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Frames are more stable as compared to bases under the action of a bounded linear operator. Sums of different frames under the action of a bounded linear operator are studied with the help of analysis, synthesis and frame operators. A simple construction of frames from the existing ones under the action of such an operator is presented here. It is shown that a frame can be added to its alternate dual frames, yielding a new frame. It is also shown that the sum of a pair of orthogonal frames is a frame. This provides an easy construction of a frame where the frame bounds can be computed easily. Moreover, for a pair of orthogonal frames, the necessary and sufficient condition is presented for their alternate dual frames to be orthogonal. This allows for an easy construction of a large number of new frames.
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  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  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  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  (Chapter 3).
Let be a Bessel sequence. The analysis and synthesis operators, denoted respectively by and are defined respectively by
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  (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  (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  (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 . 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
These are equivalent to
The sum of a pair of frames is not always a frame  (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.
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.
Let and Then and form frames for (in fact, they are orthonormal basis), where and . But the sum fails to be a frame .
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  (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 . 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  (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 . 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 . The authors of  provide a condition under which the sequences and form a Riesz basis for the space where and are Bessel sequences. In , 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  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 . The associated analysis, synthesis and frame operators of are given by the following lemma.
The analysis, synthesis and frame operators for the frame are given by respectively.
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  (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  (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].
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  (Proposition 2.8).
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
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  (Proposition 3.1), is corrected in  (Proposition 2.12).
Let and be two Bessel sequences in with analysis operators , and frame operators respectively. Let be bounded operators. Then the following statements are equivalent.
is a Riesz basis for
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 . The following proposition provides condition under which the sum in the above proposition is a frame.
Let and be bounded operators, and and be Bessel sequences in a Hilbert space Then the following statements are equivalent.
is a frame.
The frame operator is given by
The synthesis operator for the sequence is It therefore follows that (A) and (B) are equivalent. □
Let and be two Bessel sequences. Then the following are equivalent.
is a frame.
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.
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.
is a frame for
is an invertible positive operator on
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.
If and are a pair of orthogonal frames and if either or is surjective, then is surjective.
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.
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.
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.
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.
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 .
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 . Similar is the case for the system It turns out that the two systems form an orthogonal pair of frames for  (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  (Example 1.3).
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
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
Let be the dual frame of . Then is a dual to where is a surjective operator.
Since is surjective, the operator is invertible. Let So
This completes the proof. □
If the operator L is invertible, we have the following result.
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.
Let and be a pair of orthogonal frames for Then the frames and are orthogonal too.
From Lemma 1, it follows that the frame operator is given by
So and are orthogonal. □
The following is proved in , but Lemma 1 provides a very simple proof.
Let be pairwise orthogonal frames. If is surjective, then is dual to
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 . It is also known that is a Bessel sequence in . The authors of  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.
Let and be a pair of orthogonal frames and Let and respectively be their corresponding alternate dual frames, where such that and Then
The pair and is orthogonal if and only if
The pair and is orthogonal if and only if
The pair and is orthogonal if and
If is orthogonal to and is orthogonal to then is orthogonal to if and only if
(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). □
Let and be orthogonal frames with canonical dual and respectively. Then the sum is a dual to
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.
The sum is a frame.
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. □
Motivated by the earlier work on the sums of frames [7,8], an easy construction of a frame via a sum of frames is established with the aid of analysis and synthesis operators. It is also shown that a frame can be added to its alternate dual frame to yield a frame. A pair of orthogonal frames can be added to provide the sum as a frame as well. Therefore, a condition for the orthogonality of alternate dual frames for a pair of orthogonal frames is presented. Under this condition, many pairwise orthogonal frames can be constructed and their sum is always a frame. This enables us to construct a large number of frames and also allows us to compute the frame bounds.
This research received no external funding.
The author thanks anonymous referees for their comments, suggestions and corrections towards making this manuscript more readable and better organized.
Conflicts of Interest
The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
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