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Advances in Computational Mathematics
18:
357–385, 2003.
2003
Kluwer Academic Publishers. Printed in the Netherlands.
Finite normalized tight frames
John J. Benedetto
a
,
∗
and Matthew Fickus
b
a
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
Email: [email protected]
b
Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 14853, USA
Email: ±[email protected]
Received 14 September 2001; accepted 20 January 2002
Communicated by C.A. Micchelli
Frames are interesting because they provide decompositions in applications where bases
could be a liability. Tight frames are valuable to ensure fast convergence of such decom
positions. Normalized frames guarantee control of the frame elements. Finite frames avoid
the subtle and omnipresent approximation problems associated with the truncation of in±nite
frames. In this paper the theory of ±nite normalized tight frames (FNTFs) is developed. The
main theorem is the characterization of all FNTFs in terms of the minima of a potential en
ergy function, which was designed to measure the total orthogonality of a Bessel sequence.
Examples of FNTFs abound, e.g., in
R
3
the vertices of the Platonic solids and of a soccer ball
are FNTFs.
Keywords:
tight frames, potential energy, Lagrange multipliers, equidistribution, equilibrium
AMS subject classifcation:
42C99, 42C40, 65F, 74G65
1.
Introduction
We shall develop the theory of ±nite normalized tight frames (FNTFs) for real and
complex Euclidean space
K
d
,
K
=
R
,
C
, respectively; and we shall characterize all
FNTFs for
K
d
in terms of a potential energy function which we designate as the
frame
potential
, see theorem 7.1. FNTFs are developed in section 2, but, for the sake of this
preliminary discussion, an
A
FNTF for
K
d
is a ±nite sequence
{
x
n
:
n
=
1
,...,N
}
⊆
K
d
for which the Euclidean norm
k
x
n
k
is 1 for each
x
n
, i.e.,
{
x
n
}
is normalized, and
for which
∃
A>
0 such that
∀
y
∈
K
d
,y
=
1
A
N
X
n
=
1
±
y,x
n
i
x
n
.
(1.1)
∗
The author gratefully acknowledges support from DARPA Grant MDA 972011003 and the General
Research Board of the University of Maryland.
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J.J. Benedetto, M. Fickus / Finite normalized tight frames
It is elementary to see that if
{
x
n
}
N
n
=
1
is an
A
FNTF then
A
=
1 if and only if
{
x
n
}
N
n
=
1
is an orthonormal basis (ONB) for
R
d
, e.g., theorem 2.1. (Naturally we choose
the terminology “normalized” in the de±nition of an
A
FNTF because of the de±nition
of an ONB, but see remark 1.1(b) below.)
Of course, it is reasonable to ask why anyone would want to develop a theory of
FNTFs.
First, the theory of frames was initiated by Duf±n and Schaeffer [9] in 1952 as
part of an ongoing development of nonharmonic Fourier series, in which it is an im
portant, fundamental theory. The background for [9] and its relationship to nonuniform
sampling theory is the subject of chapter 1 in [2]. Further, there is a signi±cant role in
signal processing for the theory of frames. This was initiated by Daubechies et al. [8] in
1986, and since then there has been a plethora of activity in the area including a land
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