Generalized frames and their redundancy

Let h be a generalized frame in a separable Hilbert space H indexed by a me asure space (M, S, mu), and assume its analysing operator is surjective. It is shown that h is essentially discrete; that is, the corresponding index measure space (M, S, mu) can be decomposed into atoms E-1, E-2,... such tha t L-2(mu) is isometrically isomorphic to the weighted space l(w)(2) of all sequences {c(i)} of complex numbers with \ \ {c(i)}\ \ (2) = Sigma \c(i)\ ( 2)w(i) < <infinity>, where w(i) = mu (E-i), i= 1, 2,.... This provides a ne w proof for the redundancy of the windowed Fourier transform as well as any wavelet family in L-2(R).