Introduction to Finite Frame Theory - Frame Research Center

Introduction to Finite Frame Theory 7
The notion of orthogonal direct sums, which will play an essential role in Chapter
[166], can be regarded as a generalization of Parseval’s identity (Proposition 1).
Definition 3. Let (W i ) M i=1 be a family of subspaces of H N . Then their orthogonal
direct sum is defined as the space
( M∑
i=1⊕W i
)
with inner product defined by
〈x, ˜x〉 =
M
∑
i=1
l 2 := W 1 × ... × W M
〈x i , ˜x i 〉 for all x = (x i ) M i=1, ˜x = ( ˜x i ) M i=1 ∈
( M∑
i=1⊕W i
)
The extension of Parseval’s identity can be seen when choosing ˜x = x yielding
‖x‖ 2 = ∑ M i=1 ‖x i‖ 2 .
l 2 .
2.2 Review of Basics from Operator Theory
We next introduce the basic results from operator theory used throughout this book.
We first recall that each operator has an associated matrix representation.
Definition 4. Let T : H N → H K be a linear operator, let (e i ) N i=1
be an orthonormal
basis for H N , and let (g i ) K i=1 be an orthonormal basis for H K . Then the matrix
representation of T (with respect to the orthonormal bases (e i ) N i=1 and (g i) K i=1 ) is a
matrix of size K × N and is given by A = (a i j ) K , N
i=1, j=1 , where
a i j = 〈Te j ,g i 〉.
For all x ∈ H N with c = (〈x,e i 〉) N i=1
we have
T x = Ac.
2.2.1 Invertibility
We start with the following definition.
Definition 5. Let T : H N → H K be a linear operator.
(a) The kernel of T is defined by kerT := {x ∈ H N : T x = 0}. Its range is ranT :=
{T x : x ∈ H N }, sometimes also called image and denoted by imT . The rank of
T , rankT , is the dimension of the range of T .