Introduction to Finite Frame Theory - Frame Research Center

Introduction to Finite Frame Theory 21
(iii) For every x ∈ H N ,
(iv) For every x ∈ H N ,
(v) T / √ A is an isometry.
x = A −1 ·
A‖x‖ 2 =
M
∑
i=1
M
∑
i=1
〈x,ϕ i 〉ϕ i .
|〈x,ϕ i 〉| 2 .
Proof. (i) ⇔ (ii) ⇔ (iii) ⇔ (iv). These are immediate from the definition of the
frame operator and from Theorem 4.
(ii) ⇔ (v). This follows from the fact that T / √ A is an isometry if and only if
T ∗ T = A · Id. ⊓⊔
A similar result for the special case of a Parseval frame can be easily deduced
from Proposition 11 by setting A = 1.
4.2.3 Eigenvalues of the Frame Operator
Tight frames have the property that the eigenvalues of the associated frame operator
all coincide. We next consider the general situation, i.e., frame operators with
arbitrary eigenvalues.
The first and maybe even most important result shows that the largest and smallest
eigenvalues of the frame operator are the optimal frame bounds of the frame.
Optimality refers to the smallest upper frame bound and the largest lower frame
bound.
Theorem 5. Let (ϕ i ) M i=1 be a frame for H N with frame operator S having eigenvalues
λ 1 ≥ ... ≥ λ N . Then λ 1 coincides with the optimal upper frame bound and λ N
is the optimal lower frame bound.
Proof. Let (e i ) N i=1
denote the normalized eigenvectors of the frame operator S with
respective eigenvalues (λ j ) N j=1 written in decreasing order. Let x ∈ H N be arbitrarily
fixed. Since x = ∑ M j=1 〈x,e j〉e j , we obtain
By Lemma 6, this implies
M
∑
i=1
Sx =
|〈x,ϕ i 〉| 2 = 〈Sx,x〉 =
=
N
∑
j=1
〈 N∑
j=1
λ j 〈x,e j 〉e j .
λ j 〈x,e j 〉e j ,
N
∑
j=1
〈x,e j 〉e j
〉
N
∑ j |〈x,e j 〉|
j=1λ 2 N
≤ λ 1 ∑ |〈x,e j 〉| 2 = λ 1 ‖x‖ 2 .
j=1