Decompositions of frames and a new frame identity -...

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 equal-norm frame and if the frame elements have norm 1 it
is called a unit-norm 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, self-adjoint 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.: E-mail: [email protected]
P.G.C.: E-mail: [email protected]
D.E.: E-mail: [email protected]
G.K.: E-mail: [email protected]

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.