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Gabor Frames and Time-Frequency Analysis of Distributions

✍ Scribed by Hans G. Feichtinger; K Gröchenig

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
503 KB
Volume
146
Category
Article
ISSN
0022-1236

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✦ Synopsis

This paper lays the foundation for a quantitative theory of Gabor expansions f (x)= k, n c k, n e 2?in:x g(x&k;). 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.

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