A Physical Interpretation of Tight Frames
P. G. Casazza
1
,M
.F
ickus
2
,J
.Kovaˇ
cevi´
c
3
, M. T. Leon
4
, and J. C. Tremain
5
1
Mathematics Department, University of Missouri, Columbia, Missouri 65211
USA,
pete@math.missouri.edu
2
Department of Mathematics and Statistics, Air Force Institute of Technology,
Wright-Patterson AFB, Ohio 45433 USA,
Matthew.Fickus@afit.edu
3
Department of Biomedical Engineering, Carnegie Mellon University, Pittsburgh,
Pennsylvania 15213, USA,
jelenak@cmu.edu
4
Mathematics Department, University of Missouri, Columbia, Missouri 65211
USA,
mleon@math.missouri.edu
5
Mathematics Department, University of Missouri, Columbia, Missouri 65211
USA,
janet@math.missouri.edu
Summary.
We ±nd ±nite tight frames when the lengths of the frame elements are
predetermined. In particular, we derive a “fundamental inequality” which completely
characterizes those sequences which arise as the lengths of a tight frame’s elements.
Furthermore, using concepts from classical physics, we show that this characteriza-
tion has an intuitive physical interpretation.
1 Introduction
Let
H
N
be a fnite
N
-dimensional Hilbert space. A fnite sequence
{
f
m
}
M
m
=1
o± vectors is
A
-tight
±or
H
N
i± there exists
A
≥
0 such that,
A
k
f
k
2
=
M
X
m
=1
|h
f,f
m
i|
2
,
±or all
f
∈
H
N
.An
A
-tight frame
is an
A
-tight sequence ±or which
A>
0. By
polarization,
{
f
m
}
M
m
=1
is
A
-tight ±or
H
N
i± and only i±,
Af
=
M
X
m
=1
h
m
i
f
m
,
±or all
f
∈
H
N
. Clearly, any orthonormal basis is a 1-tight ±rame. However,
the converse is ±alse. For example, the vertices o± a tetrahedron, appropriately
centered and scaled, ±orm a 1-tight ±rame o± ±our elements ±or
R
3
.Moreover
,
while the elements o± an orthonormal basis are o± unit length
a priori
, there are
no explicit assumptions made about the lengths o± a tight ±rame’s elements.
This raises the question,