer, T. and Heath, Jr., R. W. (2003). Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal. 14, 257-275.
58. Sun, W. and Zhou, X. (2002). Irregular wavelet/Gabor frames, Appl. Comput. Harmon. Anal. 13, 63-76.
59. Young, R. (2001). An Introduction to Nonhannonic Fourier Series, revised 1st ed., Academic Press, San Diego.