Bracket products for Weyl-Heisenberg frames
Abstract
We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor,
Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product specifically geared to
Weyl-Heisenberg frames. This bracket product has all the properties of a standard inner product including Bessel's inequality, a Riesz Representation Theorem, and a Gram-Schmidt process which turns a sequence of functions
(
g
n
)
into a sequence
(
e
n
)
with the property that
(
E
mb
e
n
)
m,n∈Z
is orthonormal in
L
2
(R)
. Armed with this inner product, we obtain several results concerning Weyl-Heisenberg frames. First we see that fiberization in this setting takes on a particularly simple form and we use it to obtain a compressed representation of the frame operator. Next, we write down explicitly all those functions
g∈
L
2
(R)
and
ab=1
so that the family
(
E
mb
T
na
g)
is complete in
L
2
(R)
. One consequence of this is that for functions
g
supported on a half-line
[α,∞)
(in particular, for compactly supported
g
),
(g,1,1)
is complete if and only if
sup
0≤t<a
|g(t−n)|≠0
a.e. Finally, we give a direct proof of a result hidden in the literature by proving: For any
g∈
L
2
(R)
,
A≤
∑
n
|g(t−na)
|
2
≤B
is equivalent to
(
E
m/a
g)
being a Riesz basic sequence.