Characterization, Dilation, and Perturbation of Basic Continuous Frames

Zhao, Xin; Li, Pengtong

doi:https://doi.org/10.1155/2022/3270281

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 3270281 | https://doi.org/10.1155/2022/3270281

Characterization, Dilation, and Perturbation of Basic Continuous Frames

Xin Zhao¹and Pengtong Li¹

Academic Editor: Firdous A. Shah

Received01 Nov 2021

Accepted26 Feb 2022

Published29 Mar 2022

Abstract

A vector-valued function is called a basic continuous frame if it is a continuous frame for its spanning space. It is shown in this article that basic continuous frames and their oblique duals can be characterized by operators with closed ranges. Furthermore, we show that any oblique dual pair of basic continuous frames for a Hilbert space can be dilated to a Type II dual pair for a larger Hilbert space. Finally, a perturbation result for basic continuous frames is given. Since the spanning spaces of two basic continuous frames for a Hilbert space are often different, the research process is more complex than the setting of general continuous frames.

1. Introduction

The concept of discrete frames was first formally introduced by Duffin and Schaeffer [1] in 1952 and popularized greatly after the significant paper [2] by Daubechies et al. in 1986. A discrete frame is an overcomplete family of countable elements in a Hilbert space which allows every element in the space to be represented as a linear combination of the frame elements. It has been widely used in many fields such as image and signal processing, approximation theory and wireless communications. Due to different applications and theoretical goals, various generalizations of discrete frames have been presented. For example, pseudoframes [3], g-frames [4], fusion frames [5] and operator-valued frames [6]. One important generalization of discrete frames is the so-called continuous frames, which is introduced by Kaiser [7] and independently by Ali, Antonie and Gazeau [8]. We refer to [9–12] for more studies on continuous frames.

Dilation and perturbation are two significant properties for discrete and continuous frames. Gabardo and Han [13] generalized the dilation theorem for dual discrete frames (cf. [14]) to dual continuous frames. Using the method considered by Casazza and Christensen (cf. [15]), they deduced a perturbation theorem for continuous frames. Kaushik et al. [16] gave some equivalent conditions of perturbation for continuous frames and obtained a sufficient condition for the stability of a continuous frame. Since the basic continuous frame is more general in theory and more freedom when the dual frame is selected, we study basic continuous frames around their dilation and perturbation properties.

Throughout this paper, and denote Hilbert spaces over the complex field and a countable index set. For an operator , we denote its range and kernel by and , respectively. Let , be closed subspaces of . We denote by the sum of two subspaces with trivial intersection, by the orthogonal direct sum of closed subspaces and by the space . We call an operator a (oblique) projection if it satisfies . In this case, the decomposition holds. Conversely, if , then we can find a unique projection satisfying and . Let be the orthogonal projection onto and be its restriction to .

Let be a measure space where is positive. Recall that a vector-valued function is said to be a continuous frame for with respect to or a -frame if(i) is weakly measurable, i.e., for all , is a measurable function on .(ii)There exist constants , such that

The numbers , are called frame bounds. We call a -Bessel mapping if only the second inequality of (1) holds, a tight -frame if , and a Parseval -frame if . The mapping is called a basic -frame or a basic continuous frame for if it is a continuous frame for the spanning space. When and is the counting measure, the family is a discrete frame.

To a -Bessel mapping , we associate the analysis operator given by

Note that the frame condition (1) implies that . The adjoint of is called synthesis operator and is given by

In this case, we usually say that the following equation holds in the weak sense:

The frame operator is defined to be . When is a continuous frame for , is a bounded, invertible and positive operator. A -Bessel mapping is said to a dual of if , i.e.,

Every -frame always has a dual , called the canonical dual, satisfying the following reconstruction formula:

If has the unique dual, then we say is a Riesz-type-frame. We see from ([13], Proposition 1) that is a Riesz-type -frame if and only if its analysis operator is surjective. Two -frames and for and , respectively, are called similar if holds for an invertible operator . Let be a closed subspace of . We call a frame range if there exist a Hilbert space and a -frame for whose analysis operator has range space .

The paper is organized as follows. In Section 2, we present some preliminary results about basic -frames. In Section 3, we characterize basic -frames and their oblique duals. Section 4 is devoted to the dilation of oblique dual pairs of basic -frames. A perturbation theorem of basic -frames is considered in Section 5.

2. Preliminaries

This section is devoted to some preliminary results about basic continuous frames and their duals. In the rest of the paper, we agree to use the following notation:

For a given vector-valued function . For a -Bessel mapping , we always use and to denote the analysis operator and synthesis operator, respectively.

Recall that every basic continuous frame is a -Bessel mapping. Conversely, a -Bessel mapping constitutes a basic continuous frame if and only if is closed. In this case, .

For a basic -frame , the frame operator is invertible when restricted to . Its canonical dual is , where denotes the pseudoinverse of (see [17]). We recall that

Now, we give definitions on duals of basic -frames with the following. For the discrete case, we see [18–20]

Definition 1. Suppose that is a basic -frame and that is a -Bessel mapping for .(i) is a dual of if , i.e., , (ii) is a Type I dual of if is a dual of and (iii) is a Type II dual of if is a dual of and (iv) is an oblique dual of if is a basic -frame that is a dual of and is a dual of Heil et al. showed in [20] that Type I or II duals of discrete frame sequences are oblique duals. They also characterized the existence of oblique duals with respect to the direct sum decomposition of . It is easy to show that these facts still hold for basic -frames. If is a Type I dual of , then is an oblique dual of and ; if is a Type II dual of , then is an oblique dual of and . Similar to ([20], Theorem 1.4), we also have:

Proposition 1. Suppose that and are closed subspaces of and that is a -frame for . Then the following are equivalent:(i)(ii)There is a -frame for that is a Type II dual of (iii)There is a -frame for that is an oblique dual of From the above proposition, the fact that is an oblique dual of shows . For a pair of oblique duals, also have a decomposition.

Lemma 1. Suppose that is a basic -frame for and that is an oblique dual of . Assume is closed containing both and . Then,(i)(ii)

Proof. (1) By assumption, we haveHence,which implies that is a projection. Using (9), we obtainTherefore, and thus . This implies that(ii). Note that (i) shows that . Now, the conclusion follows from ([13], Lemma 3.9)

Li et al. [21] gave the decomposition of frame ranges in terms of the decomposition of analysis operators. They showed that not all closed subspaces of constitute frame ranges:

Lemma 2 (see [21]). Let be a -finite, positive measure space.(i) itself is a frame range if and only if is purely atomic(ii)Every closed subspace of a frame range is a frame range

We finish this section with the following equivalent condition about the frame range:

Lemma 3 (see [13], Corollary 2.9). For a closed subspace of , the following are equivalent:(i) is a frame range(ii)There exists an orthonormal basis for such that for a.e.

3. Characterization of Basic Continuous Frames and Their Oblique Duals

This section focuses on the characterization of basic continuous frames and their oblique duals. We begin with a characterization for basic parseval -frames.

Lemma 4. Suppose that is a measure space with positive measure . Then the following are equivalent:(i)is a basic parseval cotinuous frame for(ii) for some partial isometry whose range is a frame range and some orthonormal basis of (iii) for some orthonormal set of and some orthonormal basis of a frame range

Proof. (i) (ii). Let be the analysis operator for and . Since is a basic parseval continuous frame, is a partial isometry. By Lemma 3, we can denote an orthonormal basis for satisfying for a.e. . For any , it follows thatwhich implies .
(ii) (iii). Write and for every . Since is a partial isometry, is an isometry restricted to . Hence is an orthonormal set.
(iii) (i). For any , it follows thatimplying that is a basic parseval continuous frame.

Similarly, we can deduce the following characterization for basic -frames.

Proposition 2. Suppose that is a measure space with positive measure . Then the following are equivalent:(i) is a basic -frame for (ii) for some operator whose range is a frame range and some orthonormal basis of (iii) for some Riesz sequence of and some orthonormal basis of a frame range

Proof. (i) (ii) follows from the proof of Lemma 4. For (ii) (iii), we see that is bijective restricted to . Hence, is a Riesz sequence for .
Now, suppose (iii) holds. For any , it follows thatThe implication (iii) (i) now follows from the fact that is a Riesz sequence. □
Suppose now is a frame range and that is an orthonormal basis for . We computeimplying that forms a parseval -frame for . If is a basic -frame for with associated analysis operator , then it follows from Proposition 2 thatwhere is invertible restricted to . This means that every basic -frame is similar to a basic parseval -frame for .
Putting Lemma 2 and Proposition 2 together, we can characterize basic Riesz-type -frames:

Corollary 1. Suppose that is a purely atomic, positive measure space where is -finite, and that is an orthonormal basis for . Then, the following are equivalent:(i) is a basic Riesz-type -frame for (ii) for some surjective operator (iii) for some Riesz sequence of It is known that there is a bijective correspondence between the set of discrete frame sequences and all the operators with closed range. The following proposition derives a corresponding result for basic continuous frames. This is the essential difference between discrete frame sequences and basic continuous frames.

Proposition 3. Every basic -frame corresponds to an operator whose range is a frame range.

Proof. Suppose is an operator whose range is a frame range. Then, by Proposition 2, is a basic continuous frame for , where is an orthonormal basis for . Moreover, we computewhich implies that is the analysis operator for .

A new basic -frame can be obtained by applying a suitable operator to a basic -frame. Note that the spanning space may be changed after the action of an operator. Using Proposition 3, we can derive it in an easy way.

Corollary 2. Suppose that is a basic -frame where is -finite, and that is an operator on . Then, is a basic continuous frame if and only if has a closed range restricted to .

Proof. By assumption, is a -Bessel mapping with associated analysis operator .
Now, suppose is a basic continuous frame. Then, the analysis operator has a closed range and so is . This means that has a closed range restricted to .
Conversely, suppose has a closed range. Since and is a frame range, we see from Lemma 2 (ii) that is also a frame range. Then, by Proposition 3, is a basic continuous frame.

We finish this section with the following characterization for an oblique dual pair of basic -frames.

Proposition 4. For -Bessel mappings and of , the following are equivalent:(i) is an oblique dual of (ii)(iii)(iv), (v), (vi), where , are basic -frames(vii) where , are basic -frames

Proof. (i) (ii). By the definition of oblique duals, we get thatHence, we havewhich implies that is an oblique projection with and .
(ii) (i). Since , we getUsing Cauchy Schwarz’ inequality and that is a -Bessel mapping, we can deduce the lower frame condition for . Therefore is a -frame for and is a dual of . We can prove is a -frame and is a dual of in the same way.
(ii) (iv). The existence of implies that .
(iv) (ii). For any , we can write , where , . Hencewhich implies .
(i) (vi) follows from the proof of Lemma 1.
(vi) (ii). Since , we haveFrom the identityit follows thatThus, is an oblique projection and we haveThe fact that , are basic -frames impliesThe rest of the proof is obvious.

4. Dilation of Oblique Dual Pairs

Gabardo and Han derived a dilation theorem ([13], Theorem 1.1]) for dual pairs of continuous frames. In this section, we consider the dilation for an oblique dual pair of basic -frames. Our approach relies strongly on the following result:

Lemma 5. Suppose that is a basic -frame for and that both and are oblique duals of with . If either or holds, then .

Proof. Since and are oblique duals of , we haveNote that , which implies and . Now suppose . Thus and so for any , we can find satisfying . Therefore , and thus holds for all . This forces that , as required.

The preceding lemma also implies the uniqueness of the Type II dual for a fixed direct sum decomposition of . One can see ([22], Proposition 3.3] for a similar assertion for discrete frame sequences. Now we state this result in the following:

Corollary 3. Suppose that is a basic -frame for and that is a subspace of such that . Then there exists a unique -frame for which is the type II dual of .

Proof. From Proposition 1, we can find a -frame for that is a Type II dual of . If there exists another -frame for that is also a Type II dual of , then by Lemma 5, we get .

Now we state our main result of this section in the following:

Theorem 1. Suppose that is a basic -frame for and that is an oblique dual of . Assume both and are contained in a frame range . Then there exist a Hilbert space and a basic -frame for such that , and , where is a (unique) Type II dual of and is the orthogonal projection from onto .

Proof. Denote by an orthonormal basis for and write , where and denote orthogonal projections from onto and respectively. By Lemma 4, we know that defines a parseval -frame for . Therefore, for any ,which implies .
Let and . Clearly is a -Bessel mapping for and , where is the orthogonal projection from onto . From (29) it follows that, for any ,Then by Lemma 1, we haveSince is injective and , it follows that is bijective from onto , which means that is a continuous frame for with . Since is an oblique dual of , we see from Proposition 1 that , and thus we can writeBy Corollary 3, we can find a unique Type II dual of satisfying and . Put . Obviously, is a continuous frame for and thus it only remains to show . On the one hand, we compute, for any ,which shows that is a dual of . Thus, it follows from Proposition 4 that is an oblique dual of . On the other hand, we compute, for any ,Putting (33) and (34) together, for any , we havewhich implies thatNote that and , which yield that . Then by Lemma 5, we get the required result.
Note that the above theorem covers Theorem 1.1 in [13].

As was pointed out in the Introduction, a basic continuous frame is a basic Riesz-type continuous frame if and only if . Putting Lemma 2 and Theorem 1 together, we get the following result:

Corollary 4. Suppose that is a purely atomic, positive measure space where is -finite, and that is an oblique dual of for . Then there exist a Hilbert space and a basic Riesz-type -frame for such that and , where is the orthogonal projection from onto and is the unique dual of .

Using Theorem 1, we can investigate a dilation property for one basic continuous frame.

Corollary 5. Suppose that is a basic -frame for and that is contained in a frame range . Then there exist a Hilbert space and a basic -frame for such that and , where is the orthogonal projection from onto .

Proof. From Corollary 3, we can find a unique basic -frame which is an oblique dual of such that . Then by Theorem 1, we get the required result.

We can derive a dilation result for Type I duals of basic -frames, which is a special case of Theorem 1.

Proposition 5. Suppose that is a basic -frame for and that is a Type I dual of. Assume both and are contained in a frame range . Then, there exists a Hilbert space and a basic -frame for such that , and , where is the canonical dual of and is the orthogonal projection from onto .

Proof. Since is a Type I dual of , it follows that . Thus,which means that is a Type I dual of . Moreover, we see from Theorem 1 that is a Type II dual of . Hence is simultaneously a Type I and a Type II dual of , i.e., canonical dual of .

When is a Type II dual of in Theorem 1, we cannot get a dilation result similar to the above proposition.

Suppose that is a continuous frame and that is an orthogonal projection. It is easy to check that is a continuous frame for . When is an oblique projection, we also have the following result:

Lemma 6. Suppose that is a continuous frame for and that is an oblique projection, where and are closed subspaces of . Then is a continuous frame for .

Proof. We compute for any ,Now assume holds for any . Then we see from (38) that . Since is injective, it follows that , and thus . This shows that is injective and we have the result.

We have considered the dilation of oblique dual pairs under orthogonal projections in Theorem 1. Now we deduce a dilation result in terms of oblique projections.

Theorem 2. Suppose that is a basic -frame for and that is an oblique dual of . Assume both and are contained in a frame range . If , then there exist a -frame for with its canonical dual such that , and .

Proof. Let and be orthogonal projections from onto and respectively and write . Since , we can find a unitary operator . Let be an orthonormal basis for and define . For any , we haveWe see that is injective, and thus is a continuous frame for with frame range . Put . Clearly and is a -Bessel mapping for .
Since , for any , we can write , where , , , . Note that and , which implies thatUsing Lemma 1, we have . The fact that is a continuous frame for shows that is injective. Thus is injective, implying that is a continuous frame for . Let be the canonical dual of and . It follows from Lemma 6 that is a continuous frame for and thus it only remains to show . For any , we getwhich implies that is a dual of . Then by Proposition 4, it follows that E is an oblique dual of . For any , we haveTherefore for , and thus,Since and , we have and the theorem now follows by Lemma 5.

5. Perturbations of Basic Continuous Frames

The perturbation theorem ([13], Theorem 1.2), as a generalization of Casazza and Christensen’s result, stated a perturbation result for continuous frames. However, the spanning space of basic continuous frames may change when adding new elements. This makes the theorem in [13] no longer applicable to basic continuous frames. So we need to derive a new perturbation condition for basic continuous frames.

We begin with some concepts related to subspaces. Let , denote closed subspaces of and define two angles between and :

It is known thatThe following lemma relates the angle to projections.

Lemma 7 (see [23]). For closed subspaces and of with at least one nontrivial, the following are equivalent.(i)and(ii)(iii)is invertible(iv)is invertible(v)(vi)

We also need the following classical fact in operator theory:

Lemma 8 (see [15]). Suppose that is a linear operator on a Banach space and that there exist constants , such that

Then is bounded and invertible. Moreover, for all

Now we give a condition for the perturbation of basic continuous frames:

Theorem 3. Suppose that is a basic -frame for with frame bounds , and that is a vector-valued function. If there exist constants , and such thatfor all with , then is a -Bessel mapping with a Bessel bound

If furthermore , then is a basic -frame with a lower frame bound

Moreover, is isomorphic to and is isomorphic to .

Proof. Let and define byThen, condition (48) implies thatHence,which implies thatSince is dense in , we can extend uniquely to a bounded operator from into . Therefore (48) holds for every and is a -Bessel mapping with a Bessel boundSince is a -Bessel mapping, one can define an operator byNote that is the canonical dual of with frame bounds , . Thus for any ,Fix and write , where , . Using (48) and (57), we obtainIf furthermore , then by Lemma 7 and (45),Thus using Lemma 8, is invertible andHence, maps onto and onto . For any , it follows that , and thus . So, we can writewhich implies thatTherefore,as required.

6. Conclusion

In this paper, we investigate continuous frames for their spanning sets and call them basic continuous frames. The use of basic continuous frames allows more freedom when we design the optimal dual frame and the corresponding analysis-synthesis system. We first characterize basic continuous frames and their oblique duals by using operators with closed ranges. Then we show that an oblique dual pair of basic continuous frames for a Hilbert space can be dilated to a Type II dual pair for a larger Hilbert space. Also, we present a condition under which an oblique dual pair of basic continuous frames can be dilated to a dual pair of continuous frames for the same space. Finally, with the help of angles between different spanning subspaces, a perturbation condition for basic continuous frames is given.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11671201).

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Copyright © 2022 Xin Zhao and Pengtong Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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