Introduction to Finite Frame Theory - Frame Research Center

Introduction to Finite Frame Theory 5
2 Background Material
Let us start by recalling some basic definitions and results from Hilbert space theory
and operator theory, which will be required for all subsequent chapters. We do not
include the proofs of the presented results, and refer to the standard literature such
as, for instance, [153] for Hilbert space theory and [71, 105, 130] for operator theory.
We would like to emphasize that all following results are solely stated in the finite
dimensional setting, which is the focus of this book.
2.1 Review of Basics from Hilbert Space Theory
Letting N be a positive integer, we denote by H N a real or complex N-dimensional
Hilbert space. This will be the space considered throughout this book. Sometimes,
if it is convenient, we identify H N with R N or C N . By 〈·,·〉 and ‖ · ‖ we denote the
inner product on H N and its corresponding norm, respectively.
Let us now start with the origin of frame theory, which is the notion of an orthonormal
basis. Alongside, we recall the basic definitions we will also require in
the sequel.
Definition 1. A vector x ∈ H N is called normalized if ‖x‖ = 1. Two vectors x,y ∈
H N are called orthogonal if 〈x,y〉 = 0. A system (e i ) k i=1 of vectors in H N is called
(a) complete (or a spanning set) if span{e i } k i=1 = H N .
(b) orthogonal if for all i ≠ j, the vectors e i and e j are orthogonal.
(c) orthonormal if it is orthogonal and each e i is normalized.
(e) orthonormal basis for H N if it is complete and orthonormal.
A fundamental result in Hilbert space theory is Parseval’s Identity.
Proposition 1 (Parseval’s Identity). If (e i ) N i=1 is an orthonormal basis for H N ,
then, for every x ∈ H N , we have
‖x‖ 2 =
N
∑
i=1
|〈x,e i 〉| 2 .
Interpreting this identity from a signal processing point of view, it implies that the
energy of the signal is preserved under the map x ↦→ (〈x,e i 〉) N i=1
which we will later
refer to as the analysis map. We would also like to mention at this point, that this
identity is not only satisfied by orthonormal bases. In fact, redundant systems (“nonbases”)
such as (e 1 , √ 1
2
e 2 , √ 1
2
e 2 , √ 1 3
e 3 , √ 1
3
e 3 , √ 1 1
3
e 3 ,..., √ 1
N
e N ,..., √
N
e N ) also satisfy
this inequality, and will later be coined Parseval frames.
Parseval’s identity has the following implication, which shows that a vector x can
be recovered from the coefficients (〈x,e i 〉) N i=1
by a simple procedure. Thus, from
an application point of view, this result can also be interpreted as a reconstruction
formula.