1. Introduction
Frames are alternatives to a Riesz or orthonormal basis in Hilbert spaces. Frame theory plays an important role in signal processing, image processing, data compression and many other applied areas. Ole Christensen’s book [
1] provides a good source of theory of frames and its applications. Constructing frames and their dual frames has always been a critical point in applications. Frames are associated with operators. The properties of those operators can be found in [
2] and orthogonality of frames can be found in [
3,
4,
5,
6]. Sums of frames are studied in [
7,
8]. Sums considered in this paper consist of frames and dual frames. The sum of a pair of orthogonal frames is a frame and also the frame bounds are shown to be easy to compute. Therefore, the orthogonality of a pair of frames plays an important role in this setting and hence we characterize the orthogonality of alternate dual frames in order to obtain new frames as a sum. This allows the construction of a large number of frames from the given ones.
Note: throughout this paper the sequence of scalars will be denoted by and the sequence of vectors will be denoted by Operators involved are linear and the space is separable.
Let
be a separable Hilbert space and let
be a countable index set. A sequence
in
is called a Bessel sequence if there exists a constant
such that for all
is said to be a frame if there exist constants
such that for all
A and B are called the frame bounds. is called a tight frame or an A tight frame if It’s called a normalized tight frame if If is an orthonormal basis, it is a normalized tight frame. The left inequality in the definition of a frame implies that the sequence is complete i.e., for all implies This implies that
A complete sequence
in a Hilbert space
is a Riesz basis [
2] if there exist constants
, such that for all finite sequences
It turns out that it is precisely the image of an orthonormal basis under the action of a bounded bijective operator in a Hilbert space [
1] (Chapter 3).
Let
be a Bessel sequence. The analysis and synthesis operators, denoted respectively by
and
are defined respectively by
and
The analysis operator is actually the Hilbert space adjoint operator to the synthesis operator. These operators are well defined and bounded because
is a Bessel sequence [
1] (Lemma 5.2.1). It turns out that
is a frame if and only if the analysis operator is injective. Also, it is a frame if and only if the synthesis operator is surjective [
2] (Proposition 4.1, 4.2).
The frame operator, denoted by
is defined by
and is given by
It is known that if
is a frame, the series (
1) converges unconditionally, the operator
is bounded, self adjoint, positive and has a bounded inverse [
1] (Lemma 5.1.5). Thus we have the following reconstruction formula,
Let
be another Bessel sequence in
If the operator
given by
is an identity, then the Bessel sequences
and
are actually frames and are called dual frames [
6]. In this case, the reconstruction formula takes the form
So it follows from (
2) that the sequence
is a dual frame to
called the canonical dual frame. Besides the canonical dual, a frame has many dual frames known as alternate dual frames.
Two Bessel sequences
and
in a Hilbert space
are said to be orthogonal [
3,
4,
5,
6] if
This is equivalent to
The sum of a pair of frames is not always a frame [
9] (Proposition 6.6). For example, simply consider the sum
to see that the sum of the frames is not a frame. Moreover, here are two examples that motivate the work of this paper.
Example 1. Let be an orthonormal basis for a Hilbert space Let L be a shift operator defined by and Then is a frame for but L is not invertible.
Example 2. Let and Then and form frames for (in fact, they are orthonormal basis), where and . But the sum fails to be a frame [8].
Some known results about sums being frames are provided in
Section 2.
Section 2.1 provides conditions under which the sum of a pair of frames is a frame. In
Section 2.2, in particular, it is shown that the sum of an orthogonal pair of frames is a frame and also the frame bounds are given. We provide an easy proof of this through the use of analysis and synthesis operators (Theorem 1). This improves the result presented in [
8] (Proposition 3.1). In addition to that more sums under the action of a surjective operator are also provided. Moreover, it is shown that a frame can be added to its alternate dual frames to get a new frame (Theorem 3). An easy way to construct frames from sums is to add a pair of orthogonal frames. It is known that the canonical dual frames of a pair of orthogonal frames are orthogonal [
10]. Alternate dual frames of a pair of orthogonal frames need an extra condition to be an orthogonal pair. This condition is provided here (Theorem 2). This generalizes the results provided in [
10] (Lemma 2 and 3). This provides a large number of a pair orthogonal frames which can be added to get new frames. Some examples are provided in support of the results.
2. Sums of Frames
Frames are considerably more stable than the basis upon the action of operators [
1]. For example let
be a bounded operator,
be an orthonormal basis, and let
Then
is an orthonormal basis for
if
L is a unitary operator,
is a Riesz basis for
if
L is a bounded bijective operator,
is a Bessel sequence in
if
L is a bounded operator,
is a frame sequence (a frame sequence is a frame for its span) for
if
L is a bounded operator with closed range, and
is a frame for
if
L is a bounded surjective operator.
The operators associated with a frame are useful in the study of frames [
2,
3]. Let
and
be two bounded operators on the Hilbert space
Sums of Gabor frames are studied in [
8]. The authors of [
7] provide a condition under which the sequences
and
form a Riesz basis for the space
where
and
are Bessel sequences. In [
9], the authors study sums of frame sequences in a Hilbert space that are strongly disjoint, disjoint, a complementary pair, and weekly disjoint. The authors in [
11] study sums of frames under the same conditions.
Let
be a frame for
with frame bounds
A and
B and
L be a bounded surjective operator. Then
is a frame for
with the frame bounds
and
where
is the pseudo-inverse of
L [
7]. The associated analysis, synthesis and frame operators of
are given by the following lemma.
Lemma 1. The analysis, synthesis and frame operators for the frame are given by respectively.
Proof. Simple calculations
establish the lemma. □
Since the analysis operator is injective,
is injective and
is surjective. So it follows that
is a frame iff
L is surjective as in [
7] (Proposition 2.3). Moreover, the sequences
and
are both frames iff the operator
L is invertible. It is shown that the sequence
is a frame iff the operator
is invertible [
8] (Proposition 2.1), however it turns out to be the case when the operator is simply surjective. Calculations similar to the ones in Lemma 1 prove the following lemma [
7,
8].
Lemma 2. The analysis, synthesis, and the frame operators for the frame are given by respectively.
This lemma and the remarks before the previous lemma reveal that the frame bounds for the frame are and A special case of the above lemma is that is a also a frame. To each frame , there is a naturally associated tight frame , known as canonical Parseval frame. The system where the given frame is being added to its canonical dual and the system where the frame is being added to its canonical Parseval frame, are all frames.
For the sum to be a Riezs basis, the following proposition is taken from [
7] (Proposition 2.8).
Proposition 1. Let be a bounded operator and be a Riesz basis for where are respectively the analysis, synthesis and frame operators with Riesz basis bounds Then is a Riesz basis for with bounds and iff is invertible on
Proof. Since is a Riesz basis, is an invertible operator. If is invertible, then the operator is also invertible. But this is the analysis operator for the sequence Hence, the sequence is a Riesz basis. If is a Riesz basis, then the analysis operator is invertible. Since is invertible, so is the operator □
The following proposition, mentioned incorrectly in [
8] (Proposition 3.1), is corrected in [
7] (Proposition 2.12).
Proposition 2. Let and be two Bessel sequences in with analysis operators , and frame operators respectively. Let be bounded operators. Then the following statements are equivalent.
- (A)
is a Riesz basis for
- (B)
is an invertible operator on
A sum of two frames is not a frame in general. The fact that a sequence is a frame is equivalent to its analysis operator being injective or the synthesis operator being surjective [
3]. The following proposition provides condition under which the sum in the above proposition is a frame.
Proposition 3. Let and be bounded operators, and and be Bessel sequences in a Hilbert space Then the following statements are equivalent.
- (A)
is a frame.
- (B)
is surjective.
The frame operator is given by
Proof. The synthesis operator for the sequence is It therefore follows that (A) and (B) are equivalent. □
Corollary 1. Let and be two Bessel sequences. Then the following are equivalent.
- (A)
is a frame.
- (B)
is surjective.
The frame operator is given by
It is still difficult to verify the conditions of Proposition 3 or its corollary. The sum happens to be a frame if we impose an extra condition of orthogonality of the sequences. Assuming the Bessel sequences to be orthogonal, the following proposition is easily established.
Proposition 4. Let and be two Bessel sequences such that the frame operator is a zero operator. Let be bounded operators, and let Then the following statements are equivalent.
- (A)
is a frame for
- (B)
is surjective.
- (C)
is an invertible positive operator on
Proof. Since
and
are orthogonal frames, we have
i.e., the frame operator
The analysis operator for the sequence
is
and the frame operator
is given by
Let be a frame. Then its frame operator in an invertible and positive operator. So follows. It is straightforward to show that implies because the synthesis operator of a frame is surjective. □
In fact, one of operators or being surjective is enough, as the following Lemma states.
Lemma 3. If and are a pair of orthogonal frames and if either or is surjective, then is surjective.
Proof. Since
is a frame, the operator
is invertible and since
and
are orthogonal, we have
Let
be surjective. Then for each
there exists a
such that
Let
be such that
But then,
So the operator is surjective. □
2.1. Sums of Orthogonal Frames
Let and be a pair of orthogonal frames. Let and be the lower and and be the upper frame bounds for the frames and respectively. Then the following theorem provides a frame as a sum of two given frames and also provides the frame bounds.
Theorem 1. If the pair and is orthogonal and if one of or is surjective, then is a frame whose frame operator is the upper bound is and the lower bound is where is surjective.
Proof. Proposition 4 and Lemma 3 are enough for the above sum to be a frame. The bounds can be computed too. Let
be surjective. We note that,
and
Also, since
and
are frames, we have
since
,
we have
Note: if both
and
are surjective, then larger of
and
serves as the lower bound. For the upper bound,
So the upper bound is □
In particular if the sum is a frame iff is surjective. In addition, if in Theorem 1, then the frame operator is simply We can also obtain a Parseval frame as a sum as the following corollary suggests.
Corollary 2. If and are a pair of orthogonal Parseval frames, the sum is a Parseval frame if and only if the operators and are scaled unitary operators i.e , and where and are unitary operators.
The frame operator in this case is
This generalizes to any finite sum.
Corollary 3. Let be pairwise orthogonal Parseval frames. Let be unitary operators. Then the sum is a Parseval frame, where
The following example takes a pair of orthogonal frames for
from [
12].
Example 3. A sum of discrete Gabor frames in
Let
be the standard orthonormal basis for
Let
let
denote the
coordinate of
and let
Here
is the sequence in
whose
coordinate is
Likewise for the system
Then the systems
form Parseval frames for the space
since
is the orthogonal direct sum of
and for each fixed
m the system
is a Parseval frame for
[
12]. Similar is the case for the system
It turns out that the two systems form an orthogonal pair of frames for
[
12] (Theorem 1.4). The above corollary implies that the sum
provides a Parseval frame for
as well, i.e, the system
forms a Parseval frame for
This can be verified by the argument from [
12] (Example 1.3).
Example 4. A sum of Gabor frames in
For
let
and
be operators defined on
by
Since the polynomial
doesn’t have root on the unit circle for
the set
forms a Gabor frame wavelet set [
13,
14]. Likewise, the set
forms a Gabor frame wavelet set. Let
The families
form frames for the space
Since the
for all
it follows that for all
we have
So
and
form a pair of orthogonal frames for the space
Therefore the sum
forms a frame for
Lemma 4. Let be the dual frame of . Then is a dual to where is a surjective operator.
Proof. Since
is surjective, the operator
is invertible. Let
So
and
This completes the proof. □
If the operator L is invertible, we have the following result.
Corollary 4. Let be a dual frame of . Then is dual to
As a consequence of Lemma 1, we have the following theorem for a pair of orthogonal frames, where the operator L is assumed to be surjective.
Corollary 5. Let and be a pair of orthogonal frames for Then the frames and are orthogonal too.
Proof. From Lemma 1, it follows that the frame operator
is given by
So and are orthogonal. □
The following is proved in [
11], but Lemma 1 provides a very simple proof.
Corollary 6. Let be pairwise orthogonal frames. If is surjective, then is dual to
Proof. Use of Lemma 1 establishes this. The synthesis operator of the sum is
and the analysis operator of
is
it turns out that the composition is
In general, is dual to where or □
2.2. Orthogonality of Alternate Dual Frames
Alternate dual frames of a frame
are given by
where
(the space of bounded linear operators) such that
and
is the standard orthonormal basis of
[
15]. It is also known that
is a Bessel sequence in
[
15]. The authors of [
10] have studied the orthogonality of canonical dual frames of a pair of orthogonal frames. However, alternate dual frames of a pair of orthogonal frames need not be orthogonal. The following theorem establishes the conditions needed for the orthogonality of alternate dual frames of a pair of orthogonal frames.
Theorem 2. Let and be a pair of orthogonal frames and Let and respectively be their corresponding alternate dual frames, where such that and Then
- (A)
The pair and is orthogonal if and only if
- (B)
The pair and is orthogonal if and only if
- (C)
The pair and is orthogonal if and
- (C’)
If is orthogonal to and is orthogonal to then is orthogonal to if and only if
Proof. (A) Let
such that
for some
for some
such that
Then
For each
the sequence
provides
and for each
(A) The frame operator
is given by
and since,
(A)is established.
(B) The frame operator
is
and
Since
it follows that
for all
f iff
This establishes
(C) We notice that
since
So (C) follows using (A) and (B). (C’) follows from (A), (B) and (C). □
Corollary 7. Let and be orthogonal frames with canonical dual and respectively. Then the sum is a dual to
Proof. Lemma 1 and Theorem 2 establish this. □
Let be an alternate dual to such that as in Theorem 2. We now show that a frame can be added to any of its alternate dual frames to yield a new frame.
Theorem 3. The sum is a frame.
Proof. Let
From Proposition 4, it suffices to show that the operator
is surjective. Since
is a positive operator, the operator
is invertible. Therefore for each
there exists
such that
Let
Now,
This proves the theorem. □