Gabor Frames and Time-Frequency Analysis of Distributions*

Abstract This paper lays the foundation for a quantitative theory of Gabor expansions f ( x )=∑ k ,  n c k ,  n e 2 πinαx g ( x − kβ ). In analogy to wavelet expansions of Besov–Triebel–Lizorkin spaces, we show that the correct class of spaces which can be characterized by the magnitude of the coefficients c k ,  n is the class of modulation spaces. To analyze the behavior of the coefficients, it is necessary to invert the Gabor frame operator on these spaces. We show that the frame operator is invertible on modulation spaces if and only if it is invertible on L 2 and the atom g is in a suitable space of test functions. A similar statement for wavelet theory is false. The second part is devoted to Gabor analysis on general time–frequency lattices.

[1]  Dennis Gabor,et al.  Theory of communication , 1946 .

[2]  J. Neumann.  Mathematical Foundations of Quantum Mechanics , 1955 .

[3]  Y. Katznelson.  An Introduction to Harmonic Analysis: Interpolation of Linear Operators , 1968 .

[4]  P. Fillmore.  Notes on operator theory , 1970 .

[5]  J.B. Allen,et al.  A unified approach to short-time Fourier analysis and synthesis , 1977, Proceedings of the IEEE.

[6]  H. Feichtinger.  On a new Segal algebra , 1981 .

[7]  A. Janssen.  Gabor representation of generalized functions , 1981 .

[8]  R. Balian.  Un principe d'incertitude fort en théorie du signal ou en mécanique quantique , 1981 .

[9]  H. Triebel.  Modulation Spaces on the Euclidean $n$-Space , 1983 .

[10]  Christopher Heil,et al.  Continuous and Discrete Wavelet Transforms , 1989, SIAM Rev..

[11]  H. Feichtinger,et al.  Banach spaces related to integrable group representations and their atomic decompositions, I , 1989 .

[12]  H. Feichtinger.  Atomic characterizations of modulation spaces through Gabor-type representations , 1989 .

[13]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[14]  H. Feichtinger.  Generalized Amalgams, With Applications to Fourier Transform , 1990, Canadian Journal of Mathematics.

[15]  B. Jawerth,et al.  A discrete transform and decompositions of distribution spaces , 1990 .

[16]  Jason Wexler,et al.  Discrete Gabor expansions , 1990, Signal Process..

[17]  I. Gohberg,et al.  Classes of Linear Operators , 1990 .

[18]  K. Gröchenig.  Describing functions: Atomic decompositions versus frames , 1991 .

[19]  D. Walnut.  Continuity properties of the Gabor frame operator , 1992 .

[20]  H. Feichtinger,et al.  Gabor wavelets and the Heisenberg group: Gabor expansions and short time Fourier transform from the group theoretical point of view , 1993 .

[21]  D. Walnut,et al.  Wilson Bases and Modulation Spaces , 1992 .

[22]  Karlheinz Gröchenig,et al.  Acceleration of the frame algorithm , 1993, IEEE Trans. Signal Process..

[23]  D. Donoho.  Unconditional Bases Are Optimal Bases for Data Compression and for Statistical Estimation , 1993 .

[24]  D. Walnut.  Lattice size estimates for Gabor decompositions , 1993 .

[25]  A. Janssen.  Duality and Biorthogonality for Weyl-Heisenberg Frames , 1994 .

[26]  R. Tolimieri,et al.  Poisson Summation, the Ambiguity Function, and the Theory of Weyl-Heisenberg Frames , 1994 .

[27]  D. Walnut,et al.  Differentiation and the Balian-Low Theorem , 1994 .

[28]  I. Daubechies,et al.  Gabor Time-Frequency Lattices and the Wexler-Raz Identity , 1994 .

[29]  Augustus J. E. M. Janssen.  On rationally oversampled Weyl-Heisenberg frames , 1995, Signal Process..

[30]  O. Christensen,et al.  Group theoretical approach to Gabor analysis , 1995 .