Varying the time-frequency lattice of Gabor frames

Transactions of the American Mathematical Society

Volumn 356, Issue 5, 2004, Pages 2001-2023

  Feichtinger, Hans G ^a   Kaiblinger, Norbert ^a,b  

^a UNIVERSITY OF VIENNA   (Austria)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Coherent states; Dual atom; Gabor frame; Modulation space; Perturbation; Riesz basis; Stability; Time frequency lattice; Twisted convolution; Weyl Heisenberg frame

Indexed keywords

EID: 2142674353     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-03-03377-4     Document Type: Article

References (41)

1. P. G. Casazza and O. Christensen, Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl. 3 (1997), no. 5, 543-557. MR 98j:47028
2. P. G. Casazza, O. Christensen, and A. J. E. M. Janssen, Weyl-Heisenberg frames, translation invariant systems and the Walnut representation, J. Funct. Anal. 180 (2001), no. 1, 85-147. MR 2002b:42042
3. P. G. Casazza, O. Christensen, and M. C. Lammers, Perturbations of Weyl-Heisenberg frames, Hokkaido Math. J. 31 (2002), no. 3, 539-553. MR 2003h:42048
4. O. Christensen, Frame perturbations, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1217-1220. MR 95e:46024
5. _, Moment problems and stability results for frames with applications to irregular sampling and Gabor frames, Appl. Comput. Harmon. Anal. 3 (1996), no. 1, 82-86. MR 97f:44007
6. _, Perturbation of frames and applications to Gabor frames, in [19], pp. 193-209. MR 98j:42017
7. _, Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 3, 273-291. MR 2002c:42040
8. O. Christensen and C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr. 185 (1997), 33-47. MR 98m:42061
9. O. Christensen, C. Lennard, and C. Lewis, Perturbation of frames for a subspace of a Hilbert space, Rocky Mountain J. Math. 30 (2000), no. 4, 1237-1249. MR 2002b:42043
10. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. MR 93e:42045
11. I. Daubechies, H. J. Landau, and Z. Landau, Gabor time-frequency lattices and the Wexler-Raz identity, J. Fourier Anal. Appl. 1 (1995), no. 4, 437-478. MR 961:42021
12. S. J. Favier and R. A. Zalik, On the stability of frames and Riesz bases, Appl. Comput. Harmon. Anal. 2 (1995), no. 2, 160-173. MR 96e:42030
13. H. G. Feichtinger, On a new Segal algebra, Monatsh. Math. 92 (1981), no. 4, 269-289. MR 83a:43002
14. H. G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions I. J. Funct. Anal. 86 (1989), no. 2, 307-340. MR 91g:43011
15. _, Gabor frames and time-frequency analysis of distributions, J. Funct. Anal. 146 (1997), no. 2, 464-495. MR 98k:42041
16. H. G. Feichtinger and A. J. E. M. Janssen, Validity of WH-frame bound conditions depends on lattice parameters, Appl. Comput. Harmon. Anal. 8 (2000), no. 1, 104-112. MR 2000j:42044
17. H. G. Feichtinger and W. Kozek, Quantization of TF lattice-invariant operators on elementary LCA groups, in [19], pp. 233-266. MR 98j:42019
18. H. G. Feichtinger and K. Nowak, A first survey of Gabor multipliers, in [20], pp. 99-128.
19. H. G. Feichtinger and T. Strohmer (eds.), Gabor Analysis and Algorithms, Birkhäuser, Boston, 1998. MR 98h:42001
20. H. G. Feichtinger and T. Strohmer (eds.), Advances in Gabor Analysis, Birkhäuser, Boston, 2003.
21. H. G. Feichtinger and G. Zimmermann, A Banach space of test functions for Gabor analysis, in [19], pp. 123-170. MR 98j:42018
22. J.-P. Gabardo and D. Han, Aspects of Gabor analysis and operator algebras, in [20], pp. 129-152.
23. n) using Gabor frames, J. Fourier Anal. Appl. 7 (2001), no. 2, 101-126. MR 2002b:42045
24. N. Grip and W. Sun, Remarks on [41], J. Fourier Anal. Appl. 9 (2003), no. 1, 97-100.
25. K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001. MR 2002h:42001
26. d), Studia Math. 146 (2001), no. 1, 15-33. MR 2002b:42028
27. K. Gröchenig, C. Heil, and K. Okoudjou, Gabor analysis in weighted amalgam spaces, Sampl. Theory Signal Image Process. 1 (2002), no. 3, 225-260.
28. K. Gröchenig and A. J. E. M. Janssen, A new criterion for Gabor frames, letter to the editor, J. Fourier Anal. Appl. 8 (2002), no. 5, 507-512. MR 2003g:42048
29. K. Gröchenig and M. Leinert, Wiener's lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc., to appear.
30. C. E. Heil and D. F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), no. 4, 628-666. MR 91c:42032
31. A. J. E. M. Janssen, Duality and biorthogonality for Weyl-Heisenberg frames, J. Fourier Anal. Appl. 1 (1995), no. 4, 403-436, MR 97e:42007
32. _, On generating tight Gabor frames at critical density, J. Fourier Anal. Appl. 9 (2003), no. 2, 175-214.
33. _, Zak transforms with few zeros and the tie, in [20], pp. 31-70.
34. A. J. E. M. Janssen and T. Strohmer, Hyperbolic secants yield Gabor frames, Appl. Comput. Harmon. Anal. 12 (2002), no. 2, 259-267. MR 2002k:42073
35. K. Okoudjou, Embeddings of some classical Banach spaces into modulation spaces, Proc. Amer. Math. Soc., to appear.
36. d), Duke Math. J. 89 (1997), no. 2, 237-282. MR 981:42013
37. W. Sun and X. Zhou, On Kadec's 1/4-theorem and the stability of Gabor frames, Appl. Comput. Harmon. Anal. 7 (1999), no. 2, 239-242. MR 2000j:42047
38. _, On the stability of Gabor frames, Adv. in Appl. Math. 26 (2001), no. 3, 181-191. MR 2002c:42043
39. D. F. Walnut, Continuity properties of the Gabor frame operator, 3. Math. Anal. Appl. 165 (1992), no. 2, 479-504. MR 93f:42059
40. J. Wexler and S. Raz, Discrete Gabor expansions, Signal Process. 21 (1990), no. 3, 207-221.
41. J. Zhang, On the stability of wavelet and Gabor frames (Riesz bases), J. Fourier Anal. Appl. 5 (1999), no. 1, 105-125. MR 2000a:42055