Abstract
The operator algebras generated by translation and modulation unitary operators can play an important role in studying Gabor frames in time-frequency analysis. We present some connections between these algebras and Gabor frames, and use them to derive some well-known results and their generalizations such as the density property in Gabor analysis, as well as some new ones such as a characterization of the Gabor frames (for subspaces) admitting unique Gabor duals (within the subspaces). Finally, we provide a necessary and sufficient condition for a square-integrable function g to generate a subspace Gabor frame in the one-dimensional, rational case. The condition is phrased in terms of the Zak transform of g.
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References
O. Bratteli and D.W. Robinson. Operator algebras and quantum statistical mechanics. 1. Springer-Verlag, New York, second edition, 1987. C*- and W*-algebras, symmetry groups, decomposition of states.
P.G. Casazza. Modern tools for Weyl—Heisenberg (Gabor) frame theory. Adv. Imag. Elect. Phys., 115:1–127, 2001.
B. Christensen, O. Deng and C. Heil. Density of Gabor frames. Appl. Comp. Harm. Anal., 7:292–304, 1999.
I. Daubechies. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory, 36(5):961–1005, 1990.
I. Daubechies, H. Landau, and Z. Landau. Gabor time-frequency lattices and the Wexler—Raz identity. J. Four. Anal. Appl., 1(4):437–478, 1995.
K.R. Davidson. C* -algebras by example. American Mathematical Society, Providence, RI, 1996.
J. Dixmier. von Neumann algebras. North-Holland Publishing Co., Amsterdam, 1981. With a preface by E. C. Lance, Translated from the second French edition by F. Jellett.
H.G. Feichtinger and K. Gröchenig. Gabor frames and time-frequency analysis of distributions. J. Funct. Anal., 146(2):464–495, 1997.
H.G. Feichtinger and T. Strohmer, editors. Gabor Analysis and Algorithms: Theory and Applications. Birkhäuser, Boston, 1998.
G.B. Folland. Harmonic Analysis in Phase Space. Annals of Math. Studies. Princeton Univ. Press, Princeton (NJ), 1989.
[11] J.-P. Gabardo and D. Han. Frame representations for group-like unitary operator systems. preprint.
[12] J.-P. Gabardo and D. Han. Weyl—Heisenberg dual frames and operator algebras. preprint.
J.-P. Gabardo and D. Han. Subspace Weyl—Heisenberg frames. J. Fourier Anal. Appl., 7(4):419–433, 2001.
D. Gabor. Theory of communication. Proc. IEE (London), 93(111): 429–457, November 1946.
K. Gröchenig. Aspects of Gabor analysis on locally compact abelian groups. In H.G. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms: Theory and Applications, pages 211–231. Birkhäuser, Boston, 1998.
K.H. Gröchenig. Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston, MA, 2001.
D. Han. Wandering vectors for irrational rotation unitary systems. Trans. Amer. Math, Soc., 350(1):309–320, 1998.
D. Han and D.R. Larson. Frames, bases and group representations. Mem. Amer. Math. Soc., 147(697):x+94, 2000.
D. Han and Y. Wang. Lattice tiling and the Weyl—Heisenberg frames. Geometric and Functional Analysis, 11:742–758, 2001.
C. Heil and D. Walnut. Continuous and discrete wavelet transforms. SIAM Review, 31(4):628–666,1989.
A.J.E.M. Janssen. Signal analytic proof of two basic results on lattice expansions. Appl. Comp. Harm. Anal., 1(4):350–354, 1994.
A.J.E.M. Janssen. On rationally oversampled Weyl—Heisenberg frames. Signal Process., 47:239–245, 1995.
A.J.E.M. Janssen. The duality condition for Weyl—Heisenberg frames. In H.G. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms: Theory and Applications,pages 33–84. Birkhäuser, Boston, 1998.
R.V. Kadison and J.R. Ringrose. Fundamentals of the theory of operator algebras. Vol. I. Academic Press Inc. New York, 1983.
R.V. Kadison and J.R. Ringrose. Fundamentals of the theory of operator algebras. Vol. II. Academic Press Inc., Orlando, FL, 1986.
J. Ramanathan and T. Steger. Incompleteness of sparse coherent states. Appl. Comp. Harm. Anal., 2(2):148–153, 1995.
M.A. Rieffel. Von Neumann algebras associated with pairs of lattices in Lie groups. Math.Ann., 257:403–418, 1981.
A. Ron and Z. Shen. Weyl—Heisenberg frames and Riesz bases in L2(Rd). Duke Math. J., 89(2):237–282, 1997.
M. Takesaki. Theory of operator algebras. I. Springer-Verlag, New York, 1979.
V. S. Varadarajan. Geometry of quantum theory. Springer-Verlag, New York, second edition, 1985.
J. von Neumann. Mathematical foundations of quantum mechanics. Princeton University Press, Princeton, 1955. Translated by Robert T. Beyer.
Y.Y. Zeevi, M. Zibulski, and M. Porat. Multi-window Gabor schemes in signal and image representations. In Gabor analysis and algorithms, pages 381–407. Birkhäuser Boston, Boston, MA, 1998.
M. Zibulski and Y.Y. Zeevi. Oversampling in the Gabor scheme. IEEE Trans. SP, 41(8):2679–2687, 1993.
M. Zibulski and Y.Y. Zeevi. Discrete multi-window Gabor transforms. IEEE Trans. Signal Proc., 45:1428–1442, 1997.
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Gabardo, JP., Han, D. (2003). Aspects of Gabor Analysis and Operator Algebras. In: Feichtinger, H.G., Strohmer, T. (eds) Advances in Gabor Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0133-5_6
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DOI: https://doi.org/10.1007/978-1-4612-0133-5_6
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