Decompositions of frames and a new frame identity
Radu Balan
a
, Peter G. Casazza
b
, Dan Edidin
c
and Gitta Kutyniok
d
a
Siemens Corporate Research, 755 College Road East, Princeton, NJ 08540, USA;
b
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA;
c
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA;
d
Mathematical Institute, Justus–Liebig–University Giessen, 35392 Giessen, Germany
ABSTRACT
We analyze a fundamental question in Hilbert space frame theory:
What is the
optimal
decomposition of a
Parseval frame? We will see that this question impacts several famous unsolved problems in different areas of
mathematics. As a step towards the solution of this question, we give a new identity which holds for all Parseval
frames.
Keywords:
frame, Riesz basis, Bessel sequence
1. INTRODUCTION
Let
H
be a Hilbert space and
f
i
∈
H
for
i
∈
I
. Let
K
=
span
{
f
i
}
i
∈
I
. A number
A >
0 (respectively,
B >
0) is
called a
lower
(respectively,
upper
)
frame bound
for
F
=
{
f
i
}
i
∈
I
if for all
f
∈
K
A f
2
≤
i
∈
I

f, f
i

2
,
respectively,
i
∈
I

f, f
i

2
≤
B f
2
.
If
B <
∞
we call
F
=
{
f
i
}
i
∈
I
a
Bessel sequence with Bessel bound B
. If 0
< A
≤
B <
∞
, then
{
f
i
}
i
∈
I
is a
frame
for
K
.
If
K
=
H
we call
{
f
i
}
i
∈
I
a
frame sequence
in
H
.
The largest
A
and the smallest
B
satisfying
the above inequalities are called the
optimal
lower and upper frame bound and will be denoted
A
(
F
) and
B
(
F
)
respectively.
If
A
=
B
=
λ
we call this a
λ

tight frame
and if
λ
= 1 it is called a
Parseval frame
.
If all the
frame elements have the same norm we call this an
equalnorm
frame and if the frame elements have norm 1 it
is called a
unitnorm frame
. If
{
f
i
}
i
∈
I
is a Bessel sequence, the
synthesis operator
for
{
f
i
}
i
∈
I
is the bounded
linear operator
T
:
2
(
I
)
→
H
given by
T
(
e
i
) =
f
i
for all
i
∈
I
where
{
e
i
}
i
∈
I
is the unit vector basis of
2
(
I
).
The
analysis operator
for
{
f
i
}
i
∈
I
is
T
∗
and satisfies:
T
∗
(
f
) =
i
∈
I
f, f
i
e
i
.
Hence,
T
∗
(
f
)
2
=
i
∈
I

f, f
i

2
.
The
frame operator
for the frame is the positive, selfadjoint invertible operator
S
=
T T
∗
:
H
→
H
satisfying
Sf
=
i
∈
I
f, f
i
f
i
,
for all
f
∈
H
.
Further author information: (Send correspondence to P.G. Casazza)
R.B.: Email: [email protected]
P.G.C.: Email: [email protected]
D.E.: Email: [email protected]
G.K.: Email: [email protected]
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Finally, the family
{
f
i
}
i
∈
I
is a
Riesz basic sequence
in
H
with Riesz basis bounds
A, B
if for all sequences of
scalars
{
a
i
}
i
∈
I
we have
A
i
∈
I

a
i

2
1
/
2
≤
i
∈
I
a
i
f
i
≤
B
i
∈
I

a
i

2
1
/
2
.
If
{
f
i
}
i
∈
I
also spans
H
it is called a
Riesz basis
for
H
. For the fundamentals of frame theory we refer the reader
to Christensen.
14
A fundamental question in frame theory involves understanding the behavior of subsets of a Parseval frame.
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 Spring '10
 RunyiYu
 Linear Algebra, Frame, Hilbert space, norm Bessel, fi fi

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