When a characteristic function generates a Gabor frame. (English) Zbl 1242.42023

Gu, Qing; Han, Deguang

Appl. Comput. Harmon. Anal. 24, No. 3, 290-309 (2008).

Summary: We investigate the characterization problem which asks for a classification of all the triples \((a,b,c)\) such that the Gabor system \(\{e^{i2m\pi b t} \chi_{[na,c+na)} : m, n \in \mathbb Z \}\) is a frame for \(L^2(\mathbb R)\). We present a new approach to this problem. With the help of a set-valued mapping defined on certain union of intervals, we are able to provide a complete solution for the case of \(ab\) being a rational number. For the irrational case, we prove that the classification problem can also be completely settled if the union of some intervals obtained from the set-valued mapping becomes stabilized after finitely many times of iterations, which we conjecture is always true.

MSC:

42C10

Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

Keywords:

Gabor system; Gabor frame

References:

[1]	Casazza, P., Modern tools for Weyl-Heisenberg frame theory, Adv. Imag. Electron Phys., 115, 1-127 (2000)
[2]	Casazza, P.; Christensen, O., Weyl-Heisenberg frames for subspaces of \(L^2(R)\), Proc. Amer. Math. Soc., 129, 145-154 (2001) · Zbl 0987.42023
[3]	Casazza, P.; Kalton, N., Roots of complex polynomials and Weyl-Heisenberg frames, Proc. Amer. Math. Soc., 130, 2313-2318 (2002) · Zbl 0991.42023
[4]	P. Casazza, M. Lammers, Classifying characteristic functions giving Weyl-Heisenberg frames, preprint; P. Casazza, M. Lammers, Classifying characteristic functions giving Weyl-Heisenberg frames, preprint
[5]	Christensen, O., Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. (N.S.), 38, 273-291 (2001) · Zbl 0982.42018
[6]	Daubechies, I., The wavelet transforms, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 39, 961-1005 (1990) · Zbl 0738.94004
[7]	Daubechies, I., Ten Lectures on Wavelets, CBS-NSF Regional Conferences in Applied Mathematics, vol. 61 (1992), SIAM: SIAM Philadelphia · Zbl 0776.42018
[8]	Daubechies, I.; Grossmann, A., Frames in the Bargmann spaces of entire expansions, Comm. Pure Appl. Math., 41, 151-164 (1988) · Zbl 0632.30049
[9]	Daubechies, I.; Grossmann, A.; Meyer, Y., Painless nonorthogonal expansions, J. Math. Phys., 27, 1271-1283 (1986) · Zbl 0608.46014
[10]	Duffin, R.; Schaeffer, A., A class of non-harmonic Fourier series, Trans. Amer. Math. Soc., 72, 341-366 (1952) · Zbl 0049.32401
[11]	Feichtinger, H. G.; Strohmer, T., Gabor Analysis and Algorithms. Theory and Applications, Appl. Numer. Harmon. Anal. (1998), Birkhäuser: Birkhäuser Boston, MA · Zbl 0890.42004
[12]	Gröchenig, K., Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal. (2001), Birkhäuser: Birkhäuser Boston, MA · Zbl 0966.42020
[13]	Gröchenig, K.; Janssen, A.; Kaiblinger, N.; Pfander, G., Note on B-splines, wavelet scaling functions and Gabor frames, IEEE Trans. Inform. Theory, 49, 3318-3320 (2003) · Zbl 1286.94033
[14]	Han, D.; Larson, D., Frames, bases and group representations, Mem. Amer. Math. Soc., 697 (2000) · Zbl 0971.42023
[15]	Heil, C., History and evolution of the Density Theorem for Gabor frames, J. Fourier Anal. Appl., 13, 113-166 (2007) · Zbl 1133.42043
[16]	Heil, C.; Walnut, D., Continuous and discrete wavelet transforms, SIAM Rev., 31, 628-666 (1989) · Zbl 0683.42031
[17]	Janssen, A. J.E. M., Representations of Gabor frame operators, in: Twentieth Century Harmonic Analysis—A Celebration, (NATO Sci. Ser. II, Math. Phys. Chem., vol. 33 (2001), Kluwer Academic: Kluwer Academic Dordrecht), 73-101 · Zbl 1003.42017
[18]	Janssen, A. J.E. M., Zak transforms with few zeros and the tie, (Feichtinger, H. G.; Strohmer, T., Advances in Gabor Analysis (2003), Birkhäuser: Birkhäuser Boston), 31-70 · Zbl 1027.42025
[19]	Lyubarskii, Yu. I., Frames in the Bargmann space of entire functions, (Entire and Subharmonic Functions (1992), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 167-180 · Zbl 0770.30025
[20]	Rieffel, M., Von Neumann algebra associated with pairs of lattices in Lie groups, Math. Ann., 257, 403-418 (1981) · Zbl 0486.22004
[21]	Seip, K., Density theorems for sampling and interpolation in the Bargmann-Fock space I, J. Reine Angew. Math., 429, 91-106 (1992) · Zbl 0745.46034
[22]	Seip, K.; Wallstén, R., Sampling and interpolation in the Bargmann-Fock space II, J. Reine Angew. Math., 429, 107-113 (1992) · Zbl 0745.46033
[23]	Walnut, D., Continuity properties of the Gabor frame operator, J. Math. Anal. Appl., 165, 479-504 (1992) · Zbl 0763.47014