Let
h
h
be a generalized frame in a separable Hilbert space
H
H
indexed by a measure space
(
M
,
S
,
μ
)
(M,\mathcal { S},\mu )
, and assume its analysing operator is surjective. It is shown that
h
h
is essentially discrete; that is, the corresponding index measure space
(
M
,
S
,
μ
)
(M,\mathcal { S},\mu )
can be decomposed into atoms
E
1
,
E
2
,
⋯
E_1,E_2,\cdots
such that
L
2
(
μ
)
L^2(\mu )
is isometrically isomorphic to the weighted space
ℓ
w
2
\ell ^2_w
of all sequences
{
c
i
}
\{c_i\}
of complex numbers with
|
|
{
c
i
}
|
|
2
=
∑
|
c
i
|
2
w
i
>
∞
||\{c_i\}||^2=\sum |c_i|^2 w_i>\infty
, where
w
i
=
μ
(
E
i
)
,
i
=
1
,
2
,
⋯
.
w_i=\mu (E_i),\ i=1,2,\cdots .
This provides a new proof for the redundancy of the windowed Fourier transform as well as any wavelet family in
L
2
(
R
)
L^2(\mathbb {R})
.