Introduction to Finite Frame Theory - Frame Research Center

Introduction to Finite Frame Theory 17
4 Frames and Operators
The analysis, synthesis, and frame operator determine the operation of a frame when
analyzing and reconstructing a signal. The Grammian operator is perhaps not that
well-known, yet it crucially illuminates the behavior of a frame (ϕ i ) M i=1 embedded
as an N-dimensional subspace in the high-dimensional space R M .
For the rest of this introduction we set l M 2 := l 2({1,...,M}). Note that this space
in fact coincides with R M or C M , endowed with the standard inner product and the
associated Euclidean norm.
4.1 Analysis and Synthesis Operators
Two of the main operators associated with a frame are the analysis and synthesis
operators. The analysis operator – as the name suggests – analyzes a signal in terms
of the frame by computing its frame coefficients. We start by formalizing this notion.
Definition 15. Let (ϕ i ) M i=1 be a family of vectors in H N . Then the associated analysis
operator T : H N → l M 2 is defined by
T x := ( 〈x,ϕ i 〉 ) M
i=1 , x ∈ H N .
In the following lemma we derive two basic properties of the analysis operator.
Lemma 3. Let (ϕ i ) M i=1 be a sequence of vectors in H N with associated analysis
operator T .
(i) We have
‖T x‖ 2 =
M
∑
i=1
|〈x,ϕ i 〉| 2 for all x ∈ H N .
Hence, (ϕ i ) M i=1 is a frame for H N if and only if T is injective.
(ii) The adjoint operator T ∗ : l M 2 → H N of T is given by
T ∗ (a i ) M i=1 =
M
∑
i=1
a i ϕ i .
Proof. (i). This is an immediate consequence of the definition of T and the frame
property (2).
(ii). For x = (a i ) M i=1 and y ∈ H N , we have
〈
〈T ∗ x,y〉 = 〈x,Ty〉 = (a i ) M i=1, ( 〈y,ϕ i 〉 ) 〉
M
i=1
=
M
∑
i=1
a i 〈y,ϕ i 〉 =
〈 M∑
a i ϕ i ,y
i=1
〉
.
Thus, T ∗ is as claimed.
⊓⊔