 Research
 Open Access
 Published:
On equivalent conditions of two sequences to be Rdual
Journal of Inequalities and Applications volume 2015, Article number: 10 (2015)
Abstract
The concept of Rduals was introduced by Casazza, Kutyniok, and Lammers in 2004. In this paper, we give a condition when a Parseval frame can be dilated to an orthonormal basis of a given separable Hilbert space H. This is advantageous for deriving a condition for a sequence \(\{\omega_{j}\}_{j\in J} \) to be an Rdual of a given frame \(\{f_{j}\}_{j\in J} \).
Introduction
The concept of Rduals was first introduced by Casazza et al. in [1]. Although it is defined in general frame theory, there is a natural connection with Gabor frame theory. And it is still an open problem whether the duality principle in Gabor analysis actually can be derived from the theory of the Rdual. Lots of scholars have done much research in this area. Reference [2] introduces various alternative Rduals and shows their relations with Gabor frames. References [3] and [4] consider Rdual in Banach space. In [5], the authors give an equivalent condition for a sequence \(\{\omega_{j}\}_{j\in J} \) to be an Rdual of a given frame \(\{f_{j}\}_{j\in J} \). However, we think there is a mistake in their proof. The correction of it will be discussed in Section 3.
The dilation viewpoint on frames is introduced by Larson and Han in [6], which has a natural relation with the Rdual. They point out that any Parseval frame can be dilated to an orthonormal basis. But given a Hilbert space H and a Parseval frame of a subspace of H, can the Parseval frame be dilated to an orthonormal basis for H? This will be discussed in Section 2.
In the entire paper, we let H denote a separable Hilbert space, with the inner product \(\langle\cdot, \cdot\rangle\), and J be a countable index set.
Definition 1
A sequence \(\{f_{j}\}_{j\in J} \) of elements in H is a frame for H if there exist constants \(A , B>0 \) such that
The constants A, B are called a lower and upper frame bounds for the frame. A frame is Atight, if \(A=B \). If \(A=B=1 \), it is called a Parseval frame (a normalized tight frame in [6]).
Definition 2
A sequence \(\{\omega_{j}\}_{j\in J} \) in H is a Riesz sequence if there exist constants \(C , D>0 \) such that
for all finite sequence \(\{c_{j}\}_{j\in J} \). The numbers C, D are called Riesz bounds. A Riesz sequence is a Riesz basis for H if it is complete in H.
For more information as regards frames and Riesz bases we refer to the monograph [7]. We now state the definition of the Rdual sequence.
Definition 3
[1]
Let \(\{e_{i}\}_{i\in J} \) and \(\{h_{i}\}_{i\in J} \) denote two orthonormal bases for H, and let \(\{f_{i}\}_{i\in J} \) be any sequence in H for which
The Rdual of \(\{f_{i}\}_{i\in J} \) with respect to the orthonormal bases \(\{e_{i}\}_{i\in j} \) and \(\{h_{i}\}_{i\in J} \) is the sequence \(\{\omega_{j}\}_{j\in J} \) given by
It is well known from [1] that \(\{f_{i}\}_{i\in J} \) is a frame for H with bounds A, B if and only if \(\{\omega_{j}\}_{j\in J} \) is a Riesz sequence in H with bounds A, B. But given two sequence \(\{f_{i}\}_{i\in J} \) and \(\{\omega_{j}\}_{j\in J} \), under what conditions can we find orthonormal bases \(\{e_{i}\}_{i\in J} \) and \(\{h_{i}\}_{i\in J} \) for H such that (1.1) holds? This is the main question we want to answer in this paper. It will be discussed in Section 3 explicitly. Assume that \(\{f_{i}\}_{i\in J} \) is a frame for H. Define a sequence \(\{n_{i}\}_{i\in J} \) by
where \(\{\tilde{\omega}_{j} \}_{j\in J} \) is the canonical dual Riesz sequence of \(\{\omega_{j} \}_{j\in J} \). The construction of \(\{n_{i}\}_{i\in J} \) comes from [5]. It plays an important role in this paper.
Proposition 1
[5]
Let \(\{\omega_{j}\}_{j\in J} \) be a Riesz basis for the subspace W of H, with dual Riesz basis \(\{\tilde{\omega}_{k}\}_{k\in J} \). Let \(\{e_{i}\}_{i\in J} \) be an orthonormal basis for H. Given any sequence \(\{f_{i}\}_{i\in J} \) in H, the following hold:

(i)
There exists a sequence \(\{h_{i}\}_{i\in J} \) in H such that
$$ f_{i}=\sum_{j \in J} \langle \omega_{j}, h_{i} \rangle e_{j},\quad\forall i\in J. $$(1.3) 
(ii)
The sequence \(\{h_{i}\}_{i\in J} \) satisfying (1.3) is characterized as
$$ h_{i}=m_{i}+n_{i}, $$(1.4)where \(n_{i} \) is given by (1.2) and \(m_{i}\in W^{\perp}\).

(iii)
If \(\{\omega_{j}\}_{j\in J} \) is a Riesz basis for H, then (1.3) has the unique solution
$$h_{i}=n_{i},\quad i\in J. $$
In [5], Christensen et al. give a solution to the main question.
Theorem 1
[5]
Let \(\{\omega_{j}\}_{j\in J} \) be a Riesz sequence spanning a proper subspace W of H and \(\{e_{i}\}_{i\in J} \) an orthonormal basis for H. Given any frame \(\{f_{i}\}_{i\in J} \) for H, the following are equivalent:

(i)
\(\{\omega_{j}\}_{j\in J} \) is an Rdual of \(\{f_{i}\}_{i\in J} \) w.r.t. \(\{e_{i}\}_{i\in J}\) and some orthonormal basis \(\{h_{i}\}_{i\in J}\).

(ii)
There exists an orthonormal basis \(\{h_{i}\}_{i\in J}\) for H satisfying (1.3).

(iii)
The sequence \(\{n_{i}\}_{i\in J}\) in (1.2) is a Parseval frame.
We point out that, in fact, (iii) is not equivalent to the other items in Theorem 1. In order to clarify this, we need the following proposition from [6].
Proposition 2
[6]
Let J be a countable (or finite) index set. Suppose that \(\{x_{n} : n\in J\}\) is a Parseval frame for W. Then there exist a Hilbert space \(K\supseteq W \) and an orthonormal basis \(\{e_{n} :n\in J \} \) for K such that \(Pe_{n}=x_{n} \), where P is the orthogonal projection from K onto W.
A dilation theorem
In this section, a dilation theorem is given, which will be used in Section 3. Firstly, we give an example to show that Theorem 1 is not strictly right.
Example 1
In this example, we choose the index set \(J=\mathbb{N} \), the natural number set. Suppose \(\{z_{i}\}_{i\in J} \) is an orthonormal basis for H. Define \(f_{i}=2z_{i} \) and \(\omega_{i}=2z_{2i} \) for all \(i\in J \). Then the sequence \(\{f_{i}\}_{i\in J} \) is a Parseval frame with frame bounds 2 and \(\{\omega_{j}\}_{j\in J} \) is a Riesz sequence with bounds 2 as well. The canonical dual \(\{\tilde{\omega}_{j}\}_{j\in J} \) of \(\{\omega _{j}\}_{j\in J} \) equals \(\{\frac{1}{2}z_{2j}\}_{j\in J} \). Let
Obviously, \(\{n_{i}\}_{i\in J}\) is a Parseval frame, but \(\{\omega_{j}\}_{j\in J} \) cannot be an Rdual of \(\{f_{i}\}_{i\in J} \). If not, by (ii) of Proposition 1, an orthonormal basis \(\{h_{i}\}_{i\in J} \) for H can be characterized by
where \(m_{i}\in W^{\perp}\) for all \(i\in J \). Since \(n_{i}\in W \), we have
Since \(\Vert {z_{2i}}\Vert =1 \), one has \(m_{i}=0 \) for all \(i\in J \). Therefore \(h_{i}=n_{i}=z_{2i} \). This contradicts \(\{h_{i}\}_{i\in J} \) being an orthonormal basis for H. Thus (iii) of Proposition 1 is not right.
In fact, given any orthonormal sequence (of course a Parseval frame), it cannot be dilated to any orthonormal basis but itself. Generally, we have the following theorem.
Theorem 2
Given two separable Hilbert spaces \(H \supseteq M \), suppose that \(\{x_{n} \}_{n\in J} \) is a Parseval frame for W. Then there exists an orthonormal basis \(\{e_{n}\}_{n\in J} \) for H s.t. \(Pe_{n}=x_{n} \) if and only if
where P is an orthogonal projection from H onto W, T is the synthesis operator of \(\{x_{i}\}_{i\in J} \).
Proof
First we treat sufficiency. Since
for any \(\{c_{i}\}_{i\in J}\in\ell^{2}(J)\), a sequence \(\{c_{i}\}_{i\in J} \in\ker T\) if and only if \(\sum_{i \in J}c_{i}e_{i}\in W^{\perp}\). So (2.1) holds.
Now we treat necessity. Suppose (2.1) holds, from the proof of the Proposition 2, there exist a Hilbert space \(K=\ell^{2}(J) \), an orthogonal projection P, and an orthonormal basis \(\{e_{i}\}_{i\in J} \) for K, such that
where θ is the analysis operator of \(\{x_{i}\} \). Since θ is injective, it has inverse restricted to \(\theta (W) \). For simplicity, we just denote it by \(\theta^{1} \).
For any \(\{c_{i}\}_{i\in J} \in\ell^{2}(J)\), since
we have
Together with (2.1), we have
Therefore, there is an unitary operator η from \(W^{\perp}\) onto \((\theta(W))^{\perp}\). Combining with θ, we can define a unitary operator U from H onto K:
One can easily get
Therefore, \(U^{*}=U^{1} \). In fact, for \(t\in H \) and \(y\in K \),
where the third equation is due to the Parseval frame property of \(\{x_{n}\}_{n\in J} \) and unitarity of η. Because of the unitarity of U, also \(\epsilon_{i}= U^{1}e_{i} \) is an orthonormal basis for H.
Now, taking \(U^{1} \) on the two sides of (2.2), we have
We claim that \(U^{1}PU \) is also an orthogonal projection. In fact, by the properties of U and P, we have
and
Thus we get as desired the complete proof. □
Conditions of Rdual
In this section, we discuss under what conditions \(\{\omega_{i}\}_{i\in J} \) can be an Rdual of \(\{f_{i}\}_{i\in J} \). At first, we give two lemmata which will be used later.
Lemma 1
Let \(\{n_{i}\}_{i\in J}\) be defined as (1.2), W the close span of \(\{\omega_{j}\}_{j\in J} \), then \(\overline {\operatorname {span}}\{n_{i}\}_{i\in J}=W \).
Proof
Since \(n_{i}=\sum_{k \in J} \langle e_{k}, f_{i} \rangle\tilde{\omega}_{k}\), we have
In the opposite direction, since \(\{f_{i}\}_{i\in J} \) is a frame for H, there exists a sequence \(\{c_{\ell}\}_{\ell\in J}\in\ell^{2}(J) \) such that \(e_{m}=\sum_{\ell \in J}c_{\ell}f_{\ell}\) for \(m\in J \). Then one has
Thus \(W\subseteq \overline {\operatorname {span}}\{n_{i}\}_{i\in J}\). We have the desired result. □
Define \(S_{\omega}f=\sum_{k \in J} \langle f, \omega_{k} \rangle \omega_{k} \) and \(S_{\tilde{\omega}}f= \sum_{k \in J} \langle f, \tilde{\omega}_{k} \rangle\tilde{\omega }_{k}\), for \(f\in W \). Then \(S_{\tilde{\omega}}^{\frac{1}{2}} \tilde{\omega}_{k} \) is an orthonormal basis for W. Since \(\langle\omega_{k}, S^{1}_{\omega}\omega_{\ell}\rangle=\delta_{k,\ell } \) by [7], one has \(\tilde{\omega}_{k}=S^{1}_{\omega}\omega_{k} \). Furthermore, we have
This means the operator equation \(S_{\tilde{\omega}}=S^{1}_{\omega } \) holds.
Let \(\epsilon_{k}=S^{\frac{1}{2}}_{\tilde{\omega}}\tilde{\omega} \), then \(\tilde{\omega}_{k}=S^{\frac{1}{2}}_{\tilde{\omega}}\epsilon _{k} \). Let \(\{e_{k}\}_{k\in J} \) be an orthonormal basis for H, define an antiunitary operator \(\Lambda:H\longrightarrow W \) by
Obviously, the inverse of Λ is also an antiunitary operator and
Furthermore, the antiunitary operator Λ has the following property.
Lemma 2
Let Λ be defined as above, then \(\langle\Lambda f, g \rangle= \langle\Lambda ^{1}g, f \rangle\) for any \(f\in H \) and \(g\in W \).
Proof
By the definition of Λ, one has
□
Theorem 3
There exists an orthonormal basis \(\{e_{i}\}_{i\in J} \) such that \(\{n_{i}\}_{i\in J}\) is a Parseval frame if and only if there exists an antiunitary operator Λ such that \(S_{\omega} =\Lambda S\Lambda^{1}\), where S is the frame operator of \(\{f_{i}\}_{i\in J} \).
Proof
By the definition of \(\{n_{i}\}_{i\in J}\) and Lemma 2, we have
Suppose \(\{n_{i}\}_{i\in J}\) is a Parseval frame; then we have
By (3.1), it becomes
For arbitrary complex numbers a and b, we have
Thus \(\Lambda S\Lambda^{1} \) is a linear operator, so is the operator \(S^{\frac{1}{2}}_{\tilde{\omega}} \Lambda S\Lambda^{1}S^{\frac{1}{2}}_{\tilde{\omega}} \). This means \(S^{\frac{1}{2}}_{\tilde{\omega}} \Lambda S\Lambda^{1}S^{\frac{1}{2}}_{\tilde{\omega}}=I \) by (3.3), i.e.
On the other hand, assume there exists an antiunitary operator Λ such that \(S_{\omega}=\Lambda S\Lambda^{1} \). Define \(e_{j}=\Lambda^{1}\epsilon_{j}=\Lambda^{1} S^{\frac{1}{2}}_{\omega}\tilde{\omega}_{k} \), then (3.1) means
is a Parseval frame. □
Theorem 4
Suppose \(\{f_{i}\}_{i\in J} \) is a frame for a separable Hilbert space H and \(\{\omega_{j}\}_{j\in J} \) is a Riesz sequence in H. \(\{f_{i}\}_{i\in J} \) is an Rdual of \(\{\omega_{j}\}_{j\in J} \) if and only if the following two conditions hold:

(i)
there exists an antiunitary operator Λ s.t. \(S_{w}=\Lambda S\Lambda^{1} \);

(ii)
\(\dim(\ker T)=\dim(W^{\perp}) \).
Proof
By Proposition 1, \(\{f_{i}\}_{i \in J} \) is an Rdual of \(\{\omega_{j}\}_{j\in J} \) if and only if \(\{n_{i}\}_{i\in J}\) can be dilated to an orthonormal basis for H. By Theorem 2, this is equivalent to \(\{n_{i}\}_{i\in J}\) being a Parseval frame and (ii) holding. Using Theorem 3, we see that \(\{f_{i}\}_{i\in J} \) is an Rdual of \(\{\omega_{j}\}_{j\in J} \) if and only (i) and (ii) hold. □
We appreciate one reviewer having pointed out that Theorem 4 is of exactly the same type as the characterizations of type II/III in [2]. In the special case, if \(\{f_{i}\}_{i\in\mathbb{N}} \) is an Atight frame for a separable Hilbert space H with infinite dimension and \(\{\omega_{j}\}_{j\in\mathbb{N}} \) is an Atight Riesz sequence where ℕ denotes the natural number set, then there must be an antiunitary operator Λ form H onto W. So we have \(S=AI_{H} \), \(S_{W}=AI_{W} \), and
Thus the condition (i) of Theorem 4 holds automatically. And we get the following corollary, first given in [2].
Corollary 1
[2] Let \(\{f_{i}\}_{i\in J} \) be a tight frame for H and let \(\{\omega_{j}\}_{j\in J} \) be a tight Riesz sequence in H with the same bound. Denote the synthesis operator for \(\{f_{i}\}_{i\in J} \) by T. Then \(\{\omega_{j}\}_{j\in J} \) is an Rdual of \(\{f_{i}\} _{i\in J}\) if and only if \(\dim(\ker T)=\dim(W^{\perp}) \) holds.
Remark 1
Since \(S_{W}f=\sum_{j\in J} \langle f, \omega_{j} \rangle \omega_{j}\) and
(i) of Theorem 4 is equivalent to there existing an antiunitary operator such that
Remark 2
For parameters \(a, b \in \mathbb{R}\), define the operators \(T_{a} \) and \(E_{b} \) on \(L_{2}(\mathbb{R})\)by \(T_{a}f(x)=f(xa) \) and \(E_{b}f(x)=e^{2\pi ibx}f(x) \), respectively. From [8], we know that if \(ab<1 \) and \(\{E_{mb}T_{na}g\}_{m,n\in\mathbb{Z}} \) is a frame, then \(\{E_{mb}T_{na}g\}_{m,n\in\mathbb{Z}} \) has an infinite excess. If \(ab>1 \), then \(\{E_{mb}T_{na}\}_{m,n\in \mathbb{Z}} \) has an infinite deficit. This demonstrates that, if we want to solve the open problem, we only need (i) of Theorem 4 to hold. By Remark 1, this is equivalent to finding an antiunitary operator Λ such that
References
Casazza, PG, Kutyniok, G, Lammers, MC: Duality principles in frame theory. J. Fourier Anal. Appl. 10(4), 383408 (2004)
Stoeva, DT, Christensen, O: On Rduals and the duality principle in Gabor analysis. ArXiv eprints, April 2014
Xian, XM, Zhu, YC: Duality principles of frames in Banach spaces. Acta Math. Sci. Ser. A Chin. Ed. 29(1), 94102 (2009)
Christensen, O, Xiao, XC, Zhu, YC: Characterizing Rduality in Banach spaces. Acta Math. Sin. Engl. Ser. 29(1), 7584 (2013)
Christensen, O, Kim, HO, Kim, RY: On the duality principle by Casazza, Kutyniok, and Lammers. J. Fourier Anal. Appl. 17(4), 640655 (2011)
Larson, DR, Han, D: Frames, bases, and group representations. Mem. Am. Math. Soc. 147, 697 (first of 4 numbers) (2000)
Christensen, O: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2002)
Balan, R, Casazza, PG, Heil, C, Landau, Z: Deficits and excesses of frames. Adv. Comput. Math. 18(24), 93116 (2003)
Acknowledgements
This study was partially supported by the National Science Foundation for Young Scientists of China (Grant No. 11401433). The authors would like to express their thanks to the reviewers for their helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to each part of this work equally and read and approved the final manuscript.
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
About this article
Cite this article
Chuang, Z., Zhao, J. On equivalent conditions of two sequences to be Rdual. J Inequal Appl 2015, 10 (2015). https://doi.org/10.1186/s1366001405298
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366001405298
Keywords
 Orthonormal Basis
 Separable Hilbert Space
 Riesz Base
 Gabor Frame
 Infinite Dimension