Irregular Gabor frames and their stability

Sun, Wenchang; Zhou, Xingwei

doi:10.1090/S0002-9939-02-06931-9

Irregular Gabor frames and their stability
HTML articles powered by AMS MathViewer

by Wenchang Sun and Xingwei Zhou PDF

Proc. Amer. Math. Soc. 131 (2003), 2883-2893 Request permission

Abstract:

In this paper we give sufficient conditions for irregular Gabor systems to be frames. We show that for a large class of window functions, every relatively uniformly discrete sequence in $\mathbb R^2$ with sufficiently high density will generate a Gabor frame. Explicit frame bounds are given. We also study the stability of irregular Gabor frames and show that every Gabor frame with arbitrary time-frequency parameters is stable if the window function is nice enough. Explicit stability bounds are given.

References

H.Bacry, A.Grossmann, and J.Zak, Proofs of the completeness of lattice states in the $kq$-representation, Phys. Rev., B12(1975), 1118-1120.
V. Bargmann, P. Butera, L. Girardello, and John R. Klauder, On the completeness of the coherent states, Rep. Mathematical Phys. 2 (1971), no. 4, 221–228. MR 290680, DOI 10.1016/0034-4877(71)90006-1
John J. Benedetto, Christopher Heil, and David F. Walnut, Differentiation and the Balian-Low theorem, J. Fourier Anal. Appl. 1 (1995), no. 4, 355–402. MR 1350699, DOI 10.1007/s00041-001-4016-5
Peter G. Casazza and Ole Christensen, Weyl-Heisenberg frames for subspaces of $L^2(\mathbf R)$, Proc. Amer. Math. Soc. 129 (2001), no. 1, 145–154. MR 1784021, DOI 10.1090/S0002-9939-00-05731-2
Ole Christensen, Atomic decomposition via projective group representations, Rocky Mountain J. Math. 26 (1996), no. 4, 1289–1312. MR 1447588, DOI 10.1216/rmjm/1181071989
Ole Christensen, 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 1374398, DOI 10.1006/acha.1996.0007
Peter G. Cazassa and Ole Christensen, Perturbation of operators and applications to frame theory, J. Fourier Anal. Appl. 3 (1997), no. 5, 543–557. Dedicated to the memory of Richard J. Duffin. MR 1491933, DOI 10.1007/BF02648883
Ole Christensen, Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 3, 273–291. MR 1824891, DOI 10.1090/S0273-0979-01-00903-X
Ole Christensen, Baiqiao Deng, and Christopher Heil, Density of Gabor frames, Appl. Comput. Harmon. Anal. 7 (1999), no. 3, 292–304. MR 1721808, DOI 10.1006/acha.1999.0271
Charles K. Chui and Xian Liang Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal. 24 (1993), no. 1, 263–277. MR 1199539, DOI 10.1137/0524017
Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), no. 5, 961–1005. MR 1066587, DOI 10.1109/18.57199
Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
Ingrid Daubechies and A. Grossmann, Frames in the Bargmann space of entire functions, Comm. Pure Appl. Math. 41 (1988), no. 2, 151–164. MR 924682, DOI 10.1002/cpa.3160410203
Ingrid Daubechies, H. J. Landau, and Zeph Landau, Gabor time-frequency lattices and the Wexler-Raz identity, J. Fourier Anal. Appl. 1 (1995), no. 4, 437–478. MR 1350701, DOI 10.1007/s00041-001-4018-3
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 1325538, DOI 10.1006/acha.1995.1012
Hans G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal. 86 (1989), no. 2, 307–340. MR 1021139, DOI 10.1016/0022-1236(89)90055-4
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 1734849, DOI 10.1006/acha.2000.0281
Hans G. Feichtinger and Thomas Strohmer (eds.), Gabor analysis and algorithms, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 1998. Theory and applications. MR 1601119, DOI 10.1007/978-1-4612-2016-9
D.Gabor, Theory of communications, J. Inst. Elect. Eng.(London), 93(1943), 429–457.
Karlheinz Gröchenig, Describing functions: atomic decompositions versus frames, Monatsh. Math. 112 (1991), no. 1, 1–42. MR 1122103, DOI 10.1007/BF01321715
Karlheinz Gröchenig, Irregular sampling of wavelet and short-time Fourier transforms, Constr. Approx. 9 (1993), no. 2-3, 283–297. MR 1215773, DOI 10.1007/BF01198007
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Christopher E. Heil and David F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), no. 4, 628–666. MR 1025485, DOI 10.1137/1031129
Yu. I. Lyubarskiĭ, Frames in the Bargmann space of entire functions, Entire and subharmonic functions, Adv. Soviet Math., vol. 11, Amer. Math. Soc., Providence, RI, 1992, pp. 167–180. MR 1188007
A. M. Perelomov, Remark on the completeness of the coherent state system, Teoret. Mat. Fiz. 6 (1971), no. 2, 213–224 (Russian, with English summary). MR 475444
Jayakumar Ramanathan and Tim Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal. 2 (1995), no. 2, 148–153. MR 1325536, DOI 10.1006/acha.1995.1010
Amos Ron and Zuowei Shen, Weyl-Heisenberg frames and Riesz bases in $L_2(\mathbf R^d)$, Duke Math. J. 89 (1997), no. 2, 237–282. MR 1460623, DOI 10.1215/S0012-7094-97-08913-4
Kristian Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space. I, J. Reine Angew. Math. 429 (1992), 91–106. MR 1173117, DOI 10.1515/crll.1992.429.91
Wenchang Sun and Xingwei Zhou, On the stability of Gabor frames, Adv. in Appl. Math. 26 (2001), no. 3, 181–191. MR 1818742, DOI 10.1006/aama.2000.0715
W.Sun and X.Zhou, Irregular wavelet/Gabor frames, Appl. Comput. Harmon. Anal., 13 (2002), 63–76.
Robert M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 591684