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

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 𝐿. 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 ℋ be a Hilbert space, and 𝐽 a set which is finite or countable. A system {𝑓_𝑗}_𝑗∈𝐽⊆ℋ is called a frame (discrete frame) for ℋ, if there exist two constants 𝐶,𝐷>0 such that 𝐶𝑓²≤_𝑗∈𝐽𝑓,𝑓_𝑗²≤𝐷𝑓² (1) for all 𝑓∈ℋ. The constants 𝐶 and 𝐷 are called frame bounds. If 𝐶=𝐷, we call this frame a tight frame and if 𝐶=𝐷=1 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 Ω∶=ℕ and 𝜇 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 𝑔-frames. Fortunately, Arefijamaal and Ghasemi [29] presented a stability result for alternate duals of 𝑔-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, ℋ and 𝒦 are complex Hilbert spaces and ℒ(ℋ,𝒦) is the collection of all bounded linear operators from ℋ into 𝒦. For 𝑇∈ℒ(ℋ,𝒦), we use ℛ(𝑇) and 𝒩(𝑇) to denote the range and the null space of 𝑇, respectively. As usual we use 𝐼_ℋ to denote the identity operator on ℋ.

Definition 1. Let (Ω,𝜇) be a measure space with positive measure 𝜇. A mapping 𝐹∶Ω→ℋ is called a continuous frame with respect to (Ω,𝜇) for ℋ if it is weakly measurable and if there exist two constants 𝐴, 𝐵>0 such that 𝐴𝑓²≤_Ω𝑓,𝐹(𝜔)²𝑑𝜇(𝜔)≤𝐵𝑓²,∀𝑓∈ℋ. (2) The constants 𝐴 and 𝐵 are called the lower and upper continuous frame bounds, respectively. We call 𝐹 a tight continuous frame if 𝐴=𝐵 and a Parseval continuous frame if 𝐴=𝐵=1. The mapping 𝐹 is called Bessel if the right-hand inequality in (2) holds. In this case, 𝐵 is called the Bessel bound.

Example 2. Let 𝐷 be a bounded, Lebesgue measurable subset of ℝ^𝑛 and let ℋ=𝐿²(𝐷) and Ω=ℝ^𝑛. Define 𝐹(𝜔)(𝑡)=𝑒^{2𝜋𝑖⟨𝜔,𝑡⟩}𝜒_𝐷(𝑡), for each 𝑡∈ℝ^𝑛. Clearly, for each 𝑓∈ℋ, ⟨𝑓,𝐹(𝜔)⟩=𝑓(𝜔), for 𝜔∈Ω, where 𝑓 denotes the Fourier transform of 𝑓. By the Plancherel’s theorem, we have 𝑓,𝐹(_Ω𝜔)²𝑑𝜇(𝜔)=_Ω𝑓(𝜔)²(𝜔)=𝑑𝜇_ℝ𝑛𝑓(𝜔)²𝑑𝜇(𝜔)=𝑓², (3) which shows that 𝐹 is a Parseval continuous frame with respect to (Ω,𝜇) for ℋ, where 𝜇 is the Lebesgue measure.

By definition, if 𝐹 is a Bessel mapping with respect to (Ω,𝜇) for ℋ, then 𝑇_𝐹∶ℋ→𝐿²(Ω,𝜇) defined by 𝑇_𝐹𝑓(𝜔)=𝑓,𝐹(𝜔),𝜔∈Ω (4) is a bounded linear operator, which is called the analysis operator. It follows immediately from the observation that for all 𝑓∈ℋ, 𝜑∈𝐿²(Ω,𝜇), 𝜑,𝑇_𝐹𝑓=_Ω𝜑(𝜔)(𝜔)(𝜔)=⟨𝐹,𝑓⟩𝑑𝜇_Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔),𝑓, (5) the adjoint operator 𝑇^∗_𝐹∶𝐿²(Ω,𝜇)→ℋ is weakly given by 𝑇^∗_𝐹𝜑=_Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔),∀𝜑∈𝐿²Ω,𝜇. (6) We call 𝑇^∗_𝐹 the synthesis operator. By composing 𝑇^∗_𝐹 and 𝑇_𝐹, we obtain the frame operator 𝑆_𝐹∶ℋ→ℋ. Note that 𝑆_𝐹 is a positive, self-adjoint operator, and it is invertible if and only if 𝐹 is a continuous frame with respect to (Ω,𝜇) for ℋ. If 𝐹 is a continuous frame, then every 𝑓∈ℋ has a representation of the form 𝑓=_Ω⟨𝑓,𝑆_𝐹⁻¹𝐹(𝜔)(𝜔)(𝜔)=⟩𝐹𝑑𝜇_Ω⟨𝑓,𝐹(𝜔)⟩𝑆_𝐹⁻¹𝐹(𝜔)𝑑𝜇(𝜔). (7) 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 𝐹∶Ω→ℋ be a weakly measurable vector-valued function on Ω. Then the following conditions are equivalent. (1)𝐹is a continuous frame with respect to (Ω,𝜇)  for ℋ.(2)The analysis operator  𝑇_𝐹  is bounded and injective with closed range. (3)The synthesis operator  𝑇^∗_𝐹  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 𝐹 and 𝐺 be two continuous frames with respect to (Ω,𝜇) for ℋ and 𝒦, respectively.(1)They are said to be similar or equivalent if there is a bounded invertible operator 𝑇∶ℋ→𝒦 such that 𝐺=𝑇𝐹.(2)They are said to be unitary equivalent if there exists a unitary linear operator 𝑈∶ℋ→𝒦 such that 𝐺=𝑈𝐹.

It is easily shown that a Bessel mapping 𝐹 with respect to (Ω,𝜇) for ℋ is a continuous frame if and only if there exists a Bessel mapping 𝐺 with respect to (Ω,𝜇) for ℋ such that 𝑓=_Ω𝑓,𝐺(𝜔)𝐹(𝜔)𝑑𝜇(𝜔),∀𝑓∈ℋ. (8) In this case, we call 𝐺 a dual frame for 𝐹 and (𝐹,𝐺) a dual pair. If 𝑆_𝐹 is the frame operator for 𝐹, a continuous frame with respect to (Ω,𝜇) for ℋ, then it is easy to check that 𝑆_𝐹⁻¹𝐹 is a dual of 𝐹, 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 𝐹 to have only one dual, and if this is the case, then we call 𝐹 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 𝐹 be a continuous frame with respect to (Ω,𝜇) for ℋ with analysis operator 𝑇_𝐹, and then 𝐹 is a Riesz-type frame if and only if ℛ(𝑇_𝐹)=𝐿²(Ω,𝜇).

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

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

The definition shows that if 𝐹 is a continuous frame with respect to (Ω,𝜇) for ℋ, then 𝐹 is complete in ℋ, that is, span{𝐹(𝜔)}_𝜔∈Ω=ℋ. We say that 𝐹 is a continuous frame sequence of ℋ if it is a continuous frame for span{𝐹(𝜔)}_𝜔∈Ω.

Proposition 7. Let 𝐹 and 𝐺 be two continuous frame sequences for ℋ, and let us denote 𝑋=span{𝐹(𝜔)}_𝜔∈Ω and 𝑌=span{𝐺(𝜔)}_𝜔∈Ω and assume that ℋ=𝑋⊕𝑌^⟂. Then 𝐹 is a Riesz-type frame with respect to (Ω,𝜇) for 𝑋 if and only if 𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for 𝑌.

Proof. Denote by 𝑇_𝐹 and 𝑇_𝐺 the analysis operators of 𝐹 and 𝐺, respectively, and let 𝑇=𝑇_𝐺𝑇^∗_𝐹. Let 𝜑∈ℛ(𝑇_𝐹) and let 𝑓=𝑇^∗_𝐹𝜑. If 𝑇𝜑=0, then 0=(𝑇_𝐺𝑓)(𝜔)=⟨𝑓,𝐺(𝜔)⟩ for each 𝜔∈Ω, and thus 𝑓∈𝑌^⟂. Since 𝑓∈𝑋∩𝑌^⟂ and ℋ=𝑋⊕𝑌^⟂, it follows that 𝑓=0. Hence 𝜑∈𝒩(𝑇^∗_𝐹). Since 𝒩(𝑇^∗_𝐹)=(ℛ(𝑇_𝐹))^⟂ we know that 𝜑=0, which implies that 𝑇 is injective. Now let 𝜓∈ℛ(𝑇_𝐺). Then there is some 𝑔∈𝑋 such that 𝑇_𝐺𝑔=𝜓. Let 𝑆_𝐹 be the frame operator of 𝐹 and let 𝜙={⟨𝑆_𝐹⁻¹𝑔,𝐹(𝜔)⟩}_𝜔∈Ω, and then 𝜙∈𝐿²(Ω,𝜇) and 𝜓=𝑇_𝐺𝑔=𝑇_𝐺Ω⟨𝑆_𝐹⁻¹𝑔,𝐹(𝜔)⟩𝐹(𝜔)𝑑𝜇(𝜔)=𝑇_𝐺𝑇^∗_𝐹𝜙=𝑇𝜙. (9) Thus, we conclude that 𝑇 is an invertible operator from 𝐿²(Ω,𝜇) onto ℛ(𝑇_𝐺) and hence by Proposition 5, it follows that 𝐹 is a Riesz-type frame with respect to (Ω,𝜇) for 𝑋 if and only if 𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for 𝑌.

Let 𝐹∶Ω→ℋ be a weakly measurable vector-valued function and 𝜑∈𝐿²(Ω,𝜇). It is natural to ask whether we can find 𝑓∈ℋ such that 𝑓,𝐹(𝜔)=𝜑(𝜔),∀𝜔∈Ω. (10) 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  span{𝐹(𝜔)}_𝜔∈Ω=ℋ. Now an alternative formulation of Proposition 5 is that a continuous frame 𝐹 is a Riesz-type frame if and only if the moment problem (10) has a unique solution for all 𝜑∈𝐿²(Ω,𝜇), and furthermore we have the following.

Proposition 8. Suppose that 𝐹 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ, and let 𝜑∈𝐿²(Ω,𝜇). Then the unique solution for the moment problem (10) is exact the vector 𝑓=𝑆_𝐹⁻¹𝑇^∗_𝐹𝜑, where 𝑆_𝐹 and 𝑇_𝐹 denote the frame operator and analysis operator of 𝐹, respectively.

Proof. Let 𝑓 be an element of ℋ such that ⟨𝑓,𝐹(𝜔)⟩=𝜑(𝜔) for each 𝜔∈Ω, and let 𝑃 be the orthogonal projection from 𝐿²(Ω,𝜇) onto ℛ(𝑇_𝐹)=𝐿²(Ω,𝜇). A simple calculation shows that 𝑃𝜑(𝜔)=_Ω𝜑(𝜈)𝑆_𝐹⁻¹𝐹(𝜈),𝐹(𝜔)𝑑𝜇(𝜈),∀𝜑∈𝐿²Ω,𝜇,𝜔∈Ω. (11) Hence 𝑓,𝐹(𝜔)=𝜑(𝜔)=_Ω𝜑(𝜈)⟨𝑆_𝐹⁻¹𝐹(𝜈)(𝜔)(𝜈)=𝑆,𝐹⟩𝑑𝜇_𝐹−1𝑇^∗_𝐹𝜑,𝐹(𝜔),∀𝜔∈Ω, (12) and so 𝑓=𝑆_𝐹⁻¹𝑇^∗_𝐹𝜑 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 𝐹 be a continuous frame with respect to (Ω,𝜇) for ℋ with analysis operator 𝑇_𝐹 and the frame operator 𝑆_𝐹. Then the following are equivalent. (1)𝐹 is a Riesz-type frame. (2) There exist constants 𝐴,𝐵>0 such that 𝐴𝜑²≤𝑇^∗_𝐹𝜑²≤𝐵𝜑²,∀𝜑∈𝐿²Ω,𝜇. (13)(3) If _Ω𝜑(𝜔)⟨𝐹(𝜔),𝑓⟩𝑑𝜇(𝜔)=0 (14)

for some 𝜑∈𝐿²(Ω,𝜇) and for any 𝑓∈ℋ, then 𝜑=0.

Proof. (1)⇒(2). Assume that 𝐹 is a Riesz-type frame with frame bounds 𝐴 and 𝐵, 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 ℛ(𝑇_𝐹)=𝐿²(Ω,𝜇) and, hence, for any 𝜑∈𝐿²(Ω,𝜇), there is 𝑓∈ℋ such that 𝜑(𝜔)=⟨𝑓,𝐹(𝜔)⟩ for each 𝜔∈Ω. Then, we compute that 𝜑⁴=_Ω𝑓,𝐹(𝜔)²𝑑𝜇(𝜔)²=𝑆_𝐹𝑓,𝑓²≤𝑆_𝐹𝑓²𝑓²≤1𝐴𝑆_𝐹𝑓²_Ω𝑓,𝐹(𝜔)²𝑑𝜇(𝜔), (15) and hence 𝐴𝜑²=𝐴_Ω𝑓,𝐹(𝜔)²(𝜔)≤𝑆𝑑𝜇_𝐹𝑓²=𝑇^∗_𝐹𝑇_𝐹𝑓²=𝑇^∗_𝐹𝜑², (16) as desired.
(2)⇒(3). It follows immediately from the left-hand inequality of (13).
(3)⇒(1). In order to show that 𝐹 is a Riesz-type frame, it is sufficient to show, by Proposition 5, that 𝑇_𝐹 is surjective. The assumption implies that 𝑇^∗_𝐹 is injective, and 𝑇^∗_𝐹 is also surjective by using Proposition 3. Now the conclusion follows from the fact that 𝑆_𝐹=𝑇^∗_𝐹𝑇_𝐹 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 𝐹 be a Riesz-type frame with respect to (Ω,𝜇) for ℋ with frame bounds 𝐴, 𝐵 and the analysis operator 𝑇_𝐹. Let 𝐺∶Ω→ℋ be a weakly measurable vector-valued function and assume that there exist constants 𝜆₁, 𝜆₂, 𝜇≥0 such that max{𝜆₁+𝜇/√𝐴,𝜆₂}<1 and _Ω𝜑(𝜔)(𝐹(𝜔)−𝐺(𝜔))𝑑𝜇(𝜔)≤𝜆_1Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)+𝜆_2Ω𝜑(𝜔)𝐺(𝜔)𝑑𝜇(𝜔)+𝜇𝜑 (17) for all 𝜑∈𝐿²(Ω,𝜇) with 𝜇({𝜔∈Ω∶𝜑(𝜔)≠0})<∞. Then 𝐺 is also a Riesz-type frame with frame bounds 𝐴1−𝜆₁+𝜇/√𝐴1+𝜆₂²,𝐵1+𝜆₁+𝜇/√𝐵1−𝜆₂². (18)

Proof. It follows from [15, Theorem 1.2] that 𝐺 is a continuous frame with respect to (Ω,𝜇) for ℋ. Let 𝜑 be an arbitrary element of 𝐿²(Ω,𝜇), then _Ω𝜑(𝜔)𝐺(𝜔)(𝜔≤𝑑𝜇)_Ω𝜑(𝜔𝐹(𝜔)(𝜔(𝜔+)(−𝐺))𝑑𝜇)_Ω𝜑(𝜔)𝐹(𝜔)(𝜔≤𝑑𝜇)1+𝜆_1Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)+𝜆_2Ω𝜑(𝜔)𝐺(𝜔)𝑑𝜇(𝜔)+𝜇𝜑. (19) Since ‖∫_Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)‖=‖𝑇^∗_𝐹𝜑‖≤√𝐵‖𝜑‖, we get _Ω𝜑(𝜔)𝐺(𝜔)(𝜔≤𝑑𝜇)1+𝜆₁√𝐵1−𝜆₂𝜑𝜑=+𝜇1+𝜆₁√𝐵+𝜇1−𝜆₂𝜑, (20) and hence _Ω𝜑(𝜔)𝐺(𝜔)𝑑𝜇(𝜔)²≤1+𝜆₁√𝐵+𝜇1−𝜆₂²𝜑². (21) Similarly, we obtain _Ω𝜑(𝜔)𝐺(𝜔)(𝜔≥𝑑𝜇)_Ω𝜑(𝜔)𝐹(𝜔)(𝜔−𝑑𝜇)_Ω𝜑(𝜔𝐹(𝜔)(𝜔(𝜔≥)(−𝐺))𝑑𝜇)1−𝜆_1Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)−𝜆_2Ω𝜑(𝜔)𝐺(𝜔)𝑑𝜇(𝜔)−𝜇𝜑. (22) Since ‖∫_Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)‖≥√𝐴‖𝜑‖, we have _Ω𝜑(𝜔)𝐺(𝜔)𝑑𝜇(𝜔)²≥(1−𝜆₁)√𝐴−𝜇1+𝜆₂²𝜑². (23) If we denote by 𝑇_𝐺 the analysis operator of  𝐺, then by combing (21) with (23) we have (1−𝜆₁)√𝐴−𝜇1+𝜆₂²𝜑²≤𝑇^∗_𝐺𝜑²≤1+𝜆₁√𝐵+𝜇1−𝜆₂²𝜑²,∀𝜑∈𝐿²Ω,𝜇, (24) and so the result follows from Theorem 9.

A similar result was proved in [15]. Our next proposition points out that if 𝜇=0 in the condition (17) of Proposition 10 and if 𝐹 is a continuous frame, then 𝐺 is a Riesz-type frame if and only if 𝐹 is a Riesz-type frame.

Proposition 11. Let 𝐹 be a continuous frame with respect to (Ω,𝜇) for ℋ and let 𝐺∶Ω→ℋ be a weakly measurable vector-valued function and assume that there exist constants 𝜆₁,𝜆₂∈[0,1) such that _Ω𝜑(𝜔)(𝐹(𝜔)−𝐺(𝜔))𝑑𝜇(𝜔)≤𝜆_1Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)+𝜆_2Ω𝜑(𝜔)𝐺(𝜔)𝑑𝜇(𝜔) (25) for all 𝜑∈𝐿²(Ω,𝜇) with 𝜇({𝜔∈Ω∶𝜑(𝜔)≠0})<∞. Then 𝐹 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ if and only if 𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ.

Proof. By Theorem  1.2 in [15], we know that 𝐺 is also a continuous frame with respect to (Ω,𝜇) for ℋ. We denote by 𝑇_𝐹 and 𝑇_𝐺 the analysis operators of 𝐹 and 𝐺, respectively. We now show that ℛ(𝑇_𝐹)=ℛ(𝑇_𝐺). If 𝜑∈𝒩(𝑇^∗_𝐹), then we have ‖𝑇^∗_𝐺𝜑‖≤𝜆₂‖𝑇^∗_𝐺𝜑‖, which leads to 𝜑∈𝒩(𝑇^∗_𝐺) and so 𝒩(𝑇^∗_𝐹)⊆𝒩(𝑇^∗_𝐺). In the same manner, we can show that 𝒩(𝑇^∗_𝐺)⊆𝒩(𝑇^∗_𝐹), and thus 𝒩(𝑇^∗_𝐺)=𝒩(𝑇^∗_𝐹). It follows from Proposition 3 that both ℛ(𝑇_𝐹) and ℛ(𝑇_𝐺) are closed. Now applying the fact 𝒩𝑇^∗_𝐹⊕ℛ𝑇_𝐹=𝒩𝑇^∗_𝐹⊕𝒩𝑇^∗_𝐹^⟂=𝐿²𝑇Ω,𝜇=𝒩^∗_𝐺⊕𝒩𝑇^∗_𝐺^⟂=𝒩𝑇^∗_𝐺⊕ℛ𝑇_𝐺, (26) we see that ℛ(𝑇_𝐹)=ℛ(𝑇_𝐺). Then by Proposition 5, we can infer that 𝐹 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ if and only if 𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ.

In the following theorem we give necessary and sufficient conditions on Bessel mappings 𝐹 and 𝐺 and bounded linear operators 𝐿₁,𝐿₂∶ℋ→ℋ so that 𝐿₁𝐹+𝐿₂𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ.

Theorem 12. Let 𝐹 and 𝐺 be two Bessel mappings with respect to (Ω,𝜇) for ℋ such that 𝑓=_Ω⟨𝑓,𝐹(𝜔)⟩𝐺(𝜔)𝑑𝜇(𝜔) (27) for all 𝑓∈ℋ. Let 𝑇_𝐹 and 𝑇_𝐺 be the analysis operators of 𝐹 and 𝐺, respectively, and assume that 𝐿₁,𝐿₂∶ℋ→ℋ are bounded linear operators so that 𝐿₁𝐿^∗₂=𝐼_ℋ. Then the following statements are equivalent.(1)𝐿₁𝐹+𝐿₂𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ.(2)The operator 𝑇_𝐹𝐿^∗₁+𝑇_𝐺𝐿^∗₂ is surjective.(3)There exists a constant 𝑀>0 such that 𝑀𝜑²≤𝐿₁𝑇^∗_𝐹+𝐿₂𝑇^∗_𝐺𝜑²,∀𝜑∈𝐿²Ω,𝜇. (28)

Proof. (1)⇔(2). We first show that 𝐿₁𝐹+𝐿₂𝐺 is a continuous frame with respect to (Ω,𝜇) for ℋ. Let 𝑓 be an arbitrary element of ℋ. Then we have _Ω𝑓,𝐿₁𝐹(𝜔)+𝐿₂𝐺(𝜔)²(𝜔)=𝑑𝜇𝑓,𝐿₁𝐹(𝜔)+𝐿₂𝐺(𝜔)_𝜔∈Ω²=𝑓,𝐿₁𝐹(𝜔)+𝐿₂𝐺(𝜔)_𝜔∈Ω,𝑓,𝐿₁𝐹(𝜔)+𝐿₂𝐺(𝜔)_𝜔∈Ω=_Ω𝑓,𝐿₁𝐹(𝜔)²𝑑𝜇(𝜔)+_Ω𝑓,𝐿₂𝐺(𝜔)²(𝜔)+𝑑𝜇_Ω⟨𝑓,𝐿₁𝐹(𝜔)⟩⟨𝐿₂𝐺(𝜔)(𝜔)+,𝑓⟩𝑑𝜇_Ω⟨𝑓,𝐿₂𝐺(𝜔)⟩⟨𝐿₁𝐹(𝜔),𝑓⟩𝑑𝜇(𝜔)=_Ω𝑓,𝐿₁𝐹(𝜔)²(𝜔)+𝑑𝜇_Ω𝑓,𝐿₂𝐺(𝜔)²𝑑𝜇(𝜔)+𝐿^∗₁𝑓,𝐿^∗₂𝑓+𝐿^∗₂𝑓,𝐿^∗₁𝑓=_Ω𝑓,𝐿₁𝐹(𝜔)²𝑑𝜇(𝜔)+_Ω𝑓,𝐿₂𝐺(𝜔)²𝑑𝜇(𝜔)+2𝑓². (29) Let us denote by 𝐷_𝐹 and 𝐷_𝐺 the Bessel bounds of 𝐹 and 𝐺, respectively. For all 𝑓∈ℋ we have 2𝑓²≤_Ω𝑓,𝐿₁𝐹(𝜔)+𝐿₂𝐺(𝜔)²(𝜔)≤𝑑𝜇2+𝐷_𝐹𝐿₁²+𝐷_𝐺𝐿₂²𝑓². (30) By Proposition 5, 𝐿₁𝐹+𝐿₂𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ if and only if its analysis operator 𝑇 is surjective where 𝑇𝑓=𝑓,𝐿₁𝐹(𝜔)+𝐿₂𝐺(𝜔)_𝜔∈Ω=𝑓,𝐿₁𝐹(𝜔)_𝜔∈Ω+𝑓,𝐿₂𝐺(𝜔)_𝜔∈Ω=𝑇_𝐹𝐿^∗₁𝑓+𝑇_𝐺𝐿^∗₂𝑓. (31)
(1)⇔(3). From the proof of (1)⇔(2) we know that 𝐿₁𝐹+𝐿₂𝐺 is a continuous frame with respect to (Ω,𝜇) for ℋ. 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 𝐹 and 𝐺 be two Parseval continuous frames with respect to (Ω,𝜇) for ℋ, with analysis operators 𝑇_𝐹 and 𝑇_𝐺, respectively. Then ℛ(𝑇_𝐺)⊆ℛ(𝑇_𝐹) if and only if there exists an isometry 𝑈∶ℋ→ℋ such that 𝐺=𝑈^∗𝐹.

Proof. “⇒”. Suppose that ℛ(𝑇_𝐺)⊆ℛ(𝑇_𝐹). We already know that ℛ(𝑇_𝐹) and ℛ(𝑇_𝐺) are closed subspaces of 𝐿²(Ω,𝜇). Let us denote 𝑃=𝑇_𝐹𝑇^∗_𝐹 and 𝑄=𝑇_𝐺𝑇^∗_𝐺. It is easy to see that ℛ(𝑃)=ℛ(𝑇_𝐹) and ℛ(𝑄)=ℛ(𝑇_𝐺) since, by Proposition 3, both ℛ(𝑇^∗_𝐹) and ℛ(𝑇^∗_𝐺) are surjective. Since 𝐹 and 𝐺 are Parseval continuous frames of ℋ, it follows that 𝑃 and 𝑄 are orthogonal projections from 𝐿²(Ω,𝜇) onto ℛ(𝑇_𝐹) and ℛ(𝑇_𝐺), respectively. Let 𝑈=𝑇^∗_𝐹𝑇_𝐺, and then, for an arbitrary element 𝑓 of ℋ, recalling that 𝑇_𝐺𝑓∈ℛ(𝑇_𝐹), we have 𝑈^∗𝑈𝑓=𝑇^∗_𝐺𝑇_𝐹𝑇^∗_𝐹𝑇_𝐺𝑓=𝑇^∗_𝐺𝑇_𝐺𝑓=𝑓. (32) Thus, 𝑈 is isometric. Also 𝑈^∗𝑓=𝑇^∗_𝐺𝑇_𝐹𝑓=_Ω⟨𝑓,𝐹(𝜔)⟩𝐺(𝜔)𝑑𝜇(𝜔), (33) and so _Ω(𝜔)(𝜔)(𝜔)=⟨𝑈𝑓,𝐹⟩⟨𝐺,𝑓⟩𝑑𝜇_Ω⟨𝑈𝑓,𝐹(𝜔)⟩𝐺(𝜔)𝑑𝜇(𝜔),𝑓=⟨𝑈^∗𝑈𝑓,𝑓⟩=𝑓². (34) Note that 𝑓,𝐺(_Ω𝜔)²𝑑𝜇(𝜔)=𝑓²,_Ω𝑈𝑓,𝐹(𝜔)²𝑑𝜇(𝜔)=𝑈𝑓²=𝑓², (35) and it follows that 𝑈𝑓,𝐹(_Ω𝜔)−𝑓,𝐺(𝜔)²(𝜔)=𝑑𝜇_Ω(𝜔−(𝜔×(𝜔)(𝜔)(𝜔)=𝑈𝑓,𝐹)𝑓,𝐺)⟨𝐹,𝑈𝑓⟩−⟨𝐺,𝑓⟩𝑑𝜇_Ω𝑈𝑓,𝐹(𝜔)²𝑑𝜇(𝜔)−_Ω(𝜔)(𝜔)(𝜔)−⟨𝑈𝑓,𝐹⟩⟨𝐺,𝑓⟩𝑑𝜇_Ω(𝜔)(𝜔)(𝜔)+⟨𝑓,𝐺⟩⟨𝐹,𝑈𝑓⟩𝑑𝜇_Ω𝑓,𝐺(𝜔)²(𝜔)=𝑓𝑑𝜇₂−𝑓²−𝑓²+𝑓²=0. (36) Hence ⟨𝑓,𝑈^∗𝐹(𝜔)−𝐺(𝜔)⟩=0, for all 𝑓∈ℋ, and so 𝐺(𝜔)=𝑈^∗𝐹(𝜔), for all 𝜔∈Ω, and 𝐺=𝑈^∗𝐹 as a consequence.
“⇐”. It is straightforward.

Proposition 14. Let 𝐹 and 𝐺 be two Parseval continuous frames with respect to (Ω,𝜇) for ℋ, with analysis operators 𝑇_𝐹 and 𝑇_𝐺, respectively, and assume besides that 𝐹 is Riesz-type. Then 𝐺 is Riesz-type if and only if 𝐹 and 𝐺 are unitary equivalent.

Proof. Assume first that 𝐺 is a Riesz-type Parseval continuous frame with respect to (Ω,𝜇) for ℋ, and then ℛ(𝑇_𝐹)=ℛ(𝑇_𝐺). From the proof of Lemma 13, we know that there is a bounded linear operator 𝑈 such that 𝑈^∗𝑈=𝐼_ℋ and 𝐺=𝑈^∗𝐹, and hence 𝑇_𝐺=𝑇_𝐹𝑈. Since both 𝑇_𝐹 and 𝑇_𝐺 are invertible, it follows that 𝑈 is a unitary operator and so 𝐹 and 𝐺 are unitary equivalent.
For the other implication, let 𝑈∶ℋ→ℋ be a unitary linear operator such that 𝐺=𝑈𝐹, and then ℛ(𝑇_𝐹)=ℛ(𝑇_𝐺). By Proposition 5, 𝐺 is Riesz-type.

Furthermore, we have the following.

Proposition 15. Let 𝐹 be a Riesz-type frame with respect to (Ω,𝜇) for ℋ and 𝐺 a continuous frame with respect to (Ω,𝜇) for ℋ. Then 𝐺 is also a Riesz-type frame with respect to (Ω,𝜇) for ℋ if and only if 𝐹 and 𝐺 are similar.

Proof. Suppose first that 𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ. Let 𝑆_𝐹 and 𝑆_𝐺 be the frame operators of 𝐹 and 𝐺, respectively. It is clear that 𝐹=𝑆_𝐹^−1/2𝐹 and 𝐺=𝑆_𝐺^−1/2𝐺 are both Riesz-type Parseval continuous frames. By Proposition 14, we know that there is a unitary operator 𝑈 such that 𝐺=𝑈𝐹, that is, 𝑆_𝐺^−1/2𝐺=𝑈𝑆_𝐹^−1/2𝐹 and so 𝐺=𝑆_𝐺^1/2𝑈𝑆_𝐹^−1/2𝐹, and consequently, 𝐹 is similar to 𝐺.
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 𝐹 be a Riesz-type frame with respect to (Ω,𝜇) for ℋ and 𝐺 is a continuous frame with respect to (Ω,𝜇) for ℋ. Then the following conditions are equivalent. (1)𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ. (2) There exists a constant 𝑀>0 such that for all 𝜑∈𝐿²(Ω,𝜇), _Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)−_Ω𝜑(𝜔)𝐺(𝜔)𝑑𝜇(𝜔)²≤𝑀⋅min_Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)²,_Ω𝜑(𝜔)𝐺(𝜔)𝑑𝜇(𝜔)². (37)

Proof. (1)⇒(2). Suppose that 𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ. By Proposition 15, we know that there is a bounded and invertible operator 𝑇∶ℋ→ℋ such that 𝐺=𝑇𝐹. If 𝑓=∫_Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔) for certain 𝜑∈𝐿²(Ω,𝜇), then we have 𝑇𝑓,𝑔=_Ω𝜑(𝜔)⟨𝐹(𝜔),𝑇^∗(𝜔)=𝑔⟩𝑑𝜇_Ω𝜑(𝜔)𝑇𝐹(𝜔)𝑑𝜇(𝜔),𝑔, (38) for all 𝑔∈ℋ. Hence, 𝑇𝑓=∫_Ω𝜑(𝜔)𝑇𝐹(𝜔)𝑑𝜇(𝜔)=∫_Ω𝜑(𝜔)𝐺(𝜔)𝑑𝜇(𝜔). Therefore, 𝑓−𝑇𝑓²=++=⟨𝑓−𝑇𝑓,𝑓−𝑇𝑓⟩𝑓,𝑓𝑇𝑓,𝑇𝑓−𝑇^∗≤𝑓−𝑇𝑓,𝑓₂+‖𝑇‖²𝑓²+𝑇^∗+𝑇𝑓²≤(1+‖𝑇‖)²𝑓². (39) On the other hand, 𝑓−𝑇𝑓²=𝑓−𝑇𝑓,𝑓−𝑇𝑓=⟨𝑓,𝑓⟩+⟨𝑇𝑓,𝑇𝑓⟩+⟨−𝑇^∗≤𝑇−𝑇𝑓,𝑓⟩₋₁₂+1𝑇𝑓²+𝑇^−1∗−𝑇^∗−𝑇𝑇⁻¹≤𝑇𝑇𝑓,𝑇𝑓₋₁₂+1𝑇𝑓²+𝑇^−1∗−𝑇^∗−𝑇𝑇⁻¹𝑇𝑓²≤1+𝑇⁻¹²𝑇𝑓². (40) Hence (37) holds.
(2)⇒(1). For each 𝑓=∫_Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)∈ℋ, we define an operator 𝑇∶ℋ→ℋ as follows: 𝑇𝑓=𝑇_Ω𝜑(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)=_Ω𝜑(𝜔)𝐺(𝜔)𝑑𝜇(𝜔). (41) It is clear that 𝑇 is well defined and bounded. Let 𝑇_𝐺 be the analysis operator of 𝐺, and since 𝑇^∗_𝐺 is surjective by Proposition 3, it follows that 𝑇 is surjective. And (37) implies that 𝑇 is injective and thus 𝑇 is invertible. In order to show that 𝐺 is a Riesz-type frame, it suffices to show, by Proposition 15, that 𝐺=𝑇𝐹. To see this, let 𝑓∈ℋ, 𝜑∈𝐿²(Ω,𝜇), we have _Ω𝜑(𝜔(𝜔)(𝜔)=𝑇)𝑇𝐹,𝑓𝑑𝜇_Ω𝜑(𝜔)𝐹(𝜔)(𝜔)=𝑑𝜇,𝑓_Ω𝜑(𝜔)𝐺(𝜔)(𝜔)=𝑑𝜇,𝑓_Ω𝜑(𝜔)⟨𝐺(𝜔),𝑓⟩𝑑𝜇(𝜔) (42) that is, 0=_Ω𝜑(𝜔)(𝜔)(𝜔)(𝜔)=(𝜔)(𝜔⟨𝐺−𝑇𝐹,𝑓⟩𝑑𝜇𝜑,𝑓,𝐺−𝑇𝐹)_𝜔∈Ω. (43) Hence 𝐺(𝜔)=𝑇𝐹(𝜔) for each 𝜔∈Ω, and consequently 𝐺=𝑇𝐹, as desired.

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

Let 𝐹 and 𝐺 be two Bessel mappings with respect to (Ω,𝜇) for ℋ, and we introduce the operator 𝐿∶ℋ→ℋ in weak sense by 𝐿𝑓=_Ω𝑓,𝐹(𝜔)𝐺(𝜔)𝑑𝜇(𝜔),∀𝑓∈ℋ. (44) It is obvious that 𝐿 is well defined and bounded, and a direct calculation yields 𝐿^∗𝑓=_Ω𝑓,𝐺(𝜔)𝐹(𝜔)𝑑𝜇(𝜔),∀𝑓∈ℋ. (45)

Proposition 17. Let 𝐹 be a continuous frame with respect to (Ω,𝜇) for ℋ with analysis operator 𝑇_𝐹. Suppose that 𝐺 is a Bessel mapping with respect to (Ω,𝜇) for ℋ with analysis operator 𝑇_𝐺. If 𝐿 is surjective, then 𝐺 is a continuous frame for ℋ.
On the contrary, if 𝐹 is a Riesz-type frame for ℋ, then 𝐿 is surjective.

Proof. Suppose first that 𝐹 is a continuous frame for ℋ. It is clear that 𝐿𝑓=_Ω⟨𝑓,𝐹(𝜔)⟩𝐺(𝜔)𝑑𝜇(𝜔)=𝑇^∗_𝐺𝑇_𝐹𝑓 (46) for any 𝑓∈ℋ, namely, 𝐿=𝑇^∗_𝐺𝑇_𝐹. Since 𝐿 is surjective, then for any 𝑓∈ℋ, there exists 𝑔∈ℋ such that 𝑓=𝐿𝑔=𝑇^∗_𝐺𝑇_𝐹𝑔, and 𝑇_𝐹𝑔∈𝐿²(Ω,𝜇), and it follows that 𝑇^∗_𝐺 is surjective. By Proposition 3, we know that 𝐺 is a continuous frame with respect to (Ω,𝜇) for ℋ.
On the contrary, suppose that 𝐹 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ, and 𝐺 is a continuous frame with respect to (Ω,𝜇) for ℋ. By Proposition 6, 𝑇_𝐹 is invertible. By Proposition 3, 𝑇^∗_𝐺 is surjective, and thus 𝐿=𝑇^∗_𝐺𝑇_𝐹 is surjective.

Theorem 18. Suppose that 𝐹 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ, and 𝐺 is a continuous Bessel mapping with respect to (Ω,𝜇) for ℋ. Then 𝐿 is invertible if and only if 𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ.

Proof. We first suppose that 𝐿 is invertible. Let 𝑇_𝐹 and 𝑇_𝐺 be the analysis operators of 𝐹 and 𝐺, respectively. By Proposition 6 we know that 𝑇_𝐹 is invertible. From the proof of Proposition 17, we know that 𝐿=𝑇^∗_𝐺𝑇_𝐹. Since 𝐿 is invertible, so is 𝑇^∗_𝐺. Again by Proposition 6, we can infer that 𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ.
Now we let 𝐹 and 𝐺 be two Riesz-type frames with respect to (Ω,𝜇) for ℋ. By Proposition 6, both 𝑇_𝐹 and 𝑇_𝐺 are invertible and thus 𝐿=𝑇^∗_𝐺𝑇_𝐹 is invertible.

Theorem 19. Let 𝐹 be a continuous frame with respect to (Ω,𝜇) for ℋ, with frame bounds 𝐴,𝐵 and the frame operator 𝑆_𝐹, and 𝐺 is a Bessel mapping with respect to (Ω,𝜇) for ℋ. Suppose that there exists a number 𝜆 with 0<𝜆<𝐴 such that for all 𝑓∈ℋ, 𝐿𝑓−𝑆_𝐹𝑓≤𝜆𝑓. (47) Then 𝐹 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ if and only if 𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ.

Proof. For any 𝑓∈ℋ, we have 𝐿𝑓=𝐿𝑓−𝑆_𝐹𝑓+𝑆_𝐹𝑓≥𝑆_𝐹𝑓−𝐿𝑓−𝑆_𝐹𝑓≥(𝐴−𝜆)𝑓. (48) So 𝐿 is bounded below and is thus injective with closed range. On the other hand, since 𝐿^∗𝑓−𝑆_𝐹𝑓≤𝐿^∗−𝑆_𝐹𝑓=𝐿−𝑆_𝐹^∗𝑓≤𝜆𝑓,∀𝑓∈ℋ, (49) by the above result 𝐿^∗ is also injective with closed range. Thus, by Proposition 3, both 𝐿 and 𝐿^∗ are invertible. Let 𝑇_𝐹 and 𝑇_𝐺 be the analysis operators of 𝐹 and 𝐺, respectively, and we already know that 𝐿=𝑇^∗_𝐺𝑇_𝐹. Hence 𝑇_𝐹 is invertible if and only if 𝑇_𝐺 is invertible, and consequently 𝐹 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ if and only if 𝐺 is a Riesz-type frame with respect to (Ω,𝜇) for ℋ.

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 𝐹 be a continuous frame with respect to (Ω,𝜇) for ℋ with analysis operator 𝑇_𝐹 and frame bounds 𝐴 and 𝐵. Then there exists a one-to-one correspondence between duals of 𝐹 and operators 𝑆∈ℒ(ℋ,𝐿²(Ω,𝜇)) such that 𝑇^∗_𝐹𝑆=0.

Proof. Assume first that 𝐺 is a dual of 𝐹 with frame bounds 𝐴₁ and 𝐵₁ and let 𝑆_𝐹 be the frame operator of 𝐹. Define 𝑆∶ℋ→𝐿²(Ω,𝜇),𝑓↦𝑆𝑓 by 𝑆𝑓(𝜔)=𝑓,𝐺(𝜔)−𝑆_𝐹⁻¹𝐹(𝜔),𝑓∈ℋ,𝜔∈Ω. (50) Then 𝑆 is bounded. Indeed, 𝑆𝑓=_Ω𝑆𝑓(𝜔)²𝑑𝜇(𝜔)^1/2=_Ω𝑓,𝐺(𝜔)−𝑆_𝐹⁻¹𝐹(𝜔)²𝑑𝜇(𝜔)^1/2≤_Ω𝑓,𝐺(𝜔)²𝑑𝜇(𝜔)^1/2+_Ω𝑓,𝑆_𝐹⁻¹𝐹(𝜔)²𝑑𝜇(𝜔)^1/2≤𝐵₁𝑓+1√𝐴𝑓=𝐵₁+1√𝐴𝑓 (51) for all 𝑓∈ℋ. Moreover, by (50) we have 𝑇^∗_𝐹𝑆𝑓=_Ω(𝜔)𝐹(𝜔)(𝜔)=𝑆𝑓𝑑𝜇_Ω⟨𝑓,𝐺(𝜔)−𝑆_𝐹⁻¹𝐹(𝜔)⟩𝐹(𝜔)𝑑𝜇(𝜔)=_Ω(𝜔)(𝜔)(𝜔)−⟨𝑓,𝐺⟩𝐹𝑑𝜇_Ω𝑓,𝑆_𝐹⁻¹𝐹(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)=𝑓−𝑓=0. (52)
Conversely, let 𝑆∈ℒ(ℋ,𝐿²(Ω,𝜇)) and 𝑇^∗_𝐹𝑆=0. Let 𝐺∶Ω→ℋ be a weakly measurable vector-valued function such that for any 𝑓∈ℋ, ⟨𝑓,𝐺(𝜔)⟩=𝑆𝑓(𝜔)+⟨𝑓,𝑆_𝐹⁻¹𝐹(𝜔)⟩,𝑓∈ℋ,𝜔∈Ω. (53) It is easy to check that 𝐺 is a Bessel mapping with respect to (Ω,𝜇) for ℋ. Furthermore, _Ω(𝜔)(𝜔)(𝜔)=⟨𝑓,𝐺⟩𝐹𝑑𝜇_Ω𝑆𝑓(𝜔)𝐹(𝜔)𝑑𝜇(𝜔)+_Ω⟨𝑓,𝑆_𝐹⁻¹𝐹(𝜔)⟩𝐹(𝜔)𝑑𝜇(𝜔)=𝑇^∗_𝐹𝑆𝑓+𝑓=𝑓. (54) This follows immediately that 𝐺 is a dual of 𝐹.

Theorem 21. Let 𝐹 and 𝐺 be two continuous frames with respect to (Ω,𝜇) for ℋ with frame bounds 𝐴₁,𝐵₁ and 𝐴₂,𝐵₂, respectively. Also let 𝐹^′ be a fixed dual for 𝐹. If 𝐹−𝐺 is a Bessel mapping with sufficiently small bound 𝜖>0, then there exists a dual 𝐺^′ for 𝐺 such that 𝐹^′−𝐺^′ is also a Bessel mapping.

Proof. Let 𝑇_𝐹, 𝑇_𝐺 and 𝑆_𝐹, 𝑆_𝐺 be the analysis operators and frame operators of 𝐹 and 𝐺, respectively. It follows from Lemma 20 that there exists 𝑆∈ℒ(ℋ,𝐿²(Ω,𝜇)) such that 𝑇^∗_𝐹𝑆=0 and ⟨𝑓,𝐹^′(𝜔)⟩=⟨𝑓,𝑆_𝐹⁻¹𝐹(𝜔)⟩+𝑆𝑓(𝜔),𝑓∈ℋ,𝜔∈Ω. (55) Let 𝑀∶Ω→ℋ be a weakly measurable vector-valued function such that for any 𝑓∈ℋ, ⟨𝑓,𝑀(𝜔)⟩=⟨𝑓,𝑆_𝐺⁻¹𝐺(𝜔)⟩+𝑆𝑓(𝜔),𝜔∈Ω. (56) It is easy to see that 𝑀 is a Bessel mapping with Bessel bound (1/𝐴₂+‖𝑆‖)². If we denote the analysis operator of 𝑀 by 𝑇_𝑀, then 𝑓−𝑇^∗_𝐺𝑇_𝑀𝑓=𝑓−_Ω(𝜔)(𝜔)(𝜔=⟨𝑓,𝑀⟩𝐺𝑑𝜇)𝑓−_Ω⟨𝑓,𝑆_𝐺⁻¹𝐺(𝜔)(𝜔)(𝜔)−⟩𝐺𝑑𝜇_Ω(𝜔)𝐺(𝜔)(𝜔=𝑇𝑆𝑓𝑑𝜇)_∗𝐺𝑆𝑓=𝑇^∗_𝐺𝑆𝑓−𝑇^∗_𝐹≤𝑇𝑆𝑓_𝐺−𝑇_𝐹𝑆𝑓≤√‖‖𝜖‖𝑆‖𝑓. (57) Hence ‖𝐼_ℋ−𝑇^∗_𝐺𝑇_𝑀‖≤√𝜖‖𝑆‖ and so 𝑇^∗_𝐺𝑇_𝑀 is invertible since, by assumption, 𝜖 is sufficiently small. Therefore, every 𝑓∈ℋ can be represented by 𝑓=𝑇^∗_𝐺𝑇_𝑀𝑇^∗_𝐺𝑇_𝑀⁻¹𝑓=_Ω𝑇^∗_𝐺𝑇_𝑀⁻¹(𝜔𝐺(𝜔)(𝜔)=𝑓,𝑀)𝑑𝜇_Ω𝑓,𝑇^∗_𝑀𝑇_𝐺⁻¹𝑀(𝜔)𝐺(𝜔)𝑑𝜇(𝜔), (58) showing that 𝐺^′=(𝑇^∗_𝑀𝑇_𝐺)⁻¹𝑀 is a dual for 𝐺. In what follows, we will show that 𝐺^′ is the desired continuous frame. First note that 𝑆_𝐹−𝑆_𝐺=𝑇^∗_𝐹𝑇_𝐹−𝑇^∗_𝐹𝑇_𝐺+𝑇^∗_𝐹𝑇_𝐺−𝑇^∗_𝐺𝑇_𝐺≤𝑇_𝐹−𝑇_𝐺𝑇_𝐹+𝑇_𝐺≤√𝜖𝐵₁+𝐵₂. (59) Let 𝑇_𝐹′ be the analysis operator of 𝐹^′, and then by combining (55) with (56) we have ⟨𝑇^∗_𝐹′𝜑,𝑓⟩−⟨𝑇^∗_𝑀=𝜑,𝑓⟩_Ω𝜑(𝜔)⟨𝐹^′(𝜔)(𝜔)−,𝑓⟩𝑑𝜇_Ω𝜑(𝜔)⟨𝑀(𝜔),𝑓⟩𝑑𝜇(𝜔)=⟨𝑇^∗_𝐹𝜑,𝑆_𝐹⁻¹𝑓⟩−⟨𝑇^∗_𝐺𝜑,𝑆_𝐺⁻¹𝑓⟩=⟨𝑇^∗_𝐹𝜑,𝑆_𝐹⁻¹𝑓⟩−⟨𝑇^∗_𝐹𝜑,𝑆_𝐺⁻¹𝑓⟩+⟨𝑇^∗_𝐹𝜑,𝑆_𝐺⁻¹𝑓⟩−⟨𝑇^∗_𝐺𝜑,𝑆_𝐺⁻¹=𝑇𝑓⟩_∗𝐹𝜑,𝑆_𝐹⁻¹−𝑆_𝐺⁻¹𝑓+𝑇^∗_𝐹𝜑−𝑇^∗_𝐺𝜑,𝑆_𝐺⁻¹𝑓 (60) Therefore, 𝑇^∗_𝐹′𝜑−𝑇^∗_𝑀𝜑=sup_‖𝑓‖=1𝑇^∗_𝐹′𝜑−𝑇^∗_𝑀𝜑,𝑓=sup_‖𝑓‖=1⟨𝑇^∗_𝐹𝜑,𝑆_𝐹⁻¹−𝑆_𝐺⁻¹𝑇𝑓⟩+⟨^∗_𝐹𝜑−𝑇^∗_𝐺𝜑,𝑆_𝐺⁻¹≤𝑆𝑓⟩_𝐹−1−𝑆_𝐺⁻¹𝑇^∗_𝐹𝜑+𝑆_𝐺⁻¹𝑇^∗_𝐹𝜑−𝑇^∗_𝐺𝜑≤𝑆_𝐺⁻¹𝑆_𝐹−𝑆_𝐺𝑆_𝐹⁻¹𝑇^∗_𝐹𝜑+𝑆_𝐺⁻¹𝑇^∗_𝐹−𝑇^∗_𝐺𝜑≤1𝐴₂√𝜖𝐵₁+𝐵₂1𝐴₁𝐵₁𝜑+1𝐴₂√𝜖𝜑=√𝜖𝐴₁+𝐵₁+𝐵₁𝐵₂𝐴₁𝐴₂𝜑. (61)
If we take 𝑇=(𝑇^∗_𝐺𝑇_𝑀)⁻¹, then ‖𝑇‖≤11−𝐼_ℋ−𝑇⁻¹≤11−√𝜖‖𝑆‖, (62) and so 𝐼_ℋ−𝑇≤‖𝑇‖𝐼_ℋ−𝑇⁻¹≤√𝜖‖𝑆‖1−√𝜖‖𝑆‖. (63) Consequently, 𝑇^∗_𝐹′𝜑−𝑇^∗_𝐺′𝜑=sup_‖𝑓‖=1𝑇^∗_𝐹′𝜑−𝑇^∗_𝐺′𝜑,𝑓≤sup_‖𝑓‖=1𝐼_ℋ−𝑇^∗𝑇^∗_𝐹′+𝑇𝜑,𝑓_∗𝑇^∗_𝐹′−𝑇^∗_𝑀≤𝐼𝜑,𝑓_ℋ−𝑇𝑇^∗_𝐹′𝜑+‖𝑇‖𝑇^∗_𝐹′−𝑇^∗_𝑀𝜑≤√𝜖1−√𝜖‖𝑆‖×‖𝑆‖𝐵^′+𝐴₁+𝐵₁+𝐵₁𝐵₂𝐴₁𝐴₂𝜑, (64) where 𝐵^′ is the upper frame bound of 𝐹^′. 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).