Proceedings of the American Mathematical Society

Author(s): Wenchang Sun; Xingwei Zhou
Journal: Proc. Amer. Math. Soc. 131 (2003), 2883-2893.
MSC (2000): Primary 41A58, 42C15, 42C40
Posted: December 30, 2002
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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.

1.: H.Bacry, A.Grossmann, and J.Zak, Proofs of the completeness of lattice states in the -representation, Phys. Rev., B12(1975), 1118-1120.
2.: V.Bargmann, P.Butera, L.Girardello and J.R.Klauder, On the completeness of coherent states, Rep. Math. Phys., 2(1971), 221-228. MR 44:7860
3.: J.J.Benedetto, C.Heil, and D.Walnut, Differentiation and the Balian-Low theorem, J. Fourier Anal. Appl., 1(1995), 355-402. MR 96f:42002
4.: P.G.Casazza and O.Christensen, Weyl-Heisenberg frames for subspaces of $L^2({\mathbb{R}})$ , Proc. Amer. Math. Soc., 129(2001), 145-154. MR 2001h:42046
5.: O.Christensen, Atomic decomposition via projective group representations, Rocky Mountain J. Math., 26(1996), 1289-1312. MR 98h:43004
6.: O.Christensen, Moment problems and stability results for frames with applications to irregular sampling and Gabor frames, Appl. Comp. Harmonic Anal., 3(1996), 82-86. MR 97f:44007
7.: O.Christensen, Perturbation of operators and applications to frame theory, J. Four. Anal. Appl., 3(1997), 543 - 557. MR 98j:47028
8.: O.Christensen, Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bulletin (New series) of Amer. Math. Soc., 38(2001), 273-291. MR 2002c:42040
9.: O.Christensen, B.Deng, and C.Heil, Density of Gabor frames, Appl. Comput. Harmon. Anal., 7(1999), 292-304. MR 2000j:42043
10.: C.K.Chui and X.L.Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24(1993), 263-277. MR 94d:42039
11.: I.Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 36(1990), 961-1005. MR 91e:42038
12.: I.Daubechies, Ten Lectures on Wavelets, SIAM Philadelphia, 1992. MR 93e:42045
13.: I.Daubechies and A.Grossmann, Frames of entire functions in the Bargmann space, Comm. Pure Appl. Math., 41(1988), 151-164. MR 89e:46028
14.: I.Daubechies, H.Landau and Z.Landau, Gabor time-frequency lattices and the Wexler-Raz identity, J. Fourier Anal. Appl., 4(1995), 437-478. MR 96i:42021
15.: S.Favier and R.Zalik, On the stability of frames and Riesz bases, Appl. and Comp. Harm. Anal., 2(1995), 160-173. MR 96e:42030
16.: H.G.Feichtinger and K.Gröchenig, Banach spaces related to integrable group representations and their atomic decomposition, J. funct. Anal., 86(1989), 307-340. MR 91g:43011
17.: H.G.Feichtinger and A.J.E.M.Janssen, Validity of -frame bound conditions depends on lattice parameters, Appl. Comput. Harmon. Anal., 8(2000), 104-112. MR 2000j:42044
18.: H.G.Feichtinger and T.Strohmer, Eds., Gabor Analysis and Algorithms: Theory and Applications, Birkhäuser, Boston, 1997. MR 98h:42001
19.: D.Gabor, Theory of communications, J. Inst. Elect. Eng.(London), 93(1943), 429-457.
20.: K. Gröchenig, Describing functions: atomic decompositions versus frames, Mh. Math., 112(1991), 1-41. MR 92m:42035
21.: K. Gröchenig, Irregular sampling of wavelet and short-time Fourier transforms, Constr. Approx., 9(1993), 283-297. MR 94m:42077
22.: G.Hardy, J.E.Littlewood, and G.Pólya, Inequalities, 2nd Ed.,Cambridge: Cambridge Univ. Press, 1952. MR 13:727e
23.: C.Heil and D.Walnut, Continuous and discrete wavelet transforms, SIAM Review, 31(1989), 628-666. MR 91c:42032
24.: Y.Lyubarskii, Frames in the Bargmann space of entire functions, in Entire and subharmonic functions, vol 11 of the series Advanced in Soviet Mathematics, B.Ya. Levin, ed., Springer-verlag, Berlin, 1992. MR 93k:30036
25.: A.M.Perelomov, On the completeness of a system of coherent states, Teor. Mat. Fiz., 6(1971), 213-224. MR 57:15050
26.: J.Ramanathan and T.Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal., 2(1995), 148-153. MR 96b:81049
27.: A.Ron and Z.Shen, Weyl-Heisenberg frames and Riesz bases in $L^2(\mathbb{R}^d)$ , Duke Math. J., 89(1997), 148-153. MR 98i:42013
28.: K.Seip and R.Wallstén, Density theorems for sampling and interpolation in the Bargmann-Fock space II, J. Reine Angew. Math., 429(1992), 107-113. MR 93g:46026b
29.: W.Sun and X.Zhou, On the stability of Gabor frames, Advances in Applied Mathematics, 26(2001), 181-191. MR 2002c:42043
30.: W.Sun and X.Zhou, Irregular wavelet/Gabor frames, Appl. Comput. Harmon. Anal., 13 (2002), 63-76.
31.: R.Young, An Introduction to Non-Harmonic Fourier Series, Academic Press, New York, 1980. MR 81m:42027

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 41A58, 42C15, 42C40

Wenchang Sun
Affiliation: Department of Mathematics, Nankai University, Tianjin 300071, People's Republic of China
Email: sunwch@nankai.edu.cn

Xingwei Zhou
Affiliation: Department of Mathematics, Nankai University, Tianjin 300071, People's Republic of China
Email: xwzhou@nankai.edu.cn

DOI: 10.1090/S0002-9939-02-06931-9
PII: S 0002-9939(02)06931-9
Keywords: Gabor frames, Weyl-Heisenberg frames, stability
Received by editor(s): August 29, 2001
Received by editor(s) in revised form: March 2, 2002 and April 11, 2002
Posted: December 30, 2002
Additional Notes: This work was supported by the National Natural Science Foundation of China (Grant Nos. 10171050 and 10201014), the Mathematical Tianyuan Foundation (Grant No. TY10126007), the Research Fund for the Doctoral Program of Higher Education, and the Liuhui Center for Applied Mathematics.
Communicated by: David R. Larson
Copyright of article: Copyright 2002, American Mathematical Society

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