(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
are called "lower and upper frame bounds". The
is called "condition number" of
is called "minimal" if removing any element from it changes the linear span of
is called "tight" if
Analysis of frames and frame systems is similar. Frame systems have properties
Frames have the same properties in
For this reason we study frames in this section.
(Frame operator) Given a frame
is called "synthesis operator". The adjoint operator
is called "analysis operator". The operator
is called "frame operator".
(Properties of frame operator) Let
be a frame.
1. Analysis operator
2. The frame operator
is a symmetric positive definite invertible operator,
By definition of adjoint
where, by definition of
We have proven the claim 1.
2a is a direct verification.
To verify 2b we calculate, by
and 2b follows by the definition (
is invertible and the rest follows directly.
(Dual frame) Given a frame
and associated frame operator
is called the "dual frame". We introduce the
(Properties of dual frame) Let
are frame and frame operator.
1. For any
is a frame with a frame operator
is tight then
follows from the last result because
is symmetric. We have proven the claim 1.
We intend to derive the statement from the proposition
Properties of Schwarz
). Thus, we consider the frame
in context of the section (
). The subspaces
is a space of sequences. The operator
and is surjective by denseness of
of the section (
) is identity. The scalar product in
Hence, we take
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.