Linear Independence of Finite Gabor Systems

Applied and Numerical Harmonic Analysis

Volumn , Issue 9780817637781, 2006, Pages 171-206

  Heil, Christopher ^a  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Gabor Frame; Heisenberg Group; Riesz Basis; Tight Frame; Wavelet Frame

Indexed keywords

EID: 85047218606     PISSN: 22965009     EISSN: 22965017     Source Type: Book Series    
DOI: 10.1007/0-8176-4504-7_9     Document Type: Chapter

References (116)

1. d), Appl. Comput. Harmon. Anal., 17 (2004), pp. 119–140.
2. A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Review, 43 (2001), pp. 585–620.
3. K. I. Babenko, On conjugate functions (Russian), Doklady Akad. Nauk SSSR (N. S.), 62 (1948), pp. 157–160.
4. L. Baggett, Processing a radar signal and representations of the discrete Heisenberg group, Colloq. Math., 60/61 (1990), pp. 195–203.
5. n, J. Fourier Anal. Appl., 6 (1999), pp. 563–573.
6. R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Deficits and excesses of frames, Adv. Comput. Math., 18 (2003), pp. 93–116.
7. R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Excesses of Gabor frames, Appl. Comput. Harmon. Anal., 14 (2003), pp. 87–106.
8. R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Density, overcompleteness, and localization of frames, I. Theory, J. Fourier Anal. Appl., 12 (2006), pp. 105– 143.
9. L. Colzani and M. Vignati, The Gibbs phenomenon for multiple Fourier integrals, J. Approx. Theory, 80 (1995), pp. 119–131.
10. R. Balian, Un principe d’incertitude fort en théorie du signal ou en mécanique quantique, C. R. Acad. Sci. Paris, 292 (1981), pp. 1357–1362.
11. G. Battle, Heisenberg proof of the Balian–Low theorem, Lett. Math. Phys., 15 (1988), pp. 175–177.
12. J. J. Benedetto, Gabor representations and wavelets, in: Commutative Harmonic Analysis (Canton, NY, 1987), Contemp. Math., Vol. 91, Amer. Math. Soc., Providence, RI, 1989, pp. 9–27.
13. J. J. Benedetto, Harmonic Analysis and Applications, CRC Press, Boca Raton, FL, 1997.
14. J. J. Benedetto, W. Czaja, P. Gadziński, and A. M. Powell, The Balian–Low Theorem and regularity of Gabor systems, J. Geom. Anal., 13 (2003), pp. 217– 232.
15. 2d, J. Math. Phys., 44 (2003), pp. 1735–1750.
16. J. J. Benedetto and M. Fickus, Finite normalized tight frames, Adv. Comput. Math., 18 (2003), pp. 357–385.
17. J. J. Benedetto, C. Heil, and D. F. Walnut, Differentiation and the Balian–Low Theorem, J. Fourier Anal. Appl., 1 (1995), pp. 355–402.
18. J. J. Benedetto and M. Leon, The construction of single wavelets in D- dimensions, J. Geom. Anal., 11 (2001), pp. 1–15.
19. A. Boggess and F. J. Narcowich, A First Course in Wavelets with Fourier Analysis, Prentice-Hall, Upper Saddle River, NJ, 2001.
20. M. Bownik and Z. Rzeszotnik, The spectral function of shift-invariant spaces, Michigan Math. J., 51 (2003), pp. 387–414.
21. C. A. Cabrelli, C. Heil, and U. M. Molter, Self-similarity and multiwavelets in higher dimensions, Memoirs Amer. Math. Soc., 170, No. 807 (2004).
22. P. G. Casazza, The art of frame theory, Taiwanese J. Math., 4 (2000), pp. 129– 201.
23. P. G. Casazza, O. Christensen, A. M. Lindner, and R. Vershynin, Frames and the Feichtinger conjecture, Proc. Amer. Math. Soc., 133 (2005), pp. 1025–1033.
24. P. G. Casazza, M. Fickus, J. Kovačević, M. T. Leon, and J. C. Tremain, A physical interpretation of tight frames, Chapter 4, this volume (2006).
25. P. G. Casazza and N. J. Kalton, Roots of complex polynomials and Weyl– Heisenberg frame sets, Proc. Amer. Math. Soc., 130 (2002), pp. 2313–2318.
26. P. G. Casazza and J. C. Tremain, The Kadison–Singer problem in mathematics and engineering, Proc. Natl. Acad. Sci. USA, 103 (2006), pp. 2032–2039.
27. S. Chen, D. Donoho, and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM Review, 43 (2001), pp. 129–157.
28. O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003.
29. O. Christensen and P. G. Casazza, Frames containing a Riesz basis and preservation of this property under perturbations, SIAM J. Math. Anal., 29 (1998), pp. 266–278.
30. O. Christensen and Y. C. Eldar, Oblique dual frames and shift-invariant spaces, Appl. Comput. Harmon. Anal., 17 (2004), pp. 48–68.
31. O. Christensen and C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr., 185 (1997), pp. 33–47.
32. O. Christensen, B. Deng, and C. Heil, Density of Gabor frames, Appl. Comput. Harmon. Anal., 7 (1999), pp. 292–304.
33. O. Christensen and A. M. Lindner, Frames of exponentials: lower frame bounds for finite subfamilies, and approximation of the inverse frame operator, Linear Algebra Appl., 323 (2001), pp. 117–130.
34. O. Christensen and A. M. Lindner, Lower bounds for finite wavelet and Gabor systems, Approx. Theory Appl. (N.S.), 17 (2001), pp. 18–29.
35. O. Christensen and A. M. Lindner, Decompositions of wavelets and Riesz frames into a finite number of linearly independent sets, Linear Algebra Appl., 355 (2002), pp. 147–159.
36. E. Cordero and K. Gröchenig, Localization of frames. II, Appl. Comput. Harmon. Anal., 17 (2004), pp. 29–47.
37. W. Czaja, G. Kutyniok, and D. Speegle, Geometry of sets of parameters of wave packet framess, Appl. Comput. Harmon. Anal., 20 (2006), pp. 108–125.
38. W. Czaja and A. Powell, Recent developments in the Balian–Low Theorem, Chapter 5, this volume (2006).
39. n, J. Fourier Anal. Appl., 3 (1997), pp. 451–456.
40. I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 36 (1990), pp. 961–1005.
41. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.
42. I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27 (1986), pp. 1271–1283.
43. I. Daubechies and A. J. E. M. Janssen, Two theorems on lattice expansions, IEEE Trans. Inform. Theory, 39 (1993), pp. 3–6.
44. I. Daubechies, H. Landau, and Z. Landau, Gabor time-frequency lattices and the Wexler–Raz identity, J. Fourier Anal. Appl., 1 (1995), pp. 437–478.
45. L. Colzani and M. Vignati, The Gibbs phenomenon for multiple Fourier integrals, J. Approx. Theory, 80 (1995), pp. 119–131.
46. R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341–366.
47. K. Dykema, D. Freeman, K. Kornelson, D. Larson, M. Ordower, and E. Weber, Ellipsoidal tight frames and projection decomposition of operators, Illinois J. Math., 48 (2004), pp. 477–489.
48. G. Edgar and J. Rosenblatt, Difference equations over locally compact abelian groups, Trans. Amer. Math. Soc., 253 (1979), pp. 273–289.
49. G. B. Folland, Harmonic Analysis on Phase Space, Princeton Univ. Press, Princeton, NJ, 1989.
50. G. B. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl., 3 (1997), pp. 207–238.
51. D. Gabor, Theory of communication, J. Inst. Elec. Eng. (London), 93 (1946), pp. 429–457.
52. J.-P. Gabardo and D. Han, Aspects of Gabor analysis and operator algebras, in: Advances in Gabor Analysis, H. G. Feichtinger and T. Strohmer, eds., Birkhäuser, Boston, 2003, pp. 153–169.
53. J.-P. Gabardo and D. Han, Balian–Low phenomenon for subspace Gabor frames, J. Math. Phys., 45 (2004), pp. 3362–3378.
54. B. R. Gelbaum, Notes on Banach spaces and bases, An. Acad. Brasil. Ci., 30 (1958), pp. 29–36.
55. K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001.
56. K. Gröchenig, Uncertainty principles for time-frequency representations, in: Advances in Gabor Analysis, H. G. Feichtinger and T. Strohmer, eds., Birkhäuser, Boston, 2003, pp. 11–30.
57. K. Gröchenig, Localized frames are finite unions of Riesz sequences, Adv. Comput. Math., 18 (2003), pp. 149–157.
58. K. Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl., 10 (2004), pp. 105–132.
59. K. Gröchenig, D. Han, C. Heil, and G. Kutyniok, The Balian–Low Theorem for symplectic lattices in higher dimensions, Appl. Comput. Harmon. Anal., 13 (2002), pp. 169–176.
60. K. Gröchenig and H. Razafinjatovo, On Landau’s necessary density conditions for sampling and interpolation of band-limited functions, J. London Math. Soc. (2), 54 (1996), pp. 557–565.
61. D. Han and D. R. Larson, Frames, bases and group representations Memoirs Amer. Math. Soc., 147, No. 697 (2000).
62. E. Hayashi, S. Li, and T. Sorrells, Gabor duality characterizations, Chapter 7, this volume (2006).
63. C. Heil, Wiener Amalgam Spaces in Generalized Harmonic Analysis and Wavelet Theory, Ph.D. Thesis, University of Maryland, College Park, MD, 1990.
64. L. Colzani and M. Vignati, The Gibbs phenomenon for multiple Fourier integrals, J. Approx. Theory, 80 (1995), pp. 119–131.
65. C. Heil, An introduction to weighted Wiener amalgams, in: Wavelets and their Applications (Chennai, January 2002), M. Krishna, R. Radha and S. Thangavelu, eds., Allied Publishers, New Delhi, 2003, pp. 183–216.
66. L. Colzani and M. Vignati, The Gibbs phenomenon for multiple Fourier integrals, J. Approx. Theory, 80 (1995), pp. 119–131.
67. C. Heil and G. Kutyniok, Density of weighted wavelet frames, J. Geom. Anal., 13 (2003), pp. 479–493.
68. C. Heil and G. Kutyniok, The Homogeneous Approximation Property for wavelet frames, preprint (2005).
69. C. Heil, J. Ramanathan, and P. Topiwala, Linear independence of time-frequency translates, Proc. Amer. Math. Soc., 124 (1996), pp. 2787–2795.
70. C. Heil, J. Ramanathan, and P. Topiwala, Singular values of compact pseudodifferential operators, J. Funct. Anal., 150 (1996), pp. 426–452.
71. C. E. Heil and D. F. Walnut, Continuous and discrete wavelet transforms, SIAM Review, 31 (1989), pp. 628–666.
72. C. Heil and D. F. Walnut, eds., Fundamental Papers in Wavelet Theory, Princeton University Press, Princeton, NJ, 2006.
73. E. Hernández, D. Labate, and G. Weiss, A unified characterization of reproducing systems generated by a finite family, II, J. Geom. Anal., 12 (2002), pp. 615–662.
74. E. Hernández and G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, FL, 1996.
75. G. Higman, The units of group-rings, Proc. London Math. Soc. (2), 46 (1940), pp. 231–248.
76. J. R. Holub, Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces, Proc. Amer. Math. Soc., 122 (1994), pp. 779–785.
77. A. J. E. M. Janssen, Bargmann transform, Zak transform, and coherent states, J. Math. Phys., 23 (1982), pp. 720–731.
78. A. J. E. M. Janssen, The Zak transform: a signal transform for sampled time-continuous signals, Philips J. Res., 43 (1988), pp. 23–69.
79. A. J. E. M. Janssen, Signal analytic proofs of two basic results on lattice expansions, Appl. Comput. Harmon. Anal., 1 (1994), pp. 350–354.
80. A. J. E. M. Janssen, Duality and biorthogonality for Weyl–Heisenberg frames, J. Fourier Anal. Appl., 1 (1995), pp. 403–436.
81. A. J. E. M. Janssen, Representations of Gabor frame operators, in: Twentieth Century Harmonic Analysis—A Celebration (Il Ciocco, 2000), NATO Sci. Ser. II Math. Phys. Chem., 33, Kluwer Acad. Publ., Dordrecht, 2001, pp. 73–101.
82. A. J. E. M. Janssen, Zak transforms with few zeros and the tie, in: Advances in Gabor Analysis, H. G. Feichtinger and T. Strohmer, eds., Birkhäuser, Boston, 2003, pp. 31–70.
83. A. J. E. M. Janssen, On generating tight Gabor frames at critical density, J. Fourier Anal. Appl., 9 (2003), pp. 175–214.
84. A. J. E. M. Janssen and T. Strohmer, Hyperbolic secants yield Gabor frames, Appl. Comput. Harmon. Anal., 12 (2002), pp. 259–267.
85. R. Kadison and I. Singer, Extensions of pure states, Amer. J. Math., 81 (1959), pp. 383–400.
86. L. Colzani and M. Vignati, The Gibbs phenomenon for multiple Fourier integrals, J. Approx. Theory, 80 (1995), pp. 119–131
87. H. O. Kim and J. K. Lim, New characterizations of Riesz bases, Appl. Comput. Harmon. Anal., 4 (1997), pp. 222–229.
88. G. Kutyniok, Linear independence of time-frequency shifts under a generalized Schrödinger representation, Arch. Math. (Basel), 78 (2002), pp. 135–144.
89. G. Kutyniok, Computation of the density of weighted wavelet systems, in: Wavelets: Applications in Signal and Image Processing, M. Unser, A. Aldroubi, and A. Laine eds., SPIE Proc. Vol. 5207, SPIE, San Diego, 2003, pp. 393–404.
90. D. Labate, A unified characterization of reproducing systems generated by a finite family, J. Geom. Anal., 12 (2002), pp. 469–491.
91. H. Landau, On the density of phase-space expansions, IEEE Trans. Inform. Theory, 39 (1993), pp. 1152–1156.
92. S. Li and H. Ogawa, Pseudo-duals of frames with applications, Appl. Comput. Harmon. Anal., 11 (2001), 289–304.
93. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, I, Springer-Verlag, New York, 1977.
94. 2 (G), C. R. Acad. Sci. Paris Sér. I Math., 315 (1992), pp. 49–53.
95. P. A. Linnell, Analytic versions of the zero divisor conjecture, in: Geometry and Cohomology in Group Theory, P. H. Kropholler, G. A. Niblo, and R. Stöhr, eds., London Math. Soc. Lecture Note Series, Cambridge University Press, Cambridge, 1998, pp. 209–248.
96. P. A. Linnell, Von Neumann algebras and linear independence of translates, Proc. Amer. Math. Soc., 127 (1999), pp. 3269–3277.
97. L. Colzani and M. Vignati, The Gibbs phenomenon for multiple Fourier integrals, J. Approx. Theory, 80 (1995), pp. 119–131.
98. J. Marti, Introduction to the Theory of Bases, Springer-Verlag, New York, 1969.
99. 2 (R) of almost integer translates, C. R. Acad. Sci. Paris, 324 (1997), pp. 987–991.
100. A. Olevskii and A. Ulanovskii, Almost integer translates. Do nice generators exist?, J. Fourier Anal. Appl., 10 (2004), pp. 93–104.
101. T. E. Olson and R. A. Zalik, Nonexistence of a Riesz basis of translates, in: Approximation Theory, Lecture Notes in Pure and Applied Math., Vol. 138, Dekker, New York, 1992, pp. 401–408.
102. A. M. Perelomov, On the completeness of a system of coherent states (English translation), Theoret. Math. Phys., 6 (1971), pp. 156–164.
103. J. Ramanathan and T. Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal., 2 (1995), pp. 148–153.
104. M. Rieffel, Von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann., 257 (1981), pp. 403–418.
105. J. Rosenblatt, Linear independence of translations, J. Austral. Math. Soc. (Series A), 59 (1995), pp. 131–133.
106. Z. Rzeszotnik, private communication, 2004.
107. 2 (−π, π), J. Funct. Anal., 130 (1995), pp. 131–160.
108. I. Singer, Bases in Banach Spaces I, Springer-Verlag, New York, 1970.
109. T. Strohmer and R. W. Heath, Jr., Grassmannian frames with applications to coding and communication, Appl. Comput. Harmon. Anal., 14 (2003), pp. 257– 275.
110. W. Sun and X. Zhou, Density and stability of wavelet frames, Appl. Comput. Harmon. Anal., 15 (2003), pp. 117–133.
111. D. F. Walnut, An Introduction to Wavelet Analysis, Birkhäuser, Boston, 2002.
112. P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982.
113. Y. Wang, Sparse complete Gabor systems on a lattice, Appl. Comput. Harmon. Anal., 16 (2004), pp. 60–67.
114. R. Young, An Introduction to Nonharmonic Fourier Series, Revised First Edition, Academic Press, San Diego, 2001.
115. R. A. Zalik, On approximation by shifts and a theorem of Wiener, Trans. Amer. Math. Soc., 243 (1978), pp. 299–308.
116. R. A. Zalik, On fundamental sequences of translates, Proc. Amer. Math. Soc., 79 (1980), pp. 255–259.