The art of frame theory

Taiwanese Journal of Mathematics

Volumn 4, Issue 2, 2000, Pages 129-201

  Casazza, Peter G ^a  

^a UNIVERSITY OF MISSOURI   (United States)

Author keywords

(Gabor) frame; Fourier series; Hilbert space; Linear operator; Perturbation; Riesz basis; Weyl Heisenberg system

Indexed keywords

EID: 0001736880     PISSN: 10275487     EISSN: None     Source Type: Journal    
DOI: 10.11650/twjm/1500407227     Document Type: Article

References (140)

1. A. Aldroubi, Portraits of frames, Proc. Amer. Math. Soc. 123 (1995), 1661-1668.
2. A. Aldroubi and K. Gröchenig, Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces, J. Fourier Anal. Appl., to appear.
3. A. Aldroubi and M. Unser, Sampling procedures in function spaces and asymptotic equivalence with Shannon's sampling theory, Numer. Funct. Anal. Optim. 15 (1994), 1-21.
4. J. H. Anderson, Commutators in ideals of trace-class operators, II, Indiana Univ. Math. J. 35 (1986), 373-378.
5. L. Auslander and R. Tolimieri, Is computing with the finite Fourier transform pure or applied mathematics? Bull. Amer. Math. Soc. 1 (1979), 847-897.
6. L. Auslander and R. Tolimieri, Radar ambiguity functions and group theory, SIAM J. Math. Anal. 16 (1985), 577-599.
7. H. Bacry, A. Grossmann and J. Zak, Proof of completeness of lattice states in the kq representation, Phys. Rev. B 12 (1975), 1118-1120.
8. R. Balan, Stability theorems for Fourier frames and wavelet Riesz bases, J. Fourier Anal. Appl. 3 (1997), 499-504.
9. R. Balian, Un principe d'incertitude fort en théorie du signal on en mécanique quantique, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 1357-1362.
10. R. Balan and Z. Landau, private communication.
11. R. Balan and Z. Landau, private communication 2.
12. G. Battle, Heisenberg proof of the Balian-Low theorem, Lett. Math. Phys. 15 (1988), 175-177.
13. J. Benedetto, Gabor representations and wavelets, in: Commutative Harmonic Analysis, D. Colella, ed., Contemp. Math. 19, AMS, Providence, 1989, pp. 9-27.
14. J. Benedetto, Irregular Sampling and Frames, Wavelet Anal. Appl. 2, Academic Press, Boston, MA, 1992.
15. J. Benedetto, Frame decompositions, sampling, and uncertainty principle inequalities, preprint.
16. J. Benedetto and M. Frazier, eds., in: Wavelets: Mathematics and Applications, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1994.
17. J. Benedetto, C. Heil and D. Walnut, Differentiation and the Balian-Low Theorem, J. Fourier Anal. Appl. 1 (1995), 355-402.
18. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998), 389-427.
19. J. Benedetto and O.M. Treiber, A perfect reconstruction multirate system with narrow band filters, in: Wavelet Analysis and its Applications, L. Debnath, ed., Birkhauser, Boston, 1999.
20. J. Benedetto and O. M. Treiber, Wavelet frames: multiresolution analysis and extension principles, in: Wavelet Transforms and Time-frequency Signal Analysis, L. Debnath, ed., to appear.
21. J. Benedetto and S. Scott, Frames, irregular sampling, and a wavelet auditory model, in: Sampling Theory and Practice, F. Marvasti, ed., Plenum., to appear.
22. 2 and related spaces, in: Wavelets: Mathematics and Applications, J. J. Benedetto and M. W. Frazier, eds., CRC Press, Boca Raton, FL, Chapter 3, 1994.
23. J.W.M. Bergmans and A.J.E.M. Janssen, Robust data equalization, fractional tap spacing and the Zak transform, Philips J. Res. 42 (1987), 351-398.
24. M. Boon and J.H. Zak, Amplitudes on von Neumann lattices. J. Math. Phys. 22 (1981), 1090-1099.
25. M. Boon, J. Zak and I. J. Zucker, Rational von Neumann lattices, J. Math. Phys. 24 (1983), 316-323.
26. n), preprint.
27. P. G. Casazza, Characterizing Hilbert space frames with the subframe property, Illinois J. Math. 41 (1997), 648-666.
28. P. G. Casazza, Every frame is a sum of three (but not two) orthonormal bases and other frame representations, J. Fourier Anal. Appl. 4 (1998), 727-732.
29. P. G. Casazza, Local theory of frames and Schauder bases for Hilbert space, Illinois J. Math. 43 (1999), 291-306.
30. 0, J. Math. Anal. Appl. 202 (1996), 940-950.
31. P. G. Casazza and O. Christensen, Approximation of the frame coefficients using finite dimensional methods, J. Electron. Imaging, Special Volume "Best papers in electronic imaging 1997", 6 (1997), 479-483.
32. P. G. Casazza and O. Christensen, Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl. 3 (1997), 543-557.
33. P. G. Casazza and O. Christensen, Frames containing a Riesz basis and preservation of this property under perturbations, SIAM J. Math. Anal. 29 (1998), 266-278.
34. P. G. Casazza and O. Christensen, Riesz frames and approximation of the frame coefficients, Approx. Theory Appl. (N.S.) 14 (1998), 1-11.
35. P. G. Casazza and O. Christensen, Frames and Schauder bases, preprint.
36. 2(ℝ), preprint.
37. P. G. Casazza and O. Christensen, Approximation of the inverse frame operator and applications to Weyl-Heisenberg frames, preprint.
38. P. G. Casazza, O. Christensen and A.J.E.M. Janssen, Classifying tight Weyl-Heisenberg frames, preprint.
39. P. G. Casazza, O. Christensen and A.J.E.M. Janssen, Weyl-Heisenberg frames, translation-invariant systems, and the Walnut representation, preprint.
40. P. G. Casazza, O. Christensen and N. J. Kalton, Frames of translates, preprint.
41. P. G. Casazza, D. Han and D. R. Larson, Frames for Banach and Hilbert spaces, preprint.
42. P. G. Casazza and M. C. Lammers, Pointwise inner products and Weyl-Heisenberg frames, preprint.
43. P. G. Casazza and N. J. Kalton, Generalizing the Paley-Wiener perturbation theory for Banach spaces, Proc. Amer. Math. Soc. 127 (1999), 519-527.
44. O. Christensen, Frames and the projection method, Appl. Comput. Harmon. Anal. 1 (1993), 50-53.
45. O. Christensen, A Paley-Wiener theorem for frames, Proc. Amer. Math. Soc. 123 (1995), 2199-2202.
46. O. Christensen, Frames containing a Riesz basis and approximation of the frame coefficients using finite-dimensional methods, J. Math. Anal. Appl. 199 (1996), 256-270.
47. O. Christensen, Atomic decomposition via projective group representations, Rocky Mountain J. Math. 26 (1996), 1289-1312.
48. O. Christensen, B. Deng and C. Heil, Density of Gabor frames, Appl. Comput. Harmon. Anal., to appear.
49. O. Christensen and C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr. 85 (1997), 33-47.
50. T.A.C.M. Claasen and W.F.G. Mecklenbrauker, Wigner distribution - a tool for time-frequency signal analysis, Philips J. Res. 35 (1980), 217-250.
51. R. Coifman and Y. Zeevi, eds., Signal and Image Representation in Combined Spaces, Wavelet Ana. Appl., Vol 7, Academic Press, Boston, MA, 1988.
52. I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996.
53. I. Daubechies, Time-frequency localization operators: a geometric phase space approch, IEEE Trans. Inform. Theory 34 (1988), 605-612.
54. I. Daubechies, The wavelet transform, time-frequency locaization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), 961-1005.
55. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Math., Philadelphia, 1992.
56. I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271-1283.
57. I. Daubechies, S. Jaffard and J. L. Journé, A simple Wilson orthonormal basis with exponential decay, SIAM J. Math. Anal. 22 (1991), 554-572.
58. I. Daubechies and A.J.E.M. Janssen, Two theorems on lattice expansions, IEEE Trans. Inform. Theory 39 (1993), 3-6.
59. I. Daubechies, H. Landau and Z. Landau, Gabor time-frequency lattices and the Wexler-Rax identity, J. Fourier Anal. Appl. 1 (1995), 437-478.
60. I. Daubechies and T. Paul, Time-frequency localization operators - a geometric phase space approach: II: The use of dilations, Inverse Problems 4 (1988), 661-680.
61. J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math., Springer-Verlag, 1984.
62. R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366.
63. p Spaces, Academic Press, Boston, MA, 1970.
64. H. G. Feichtinger, Banach convolution algebras of Wiener type, in: Functions, Series, Operators, Proc. Conf. Budapest 38 Colloq. Math. Soc. János Bolyai, 1980, pp. 509-524.
65. H. G. Feichtinger, Atomic characterizations of modulation spaces through Gabor-type representations, Constructive Function Theory - (Edmonton, 1986), Rocky Mountain J. Math. 19 (1989), 113-125.
66. H. G. Feichtinger, O. Christensen and T. Strohmer, Group theoretical approach to Gabor analysis, Opt. Engrg. 34 (1995), 1697-1704.
67. H. G. Feichtinger and P. Grobner, Banach spaces of distributions defined by decomposition methods, I, Math. Nachr. 123 (1985), 97-120.
68. H. G. Feichtinger and K. Gröchenig, A unified approach to atomic decompositions through integrable group representations, in: Function Spaces and Applications, M. Cwikel et al., eds., Lecture Notes in Math. 1302, Springer-Verlag, Berlin, 1988, pp. 52-73.
69. H. G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal. 86 (1989), 307-340.
70. H. G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, H, Monatsh. Math. 108 (1989), 129-148.
71. H. G. Feichtinger and K. Gröchenig, Gabor frames and time-frequency analysis of distributions, J. Funct. Anal. 146 (1997), 464-495.
72. H. G. Feichtinger and A.J.E.M. Janssen, Validity of WH-frame bound conditions depends on lattice parameters, Appl. Comput. Harmon. Anal., to appear.
73. H. G. Feichtinger and T. Strohmer, eds., Gabor Analysis and Algorithms: Theory and Applications, Birkhäuser, Boston, 1998.
74. H. G. Feichtinger and G. Zimmermann, A Banach space of test functions for Gabor analysis, in: Gabor Analysis and Algorithms: Theory and Applications, H. G. Feichtinger and T. Strohmer, eds., Birkhäuser, Boston, 1988, pp. 123-170.
75. M. Frank and D. R. Larson, Frames in Hilbert C*-modules and C*-algebras, preprint.
76. M. Frank, V. I. Paulsen and T. R. Tiballi, Symmetric approximation of frames and bases in Hilbert spaces, preprint.
77. M. Frazier and B. Jawerth, Decompositions of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799.
78. M. Frazier and B. Jawerth, The φ-transform and applications to distribution spaces, in: Function Spaces and Applications (Lund, 1986), Lecture Notes in Math., 1302, Springer, Berlin-New York, 1988, pp. 223-246.
79. M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34-170.
80. p, Bull. Amer. Math. Soc. 13 (1985), 1-21.
81. J. P. Gabardo and D. Han, Subspace Weyl-Heisenberg frames, preprint.
82. J. P. Gabardo and D. Han, private communication.
83. D. Gabor, Theory of communications, J. Inst. Elec. Engrg. 93 (1946), 429-457.
84. I. Gohberg and S. Goldberg, Basic Operator Theory, Birkhäuser, Boston, 1981.
85. P. Goupillaud, A. Grossmann and J. Morlet, Cycle-octave and related transforms in seismic signal analysis, Geoexploration 23 (1985), 85-102.
86. K. Gröchenig, Describing functions: Atomic decompositions versus frames, Monatsh. Math. 112 (1991), 1-41.
87. K. Grochenig, Foundations of Time-Frequency Analysis, Birkhäuser, 2000, to appear.
88. A. Grossmann, Wavelet transforms and edge detection, in: Stochastic Processing in Physics and Engineering, S. Albeverio et al., eds., D. Reidel, Dordrecht, the Netherlands, 1988, pp. 149-157.
89. A. Grossmann and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984), 723-736.
90. A. Grossmann, J. Morlet and T. Paul, Transforms associated to square integrable group representations, I, General results, J. Math. Phys. 26 (1985), 2473-2479.
91. P. R. Halmos, A Hilbert Space Problem Book, Van Nostrand, Princeton, N. J., 1967.
92. D. Han and D. Larson, Frames, Bases and Group Representations, Mem. Amer. Math. Soc., to appear.
93. C. Heil, Wiener Amalgam Spaces in Generalized Harmonic Analysis and Wavelet Theory, Ph.D. Thesis, University of Maryland, College Park, MD, 1990.
94. C. Heil, J. Ramanathan and P. Topiwala, Linear independence of time-frequency shifts, Proc. Amer. Math. Soc. 124 (1996), 2787-2795.
95. C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), 628-666.
96. S. H. Hilding, Note on completeness theorems of Paley-Wiener type, Ann. Math. 48 (1948), 953-955.
97. J. R. Holub, Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces, Proc. Amer. Math. Soc. 122 (1994), 779-785.
98. A.J.E.M. Janssen, Bergmann transform, Zak transform, and coherent states, J. Math. Phys. 23 (1982), 720-731.
99. A.J.E.M. Janssen, The Zak transform: a signal transorm for sampled time-continuous signals, Philips J. Res. 43 (1988), 720-731.
100. A.J.E.M. Janssen, Signal analytic proofs of two basic results on lattice expansions, Appl. Comput. Harmon. Anal. 1 (1994), 350-354.
101. A.J.E.M. Janssen, Duality and biorthogonality for Weyl-Heisenberg frames, J. Fourier Anal. Appl. 1 (1995), 403-436.
102. A.J.E.M. Janssen, The duality condition for Weyl-Heisenberg frames, in: Gabor Analysis and Algorithms: Theory and Applications, H. G. Feichtinger and T. Strohmer, eds., Birkhäuser, Boston 1998, pp. 33-84.
103. Y. Katznelson, An Introduction to Harmonic Analysis, John Wiley, New York, 1968.
104. H. O. Kim and J. K. Lim, Frame multiresolution analysis, preprint.
105. H. O. Kim and J. K. Lim, New characterizations of Riesz bases, Appl. Comput. Harmon. Anal., to appear.
106. W. Kohn, Analytic properties of Bloch waves and Wannier functions, Phys. Rev. 115 (1959), 809-821.
107. S. Li, On general frame decompositions, Mumer. Funct. Anal. Optim. 16 (1995), 1181-1191.
108. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Ergegnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1977.
109. P. Linnell, Von Neumann algebras and linear independence of translates, Proc. Amer. Math. Soc., to appear.
110. F. Low, Complete sets of wave packets, in: A Passion for Physics - Essays in Honor of Geoffrey Chew, World Scientific, Singapore, 1985, pp. 17-22.
111. Y. I. Lyubarskii, Frames in the Bargmann space of entire functions, Adv. Soviet Math. 429 (1992), 107-113.
112. Y. I. Lyubarskii and K. Seip, Sampling and interpolation sequences for multiband-limited functions and exponential bases on disconnected sets, J. Fourier Anal. Appl. 3 (1997), 597-615.
113. Y. I. Lyubarskii and K. Seip, Convergence and summability of Gabor expansions at the Nyquist density, J. Fourier Anal. Appl. 5 (1999), 127-157.
114. 2(ℝ), Trans. Amer. Math. Soc. 315 (1989), 69-87.
115. Y. Meyer, Principe d'incertitude, Bases Hilbertiennes et Algébres d'opeérateurs, Séminaire Bourbaki 662, 1985/86.
116. Y. Meyer, Ondelettes et Fonctions Splines, Séminaire EDP, Ecole Polytechnique, Paris, 1986.
117. Y. Meyer, Wavelets with Compact Support, Zygmund Lectures, University of Chicago, 1987.
118. C. Neumann, Untersuchungen uber das Logarithmische und Newtonsche Potential, Teubner, Leipzig, 1877.
119. T. E. Olson and R. A. Zalik, Nonexistence of a Riesz basis of translates, in: Approx. Theory, Proceedings of the Sixth Southeastern Approximation Theorists Annual Conference, G. A. Anastassiou, ed., Marcel Dekker, New York, 1992.
120. R.E.A.C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, AMS Colloquium Publications, 19, 1934.
121. T. Rado and P. Reichelderfer, Continuous Transformations in Analysis, Springer-Verlag, Berlin, N.Y., 1955.
122. J. Ramanathan and T. Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal. 2 (1995), 148-153.
123. J. J. Rehr, D. J. Sullivan, J. W. Wilkins and K. G. Wilson, Phase space Wannier functions in electronic structure calculations, Cornell University preprint, 1987.
124. M. A. Rieffel, Von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257 (1981), 403-418.
125. d), Duke Math. J. 89 (1997), 237-281.
126. d), Canad. J. Math. 47 (1995), 1051-1094.
127. W. Schempp, Radar ambiguity functions, the Heisenberg group, and holomorphic theta series, Proc. Amer. Math. Soc. 92 (1984), 103-110.
128. O. Scherzer and T. Strohmer, A multi-level algorithm for the solution of moment problems, preprint.
129. 2(-π, π), J. Funct. Anal. 130 (1995), 131-160.
130. K. Seip and R. Wallsten, Density theorems for sampling and interpolation in the Bergmann-Fock space II, J. Reine Angew. Math. 429 (1992), 107-113.
131. C. E. Shannon, Communications in the presence of noise, Proc. IRE 37 (1949), 10-21.
132. R. Vershynin, Frames and contact points of convex bodies, preprint.
133. D. Walnut, Weyl-Heisenberg Wavelet Expansions: Existence and Stability in Weighted Spaces, Ph.D. thesis, University of Maryland, College Park, MD, 1989.
134. J. Wexler and S. Raz, Discrete Gabor expansions, Signal Process. Digit. Filtering 21 (1990), 207-220.
135. J. M. Whittaaker, Interpolation Function Theory, Cambridge University Press, Cambridge, 1935.
136. K. G. Wilson, Generalized Wannier functions, Cornell University preprint, 1987.
137. The Wutam Consortium, Basic properties of wavelets, J. Fourier Anal. Appl., to appear.
138. R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.
139. J. Zak, Finite translations in solid state physics, Phys. Rev. Lett. 19 (1967), 1385-1397.
140. A. Zygmund, Trigonometric Series, Vol. I, Cambridge University Press, 1959.