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On equivalent conditions of two sequences to be R-dual
- Zhitao Chuang^{1}Email author and
- Junjian Zhao^{2}
https://doi.org/10.1186/s13660-014-0529-8
© Chuang and Zhao; licensee Springer 2015
- Received: 28 September 2014
- Accepted: 13 December 2014
- Published: 13 January 2015
Abstract
The concept of R-duals 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 R-dual of a given frame \(\{f_{j}\}_{j\in J} \).
Keywords
- Orthonormal Basis
- Separable Hilbert Space
- Riesz Base
- Gabor Frame
- Infinite Dimension
1 Introduction
The concept of R-duals 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 R-dual. Lots of scholars have done much research in this area. Reference [2] introduces various alternative R-duals and shows their relations with Gabor frames. References [3] and [4] consider R-dual in Banach space. In [5], the authors give an equivalent condition for a sequence \(\{\omega_{j}\}_{j\in J} \) to be an R-dual 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 R-dual. 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
Definition 2
For more information as regards frames and Riesz bases we refer to the monograph [7]. We now state the definition of the R-dual sequence.
Definition 3
[1]
Proposition 1
[5]
- (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)
- (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]
- (i)
\(\{\omega_{j}\}_{j\in J} \) is an R-dual 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.
2 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 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
Proof
3 Conditions of R-dual
In this section, we discuss under what conditions \(\{\omega_{i}\}_{i\in J} \) can be an R-dual 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
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
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
Theorem 4
- (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 R-dual 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 R-dual of \(\{\omega_{j}\}_{j\in J} \) if and only (i) and (ii) hold. □
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 R-dual of \(\{f_{i}\} _{i\in J}\) if and only if \(\dim(\ker T)=\dim(W^{\perp}) \) holds.
Remark 1
Remark 2
Declarations
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.
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.
Authors’ Affiliations
References
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