(Frame system) Let
be a separable Hilbert space. A finite or countable subset
,
,
is called a "frame system" if there are constants
:
such that for any
is a called "frame" if its linear span is dense in
:
Let
,
are called "lower and upper frame bounds". The
ratio
is called "condition number" of
.
The system
is called "minimal" if removing any element from it changes the linear span of
.
The system
is called "tight" if
.

Remark

Analysis of frames and frame systems is similar. Frame systems have properties
in
.
Frames have the same properties in
.
For this reason we study frames in this section.

Definition

(Frame operator) Given a frame
the
operator
is called "synthesis operator". The adjoint operator
is called "analysis operator". The operator
is called "frame operator".

Proposition

(Properties of frame operator) Let
be a frame.

1. Analysis operator
takes the
form

2. The frame operator
is a symmetric positive definite invertible operator,

2a.
,

2b.
,

2c.
,

2d.
,

2e.
.

Proof

By definition of adjoint
operator
We set
for every
s.t.
:
where, by definition of
,
Thus
or
We have proven the claim 1.

2a is a direct verification.

To verify 2b we calculate, by
definitions,
and 2b follows by the definition (
Frame system
).
Hence,
is invertible and the rest follows directly.

Definition

(Dual frame) Given a frame
and associated frame operator
the
system
is called the "dual frame". We introduce the
notation

Proposition

(Properties of dual frame) Let
and
are frame and frame operator.

1. For any

2.
is a frame with a frame operator
.

3. If
is tight then
.

Proof

We
have
and
follows from the last result because
(and thus
)
is symmetric. We have proven the claim 1.

Note
that
We intend to derive the statement from the proposition
(
Properties of Schwarz
operator
). Thus, we consider the frame
in context of the section (
Stable
space splittings
). The subspaces
are one-dimensional.
is a space of sequences. The operator
acts
and is surjective by denseness of
.
The operator
of the section (
Stable space
splittings
) is identity. The scalar product in
is given
by
Hence, we take
:
and
achieve
The existence of
has already been established in the proposition
(
Properties of frame operator
).
In the same proposition, part 2b, stability of splitting was established.
Thus