GABOR FRAMES AND TIME-FREQUENCY ANALYSIS OF DISTRIBUTIONS

This gaper lays the foundation Tor a quantitative theory of Gabor expa nsions f(x) = Sigma(k,n)c(k,n)e(2 pi in alpha alpha)g(x-k beta). III a nalogy to wavelet expansions of Besov-Triebel-Lizorkin spaces, we show that the correct class of spaces which can be characterized by the ma gnitude of the coefficients c(k,n) is the class of modulation spaces. To analyze the behavior of the coefficients, it is necessary to invert tho Gabor frame operator on these spaces. We show that the frame oper ator is invertible on modulation spaces if and only if it is invertibl e on L-2 and tile atom g is in a suitable space of test functions. A s imilar statement for wavelet theory; is false. The second part is devo ted to Gabor analysis on general time-frequency lattices. (C) 1997 Aca demic Press.