Advances in Mathematical Physics

Volume 2013 (2013), Article ID 298982, 11 pages

http://dx.doi.org/10.1155/2013/298982

Research Article

New Characterizations of Riesz-Type Frames and Stability of Alternate Duals of Continuous Frames

Zhong-Qi Xiang

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received 11 April 2013; Accepted 21 May 2013

Academic Editor: Remi Leandre

Copyright © 2013 Zhong-Qi Xiang. 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.

Abstract.

We give new characterizations of Riesz-type frames, on equivalent conditions for a continuous frame to be a Riesz-type frame and on equivalency relations between Riesz-type frames and continuous frames. We characterize also the Riesz-type frames by using a bounded linear operator . Finally, we study the stability of alternate duals of continuous frames and we prove that if two continuous frames are close to each other, then we can find alternate duals of them which are close to each other.

1. Introduction

Frames were first introduced in 1952 by Duffin and Schaeffer [1]. It seems, however, that Duffin-Scheaffer ideas did not attract much interest outside the realm of nonharmonic Fourier series until the fundamental paper [2] by Daubechies et al. in 1986. During the last 20 years, the theory of frames has been growing rapidly, since several new applications have been developed. For example, besides traditional applications as signal processing, image processing, data compression, and sampling theory, frames are now used to mitigate the effect of losses in packet-based communication systems and hence to improve the robustness of data transmission [3, 4] and to design high-rate constellations with full diversity in multiple-antenna code design [5]. We refer readers to the papers [6–12] for more details on frame theory and its applications.

Let be a Hilbert space, and a set which is finite or countable. A system is called a frame (discrete frame) for , if there exist two constants such that

for all . The constants and are called frame bounds. If , we call this frame a tight frame and if it is called a Parseval frame.

The concept of a generalization of frames to a family indexed by some locally compact space endowed with a Radon measure was proposed by Kaiser [13] and independently by Ali et al. [14]. These frames are known as continuous frames. Gabardo and Han in [15] called these frames Frames associated with measurable spaces, and Askari-Hemmat et al. in [16] called them generalized frames and in mathematical physics they are referred to as coherent states [17]. Continuous frames have been widely applied in continuous wavelets transform [17, 18] and the short time Fourier transform [19]. For more details, the interested readers can refer to [20–22].

If in the definition of continuous frame, the measure space and is the counting measure, then the continuous frame will be a discrete frame and so some known results in frame theory can be generalized to continuous frames. However, the Riesz basis, which is a basic concept in frame theory, has no counterpart in continuous frame theory. This motivates us to pay attention to what kinds of continuous frames that are closer in properties to Riesz bases. We found that the Riesz-type frames introduced by Gabardo and Han [15] are more reasonable. And in the same paper, they made a discussion of some properties of Riesz-type frames in some aspects. In this paper, we give new characterizations of Riesz-type frames, on equivalent conditions for a continuous frame to be a Riesz-type frame, and on equivalency relations between Riesz-type frames and continuous frames. We characterize also the Riesz-type frames by means of a bounded linear operator.

The stability of frames is important in practice and is therefore studied widely by many authors [20, 23–25]. The stability of dual frames is also needed in practice. However, most of the known results on this topic are stated about canonical dual; see [26] for frames and [27, 28] for -frames. Fortunately, Arefijamaal and Ghasemi [29] presented a stability result for alternate duals of -frames by observing the difference between an alternate dual and the canonical dual. Inspired by their idea we discuss the stability of alternate duals of continuous frames.

The paper is organized in the following manner. Section 2 investigates some basic properties of Riesz-type frames. The main results of the paper are included in Section 3, where some new characterizations of Riesz-type frames are obtained. Section 4 studies the stability of alternate duals of continuous frames.

2. The Definitions and Some Basic Results

In the following we briefly recall some definitions and results about continuous frames and Riesz-type frames in Hilbert spaces.

We first give some notations which we will need later. Throughout this paper, and are complex Hilbert spaces and is the collection of all bounded linear operators from into . For , we use and to denote the range and the null space of , respectively. As usual we use to denote the identity operator on .

Definition 1. Let be a measure space with positive measure . A mapping is called a continuous frame with respect to for if it is weakly measurable and if there exist two constants , such that

The constants and are called the lower and upper continuous frame bounds, respectively. We call a tight continuous frame if and a Parseval continuous frame if . The mapping is called Bessel if the right-hand inequality in (2) holds. In this case, is called the Bessel bound.

Example 2. Let be a bounded, Lebesgue measurable subset of and let and . Define , for each . Clearly, for each , , for , where denotes the Fourier transform of . By the Plancherel’s theorem, we have

which shows that is a Parseval continuous frame with respect to for , where is the Lebesgue measure.

By definition, if is a Bessel mapping with respect to for , then defined by

is a bounded linear operator, which is called the analysis operator. It follows immediately from the observation that for all , ,

the adjoint operator is weakly given by

We call the synthesis operator. By composing and , we obtain the frame operator . Note that is a positive, self-adjoint operator, and it is invertible if and only if is a continuous frame with respect to for . If is a continuous frame, then every has a representation of the form

Remember that the representations above have to be interpreted in the weak sense.

We can also characterize the continuous frames in terms of the associated analysis and synthesis operators as in frame theory.

Proposition 3. Let be a weakly measurable vector-valued function on . Then the following conditions are equivalent.

(1)is a continuous frame with respect to   for .(2)The analysis operator    is bounded and injective with closed range. (3)The synthesis operator    is bounded, linear, and onto.

Proof. This claim holds in an analogous way as in frame theory.

The definitions of similar and unitary equivalent frames give rise to the definitions of similar and unitary equivalent continuous frames.

Definition 4. Let and be two continuous frames with respect to for and , respectively.

(1)They are said to be similar or equivalent if there is a bounded invertible operator such that .(2)They are said to be unitary equivalent if there exists a unitary linear operator such that .

It is easily shown that a Bessel mapping with respect to for is a continuous frame if and only if there exists a Bessel mapping with respect to for such that

In this case, we call a dual frame for and a dual pair. If is the frame operator for , a continuous frame with respect to for , then it is easy to check that is a dual of , and it is called the canonical dual, but a dual which is not the canonical dual is called an alternate dual, or simply a dual. It is certainly possible for a continuous frame to have only one dual, and if this is the case, then we call a Riesz-type frame.

The following proposition is a useful tool and will be used frequently in the rest of the paper. The reader can find a proof in [15].

Proposition 5. Let be a continuous frame with respect to for with analysis operator , and then is a Riesz-type frame if and only if .

The following is an immediate consequence of Propositions 3 and 5.

Proposition 6. A weakly measurable vector-valued function is a Riesz-type frame with respect to for if and only if the operator defined by (4) is well defined, bounded, and invertible.

The definition shows that if is a continuous frame with respect to for , then is complete in , that is, . We say that is a continuous frame sequence of if it is a continuous frame for .

Proposition 7. Let and be two continuous frame sequences for , and let us denote and and assume that . Then is a Riesz-type frame with respect to for if and only if is a Riesz-type frame with respect to for .

Proof. Denote by and the analysis operators of and , respectively, and let . Let and let . If , then for each , and thus . Since and , it follows that . Hence . Since we know that , which implies that is injective. Now let . Then there is some such that . Let be the frame operator of and let , and then and

Thus, we conclude that is an invertible operator from onto and hence by Proposition 5, it follows that is a Riesz-type frame with respect to for if and only if is a Riesz-type frame with respect to for .

Let be a weakly measurable vector-valued function and . It is natural to ask whether we can find such that

A problem of this type is called a moment problem. It is clear that if the moment problem (10) has a solution, and it is unique if and only if  . Now an alternative formulation of Proposition 5 is that a continuous frame is a Riesz-type frame if and only if the moment problem (10) has a unique solution for all , and furthermore we have the following.

Proposition 8. Suppose that is a Riesz-type frame with respect to for , and let . Then the unique solution for the moment problem (10) is exact the vector , where and denote the frame operator and analysis operator of , respectively.

Proof. Let be an element of such that for each , and let be the orthogonal projection from onto . A simple calculation shows that

Hence

and so as desired.

3. New Characterizations of Riesz-Type Frames

In this section, we get several characterizations of Riesz-type frames. We start with the following equivalent conditions under which a continuous frame is a Riesz-type frame.

Theorem 9. Let be a continuous frame with respect to for with analysis operator and the frame operator . Then the following are equivalent.

(1) is a Riesz-type frame. (2) There exist constants such that

(3) If

for some and for any , then .

Proof. (1)(2). Assume that is a Riesz-type frame with frame bounds and , and then the right-hand inequality in (13) holds. To complete the proof, it remains to establish the left-hand inequality. By Proposition 5, we know that and, hence, for any , there is such that for each . Then, we compute that

and hence

as desired.

(2)(3). It follows immediately from the left-hand inequality of (13).

(3)(1). In order to show that is a Riesz-type frame, it is sufficient to show, by Proposition 5, that is surjective. The assumption implies that is injective, and is also surjective by using Proposition 3. Now the conclusion follows from the fact that is invertible.

The following perturbation result for Riesz-type frames is due to Gabardo and Han in [15]. Here, as an application of Theorem 9, we present a new approach to its proof.

Proposition 10. Let be a Riesz-type frame with respect to for with frame bounds , and the analysis operator . Let be a weakly measurable vector-valued function and assume that there exist constants , , such that and

for all with . Then is also a Riesz-type frame with frame bounds

Proof. It follows from [15, Theorem 1.2] that is a continuous frame with respect to for . Let be an arbitrary element of , then

Since , we get

and hence

Similarly, we obtain

Since , we have

If we denote by the analysis operator of  , then by combing (21) with (23) we have

and so the result follows from Theorem 9.

A similar result was proved in [15]. Our next proposition points out that if in the condition (17) of Proposition 10 and if is a continuous frame, then is a Riesz-type frame if and only if is a Riesz-type frame.

Proposition 11. Let be a continuous frame with respect to for and let be a weakly measurable vector-valued function and assume that there exist constants such that

for all with . Then is a Riesz-type frame with respect to for if and only if is a Riesz-type frame with respect to for .

Proof. By Theorem  1.2 in [15], we know that is also a continuous frame with respect to for . We denote by and the analysis operators of and , respectively. We now show that . If , then we have , which leads to and so . In the same manner, we can show that , and thus . It follows from Proposition 3 that both and are closed. Now applying the fact

we see that . Then by Proposition 5, we can infer that is a Riesz-type frame with respect to for if and only if is a Riesz-type frame with respect to for .

In the following theorem we give necessary and sufficient conditions on Bessel mappings and and bounded linear operators so that is a Riesz-type frame with respect to for .

Theorem 12. Let and be two Bessel mappings with respect to for such that

for all . Let and be the analysis operators of and , respectively, and assume that are bounded linear operators so that . Then the following statements are equivalent.

(1) is a Riesz-type frame with respect to for .(2)The operator is surjective.(3)There exists a constant such that

Proof. (1)(2). We first show that is a continuous frame with respect to for . Let be an arbitrary element of . Then we have

Let us denote by and the Bessel bounds of and , respectively. For all we have

By Proposition 5, is a Riesz-type frame with respect to for if and only if its analysis operator is surjective where

(1)(3). From the proof of (1)(2) we know that is a continuous frame with respect to for . Now the result follows immediately from Theorem 9.

We now turn our attention to the characterizations of the equivalency relations between Riesz-type frames and continuous frames. To prove our first result, we need the following.

Lemma 13. Let and be two Parseval continuous frames with respect to for , with analysis operators and , respectively. Then if and only if there exists an isometry such that .

Proof. “”. Suppose that . We already know that and are closed subspaces of . Let us denote and . It is easy to see that and since, by Proposition 3, both and are surjective. Since and are Parseval continuous frames of , it follows that and are orthogonal projections from onto and , respectively. Let , and then, for an arbitrary element of , recalling that , we have

Thus, is isometric. Also

and so

Note that

and it follows that

Hence , for all , and so , for all , and as a consequence.

“”. It is straightforward.

Proposition 14. Let and be two Parseval continuous frames with respect to for , with analysis operators and , respectively, and assume besides that is Riesz-type. Then is Riesz-type if and only if and are unitary equivalent.

Proof. Assume first that is a Riesz-type Parseval continuous frame with respect to for , and then . From the proof of Lemma 13, we know that there is a bounded linear operator such that and , and hence . Since both and are invertible, it follows that is a unitary operator and so and are unitary equivalent.

For the other implication, let be a unitary linear operator such that , and then . By Proposition 5, is Riesz-type.

Furthermore, we have the following.

Proposition 15. Let be a Riesz-type frame with respect to for and a continuous frame with respect to for . Then is also a Riesz-type frame with respect to for if and only if and are similar.

Proof. Suppose first that is a Riesz-type frame with respect to for . Let and be the frame operators of and , respectively. It is clear that and are both Riesz-type Parseval continuous frames. By Proposition 14, we know that there is a unitary operator such that , that is, and so , and consequently, is similar to .

The opposite implication is obvious.

As an application of Proposition 15, we have the following equivalent condition for a continuous frame to be a Riesz-type frame.

Theorem 16. Let be a Riesz-type frame with respect to for and is a continuous frame with respect to for . Then the following conditions are equivalent.

(1) is a Riesz-type frame with respect to for . (2) There exists a constant such that for all ,

Proof. (1)(2). Suppose that is a Riesz-type frame with respect to for . By Proposition 15, we know that there is a bounded and invertible operator such that . If for certain , then we have

for all . Hence, . Therefore,

On the other hand,

Hence (37) holds.

(2)(1). For each , we define an operator as follows:

It is clear that is well defined and bounded. Let be the analysis operator of , and since is surjective by Proposition 3, it follows that is surjective. And (37) implies that is injective and thus is invertible. In order to show that is a Riesz-type frame, it suffices to show, by Proposition 15, that . To see this, let , , we have

that is,

Hence for each , and consequently , as desired.

Next we will introduce a bounded linear operator about two Bessel mappings from to . The idea is derived from the operator which was considered for fusion frames by Găvruţa in [30]. In this paper, we will use the operator to characterize the Riesz-type frames further.

Let and be two Bessel mappings with respect to for , and we introduce the operator in weak sense by

It is obvious that is well defined and bounded, and a direct calculation yields

Proposition 17. Let be a continuous frame with respect to for with analysis operator . Suppose that is a Bessel mapping with respect to for with analysis operator . If is surjective, then is a continuous frame for .

On the contrary, if is a Riesz-type frame for , then is surjective.

Proof. Suppose first that is a continuous frame for . It is clear that

for any , namely, . Since is surjective, then for any , there exists such that , and , and it follows that is surjective. By Proposition 3, we know that is a continuous frame with respect to for .

On the contrary, suppose that is a Riesz-type frame with respect to for , and is a continuous frame with respect to for . By Proposition 6, is invertible. By Proposition 3, is surjective, and thus is surjective.

Theorem 18. Suppose that is a Riesz-type frame with respect to for , and is a continuous Bessel mapping with respect to for . Then is invertible if and only if is a Riesz-type frame with respect to for .

Proof. We first suppose that is invertible. Let and be the analysis operators of and , respectively. By Proposition 6 we know that is invertible. From the proof of Proposition 17, we know that . Since is invertible, so is . Again by Proposition 6, we can infer that is a Riesz-type frame with respect to for .

Now we let and be two Riesz-type frames with respect to for . By Proposition 6, both and are invertible and thus is invertible.

Theorem 19. Let be a continuous frame with respect to for , with frame bounds and the frame operator , and is a Bessel mapping with respect to for . Suppose that there exists a number with such that for all ,

Then is a Riesz-type frame with respect to for if and only if is a Riesz-type frame with respect to for .

Proof. For any , we have

So is bounded below and is thus injective with closed range. On the other hand, since

by the above result is also injective with closed range. Thus, by Proposition 3, both and are invertible. Let and be the analysis operators of and , respectively, and we already know that . Hence is invertible if and only if is invertible, and consequently is a Riesz-type frame with respect to for if and only if is a Riesz-type frame with respect to for .

4. Stability of Duals of Continuous Frames

We begin with showing that the difference between an alternate dual and the canonical dual can be considered as a bounded operator.

Lemma 20. Let be a continuous frame with respect to for with analysis operator and frame bounds and . Then there exists a one-to-one correspondence between duals of and operators such that .

Proof. Assume first that is a dual of with frame bounds and and let be the frame operator of . Define by

Then is bounded. Indeed,

for all . Moreover, by (50) we have

Conversely, let and . Let be a weakly measurable vector-valued function such that for any ,

It is easy to check that is a Bessel mapping with respect to for . Furthermore,

This follows immediately that is a dual of .

Theorem 21. Let and be two continuous frames with respect to for with frame bounds and , respectively. Also let be a fixed dual for . If is a Bessel mapping with sufficiently small bound , then there exists a dual for such that is also a Bessel mapping.

Proof. Let , and , be the analysis operators and frame operators of and , respectively. It follows from Lemma 20 that there exists such that and

Let be a weakly measurable vector-valued function such that for any ,

It is easy to see that is a Bessel mapping with Bessel bound . If we denote the analysis operator of by , then

Hence and so is invertible since, by assumption, is sufficiently small. Therefore, every can be represented by

showing that is a dual for . In what follows, we will show that is the desired continuous frame. First note that

Let be the analysis operator of , and then by combining (55) with (56) we have

Therefore,

If we take , then

and so

Consequently,

where is the upper frame bound of . The proof is completed.

Acknowledgments

The author thanks the referee(s) and the editor(s) for their valuable comments and suggestions which improved the quality of the paper. The author was partially supported by the National Natural Science Foundation of China (Grant no. 11271148).

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