Open Access

Some equalities and inequalities for fusion frames

SpringerPlus20165:121

https://doi.org/10.1186/s40064-016-1685-8

Received: 22 September 2015

Accepted: 6 January 2016

Published: 12 February 2016

Abstract

Fusion frames have some properties similar to those of frames in Hilbert spaces, but not all of their properties are similar. Some authors have established some equalities and inequalities for conventional frames. In this paper, we give some equalities and inequalities for fusion frames. Our results generalize and improve the remarkable results which have been obtained by Balan, Casazza and Gǎvruta etc.

Keywords

Fusion frameEqualityFrame operator

Background

Frames, which generalize the concept of bases, can take on infinitely many different representations for a given vector (Christensen 2008). Duffin and Schaeffer (1952) introduced the concept of frame to study some deep problems in nonharmonic Fourier series. After the fundamental paper by Daubechies et al. (1986), frame was popularized from then on. Now, frames are useful in some areas such as the signal and image processing, neural networks, data compression and sampling theory, among others. For signal processing frames can provide resilience to additive noise (Daubechies 1992), resilience to quantization (Goyal et al. 1998), numerical stability of reconstruction (Daubechies 1992), and greater freedom to capture signal characteristic (Benedetto and Colella 1995; Benedetto and Pfander 1998; Unser 1995).

Later on, being the generalization of the frames, fusion frames were introduced by Casazza and Kutyniok (2004) and Fornasier (2002) to handle some large systems which are impossible to handle effectively by just a simple frame. The essence of fusion frame is the construction of global frames from local frames in Hilbert space. So the characteristic fusion frame is special suiting for application such as distributing sensing, parallel processing and packet encoding, and so on. Now, many excellent results of conventional frames have been achieved and applied successfully, which properties of the conventional frames may be extended to the fusion frames? It is a tempting subject because of the complexity of the structure of fusion frames compared with conventional frames.

In this paper, we mainly study the equalities and inequalities of fusion frames. On some equalities for conventional frames were first found by Balan et al. (2007) when the authors studied the optimal decomposition of a Parseval frame. Later on, many authors such as Gǎvruţa (2006) and Zhu and Wu (2010) developed or improved some equalities or inequalities of the conventional frames on the basis of the work in originally in Balan et al. (2007).

Preliminaries

First we will briefly recall the definitions and basic properties of fusion frames. For more details we refer to Casazza and Kutyniok (2004) and Asgari and Khosravi (2005). Throughout the paper, \(\mathcal {H}\) is a Hilbert spaces, and \(I=\{1,2,\ldots ,M\}\) is a subset of \(\mathbf {N}\), \(I_{\mathcal {H}}\) denotes the identity operator on \(\mathcal {H}\).

A family of the vector \(\Phi =\{\varphi _{i}\}_{i\in I}\subset \mathcal {H}\) is called a frame, if there exist constants \(0<A\le B<\infty\) such that for any \(f\in \mathcal {H}\),
$$\begin{aligned} A\Vert f\Vert ^{2}\le \sum \limits _{i\in I}|\langle f,\varphi _{i}\rangle |^{2}\le B\Vert f\Vert ^{2}. \end{aligned}$$
(1)
The constants A and B are known respectively as the lower and upper frame bounds.

Definition 1

Let \(\{W_{i}\}_{i\in I}\) be a sequence of closed subspaces in \(\mathcal {H}\), and \(\{w_{i}\}_{i\in I}\) be a family of weights, i.e., \(w_{i}>0\) for all \(i\in I\). Then \({\mathbf {W}}=\{(W_{i},w_{i})\}_{i\in I}\) is a fusion frame, if there exist constants \(0<C\le D<\infty\) such that for any \(f\in \mathcal {H}\)
$$\begin{aligned} C\Vert f\Vert ^{2}\le \sum \limits _{i\in I}w_{i}\Vert \pi _{W_{i}}(f)\Vert ^{2}\le D\Vert f\Vert ^{2}, \end{aligned}$$
(2)
where \(\pi _{W_{i}}\) denotes the orthogonal projection of \(\mathcal {H}\) onto \(W_{i}\). We call \(C,\,D\) the fusion frame bounds. The frame \({\mathbf {W}}=\{(W_{i},w_{i})\}_{i\in I}\) is called a tight fusion frame if \(C=D\), and is called a Parseval fusion frame if \(C=D=1\). If we only know that \({\mathbf {W}}=\{(W_{i},w_{i})\}_{i\in I}\) satisfies the upper inequality in (2), then \({\mathbf {W}}=\{(W_{i},w_{i})\}_{i\in I}\) is called a Bessel fusion sequence with Bessel bound D.
Let \(\mathbf {W}\) be a Bessel fusion sequence for \(\mathcal {H}\). The synthesis operator \(T^{*}:l^{2}(I)\rightarrow \mathcal {H}\) is defined by
$$\begin{aligned} T^{*}(\{f_{i}\}_{i\in I})=\sum \limits _{i\in I}w_{i}\pi _{W_{i}}(f_{i}),\quad \forall \{f_{i}\}_{i\in I}\in l^{2}(I). \end{aligned}$$
The adjoint operator \(T:\mathcal {H}\rightarrow l^{2}(I)\) given by \(T(f)=\{w_{i}\pi _{W_{i}}(f)\}_{i\in I}\) is called the analysis operator. In Casazza and Kutyniok (2004) we know that
$$\begin{aligned} S:\mathcal {H}\rightarrow \mathcal {H},\;Sf=\sum \limits _{i\in I}w_{i}^{2}\pi _{W_{i}}f, \end{aligned}$$
which is a bounded, self-adjoint, positive and invertible operator with \(CI_{\mathcal {H}}\le S\le DI_{\mathcal {H}}\), and satisfies
$$\begin{aligned} \langle Sf,f\rangle =\sum \limits _{i\in I}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}. \end{aligned}$$
Then the following standard reconstruction formula takes places for all \(f\in \mathcal {H}\),
$$\begin{aligned} f=SS^{-1}f=\sum \limits _{i\in I}w_{i}^{2}\pi _{W_{i}}(S^{-1}f), \end{aligned}$$
and
$$\begin{aligned} \langle S^{-1}f{,}\,f\rangle =\sum \limits _{i\in I}w_{i}^{2}\Vert \pi _{W_{i}}S^{-1}f\Vert ^{2}. \end{aligned}$$
Casazza and Kutyniok (2004) define the dual fusion frame of fusion frame, which is similar to the canonical dual frame in the classical frame theory.

Definition 2

(Casazza and Kutyniok 2004) Let \(\{(W_{i},w_{i})\}_{i\in I}\) be a fusion frame with fusion frame operator S. Then \(\{(S^{-1}W_{i},w_{i})\}_{i\in I}\) is called the dual fusion frame of \(\{(W_{i},w_{i})\}_{i\in I}\).

If \({\mathbf {W}}=\{(W_{i},w_{i})\}_{i\in I}\) is a Bessel fusion sequence in \(\mathcal {H}\), for every \(J\subset I\) we define the operator \(S_{J}\) by
$$\begin{aligned} S_{J}f=\sum \limits _{i\in J}w_{i}^{2}\pi _{W_{i}}f, \end{aligned}$$
(3)
it is trivial to show that \(S_{J}\) is a self-adjoint, bounded linear operator in \(\mathcal {H}\), and denote \(J^{c}=I\setminus J\).
Gǎvruţa (2007) gives a more general alternate dual reconstruction formula, that is, given a fusion frame \({\mathbf {W}}=\{(W_{i},w_{i})\}_{i\in I}\) with frame operator S and a Bessel sequence \({\mathbf {V}}=\{(V_{i},v_{i})\}_{i\in I}\), there is
$$\begin{aligned} f=\sum \limits _{i\in I}v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f,\quad \forall f\in \mathcal {H}. \end{aligned}$$

In this case we call \({\mathbf {V}}=\{(V_{i},v_{i})\}_{i\in I}\) an alternate dual fusion frame of \({\mathbf {W}}=\{(W_{i},w_{i})\}_{i\in I}\).

In the study of longstanding conjecture of signal processing community: a signal can be reconstructed without information about the phase. Balan et al. (2006) found some new frame equalities. In order to compare with the “Main results” section, we list the important equalities in Balan et al. (2007) as follows.

Theorem 1

(Balan et al. 2007) Let \(\{f_{i}\}_{i\in I}\) be a Parseval frame for \(\mathcal {H}\). Then for any \(J\subset I\) and \(f\in \mathcal {H}\) we have
$$\begin{aligned}&\sum \limits _{i\in J}|\langle f,f_{i}\rangle |^{2}-\Vert \sum \limits _{i\in J}\langle f,f_{i}\rangle f_{i}\Vert ^{2}\nonumber \\&=\sum \limits _{i\in J^{c}}|\langle f,f_{i}\rangle |^{2}-\Vert \sum \limits _{i\in J^{c}}\langle f,f_{i}\rangle f_{i}\Vert ^{2}. \end{aligned}$$
(4)

Remark 1

A frame \(\{g_{i}\}_{i\in I}\) is called alternate dual frame of \(\{f_{i}\}_{i\in I}\) and \(f=\sum \limits _{i\in I}\langle f,g_{i}\rangle f_{i}\), \(f\in \mathcal {H}\). Then we get a more general result about the alternate dual frame (Gǎvruţa 2006).

Theorem 2

Let \(\{f_{i}\}_{i\in I}\) be a frame for \(\mathcal {H}\) with an alternate dual frame \(\{g_{i}\}_{i\in I}\subset \mathcal {H}\). Then for any \(J\subset I\) and any \(f\in \mathcal {H}\) we have
$$\begin{aligned} \begin{array}{l} Re\left( \sum \limits _{i\in J}\langle f,g_{i}\rangle \overline{\langle f,f_{i}\rangle }\right) - \Vert \sum \limits _{i\in J}\langle f,g_{i}\rangle f_{i}\Vert ^{2}\\ =Re\left( \sum \limits _{i\in J^{c}}\langle f,g_{i}\rangle \overline{\langle f,f_{i}\rangle }\right) -\Vert \sum \limits _{i\in J^{c}}\langle f,g_{i}\rangle f_{i}\Vert ^{2}. \end{array} \end{aligned}$$
(5)

Zhu and Wu (2010) generalized the equality (5) to a more general form which does not involve the real parts of the complex numbers.

Theorem 3

Let \(\{f_{i}\}_{i\in I}\) be a frame for \(\mathcal {H}\) and \(\{g_{i}\}_{i\in I}\subset \mathcal {H}\) is an alternate dual frame of \(\{f_{i}\}_{i\in I}\). Then for any \(J\subset I\) and \(f\in \mathcal {H}\) we have
$$\begin{aligned} \begin{array}{l} \left( \sum \limits _{i\in J}\langle f,g_{i}\rangle \overline{\langle f,f_{i}\rangle }\right) -\Vert \sum \limits _{i\in J}\langle f,g_{i}\rangle f_{i}\Vert ^{2}\\ =\overline{\left( \sum \limits _{i\in J^{c}}\langle f,g_{i}\rangle \overline{\langle f,f_{i}\rangle }\right) } -\Vert \sum \limits _{i\in J^{c}}\langle f,g_{i}\rangle f_{i}\Vert ^{2}. \end{array} \end{aligned}$$
(6)
Next, we extended this equality to fusion frame.

Main results

Motivated by the work of Balan et al. (2007) and Gǎvruţa (2006), in this section, we continue this work about fusion frames and get some important equalities and inequalities of these frames in a different case.

Lemma 1

(Zhu and Wu 2010) Let P and Q be two linear bounded operators on \(\mathcal {H}\) such that \(P+Q=I_{\mathcal {H}}\). Then \(P-P^{*}P=Q^{*}-Q^{*}Q\).

Now, we present main theorems of this section.

Theorem 4

Let \(\{(W_{i},w_{i})\}_{i\in I}\) be a fusion frame for \(\mathcal {H}\) with the fusion frame operator S, \(\{(V_{i},v_{i})\}_{i\in I}\) is the alternate dual fusion frame of \(\{(W_{i},w_{i})\}_{i\in I}\). Then, for any \(J\subset I\) and any \(f\in \mathcal {H}\),
$$\begin{aligned} \begin{array}{lll} \sum \limits _{i\in J}v_{i}w_{i}\langle S^{-1}\pi _{W_{i}}(f),\pi _{V_{i}}f\rangle -\Vert \sum \limits _{i\in J}v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f\Vert ^{2}&{}\\ = \sum \limits _{i\in J^{c}}v_{i}w_{i}\langle \pi _{V_{i}}f,S^{-1}\pi _{W_{i}}f\rangle -\Vert \sum \limits _{i\in J^{c}}v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f\Vert ^{2}. \end{array} \end{aligned}$$

Proof

For any \(J\subset I\), we define a bounded linear operator \(S_{J}\) as
$$\begin{aligned} S_{J}f=\sum \limits _{i\in J}v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f,\quad \forall f\in \mathcal {H}. \end{aligned}$$
Clearly, \(S_{J}+S_{J^{c}}=I_{\mathcal {H}}\). This, together with Lemma 1, implies that
$$\begin{aligned} \begin{array}{l} \sum \limits _{i\in J}v_{i}w_{i}\langle S^{-1}\pi _{W_{i}}(f),\pi _{V_{i}}f\rangle -\Vert \sum \limits _{i\in J}v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f\Vert ^{2}\\ =\sum \limits _{i\in J}v_{i}w_{i}\langle S^{-1}\pi _{W_{i}}(f),\pi _{V_{i}}f\rangle -\langle S_{J}f,S_{J}f\rangle \\ =\langle S_{J}f,f\rangle -\langle S_{J}^{*}S_{J}f,f\rangle \\ =\langle S_{J^{c}}^{*}f,f\rangle -\langle S_{J^{c}}^{*}S_{J^{c}}f,f\rangle \\ =\langle f,S_{J^{c}}f\rangle -\langle S_{J^{c}}f,S_{J^{c}}f\rangle \\ =\langle f,\sum \limits _{i\in J^{c}}v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f\rangle \\ \quad -\,\Vert \sum \limits _{i\in J^{c}}v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f\Vert ^{2}\\ =\sum \limits _{i\in J^{c}}v_{i}w_{i}\langle \pi _{V_{i}}f,S^{-1}\pi _{W_{i}}f\rangle \\ \quad -\,\Vert \sum \limits _{i\in J^{c}}v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f\Vert ^{2}. \end{array} \end{aligned}$$
(7)

In the situation of Parseval fusion frames the equality is of special form.□

Corollary 1

Let \(\{(W_{i},w_{i})\}_{i\in I}\) be a Parseval fusion frame for \(\mathcal {H}\) with the fusion frame operator \(S=I_{\mathcal {H}}\), \(\{(V_{i},v_{i})\}_{i\in I}\) is the alternate dual fusion frame of \(\{(W_{i},w_{i})\}_{i\in I}\). Then, for any \(J\subset I\) and any \(f\in \mathcal {H}\),
$$\begin{aligned} \begin{array}{lll} \sum \limits _{i\in J}v_{i}w_{i}\langle \pi _{W_{i}}(f),\pi _{V_{i}}f\rangle -\Vert \sum \limits _{i\in J}v_{i}w_{i}\pi _{V_{i}}\pi _{W_{i}}f\Vert ^{2}&{}\\ \quad = \sum \limits _{i\in J^{c}}v_{i}w_{i}\langle \pi _{V_{i}}f,\pi _{W_{i}}f\rangle -\Vert \sum \limits _{i\in J^{c}}v_{i}w_{i}\pi _{V_{i}}\pi _{W_{i}}f\Vert ^{2}. \end{array} \end{aligned}$$

Remark 2

Clearly, when the dual fusion frame of \(\{(W_{i},w_{i})\}_{i\in I}\) is itself, i.e., \(\{(V_{i},v_{i})\}_{i\in I}=\{(W_{i},w_{i})\}_{i\in I}\), which was obtained Theorem 2.2 in Xiyan et al. (2009) as a particular case from the above result.

In fact, similarly to the proof of Theorem 4, we can give a more general result as follow. Moreover, the result has another proof in Xiao et al. (2014).

Theorem 5

Let \(\{(W_{i},w_{i})\}_{i\in I}\) be a fusion frame for \(\mathcal {H}\) with the fusion frame operator S, \(\{(V_{i},v_{i})\}_{i\in I}\) is the alternate dual fusion frame of \(\{(W_{i},w_{i})\}_{i\in I}\). Then, for any \(f\in \mathcal {H}\) and any \(\{b_{i}\}_{i\in I}\in l^{\infty }(I)\),
$$\begin{aligned} \begin{array}{l} \sum \limits _{i\in I}b_{i}v_{i}w_{i}\langle S^{-1}\pi _{W_{i}}f,\pi _{V_{i}}f\rangle \\ \quad -\,\Vert \sum \limits _{i\in I}b_{i}v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f\Vert ^{2}\\ =\sum \limits _{i\in I}(1-\bar{b_{i}})v_{i}w_{i}\langle \pi _{V_{i}}f,S^{-1}\pi _{W_{i}}f\rangle \\ \quad -\,\Vert \sum \limits _{i\in I}(1-b_{i})v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f\Vert ^{2}. \end{array} \end{aligned}$$
(8)
where \(\bar{b_{i}}\) is the conjugata of \(b_{i}\).

Remark 3

Let \(\{(W_{i},w_{i})\}_{i\in I}\) be a tight fusion frame for \(\mathcal {H}\) with the fusion frame bound A, and \(b_{i}\) is real for any \(i\in I\). In this case, using the Theorem 5, we obtain
$$\begin{aligned} \begin{array}{l} A\sum \limits _{i\in I}b_{i}v_{i}w_{i}\langle S^{-1}\pi _{W_{i}}f,\pi _{V_{i}}f\rangle \\ \quad -\,\Vert \sum \limits _{i\in I}b_{i}v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f\Vert ^{2}\\ =A\sum \limits _{i\in I}(1-b_{i})v_{i}w_{i}\langle \pi _{V_{i}}f,S^{-1}\pi _{W_{i}}f\rangle \\ \quad -\,\Vert \sum \limits _{i\in I}(1-b_{i})v_{i}w_{i}\pi _{V_{i}}S^{-1}\pi _{W_{i}}f\Vert ^{2}. \end{array} \end{aligned}$$

Lemma 2

(Gǎvruţa 2006) Let P and Q are two self-adjoint bounded linear operators in \(\mathcal {H}\) and \(P+Q=I_{\mathcal {H}}\). Then we have
$$\begin{aligned} \langle Pf,f\rangle +\Vert Qf\Vert ^{2}=\langle Qf,f\rangle +\Vert Pf\Vert ^{2}\ge \frac{3}{4}\langle f,f\rangle . \end{aligned}$$

Theorem 6

Let \(\{(W_{i},w_{i})\}_{i\in I}\) be a fusion frame for \(\mathcal {H}\) with the fusion frame operator S, \(\{(S^{-1}W_{i},w_{i})\}_{i\in I}\) is the dual fusion frame of \(\{(W_{i},w_{i})\}_{i\in I}\). Then, for any \(J\subset I\) and any \(f\in \mathcal {H}\), we have
$$\begin{aligned} \begin{array}{l} \sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}+\sum \limits _{i\in I}w_{i}^{2}\Vert \pi _{W_{i}}S^{-1}S_{J^{c}}f\Vert ^{2}\\ =\sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}+\sum \limits _{i\in I}w_{i}^{2}\Vert \pi _{W_{i}}S^{-1}S_{J}f\Vert ^{2}\\ \ge \displaystyle \frac{3}{4}\langle Sf,f\rangle . \end{array} \end{aligned}$$

Proof

Applying \(S=S_{J}+S_{J^{c}}\), we have that \(I_{\mathcal {H}}=S^{-\frac{1}{2}}S_{J}S^{-\frac{1}{2}}+S^{-\frac{1}{2}}S_{J^{c}}S^{-\frac{1}{2}}\). Combining this with Lemma 2, it follows that
$$\begin{aligned} \begin{array}{l} \langle S^{-\frac{1}{2}}S_{J}S^{-\frac{1}{2}}f,f\rangle +\Vert S^{-\frac{1}{2}}S_{J^{c}}S^{-\frac{1}{2}}f\Vert ^{2} \\=\langle S^{-\frac{1}{2}}S_{J^{c}}S^{-\frac{1}{2}}f,f\rangle +\Vert S^{-\frac{1}{2}}S_{J}S^{-\frac{1}{2}}f\Vert \\\ge \frac{3}{4}\langle f,f\rangle . \end{array} \end{aligned}$$
(9)
Replacing f by \(S^{\frac{1}{2}}f\), one has
$$\begin{aligned} \begin{array}{l} \langle S_{J}f,f\rangle +\langle S^{-1}S_{J^{c}}f,S_{J^{c}}f\rangle \\=\langle S_{J^{c}}f,f\rangle +\langle S^{-1}S_{J}f,S_{J}f\rangle \ge \frac{3}{4}\langle Sf,f\rangle . \end{array} \end{aligned}$$
Combining this with \(\langle S_{J}f,f\rangle =\sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{w_{i}}f\Vert ^{2}\) and \(\langle S^{-1}f,f\rangle =\sum \limits _{i\in I}w_{i}^{2}\Vert \pi _{W_{i}}S^{-1}f\Vert ^{2}\), the proof is completed.□

Remark 4

The identity of above was established Theorem 2.1 in Xiyan et al. (2009), but the inequality in this form is a new result.

Corollary 2

Let \(\{(W_{i},w_{i})\}_{i\in I}\) be a tight fusion frame for \(\mathcal {H}\) with the fusion frame bound A. Then
$$\begin{aligned} \begin{array}{l} A\sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}+\Vert S_{J^{c}}f\Vert ^{2} \\=A\sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}(f)\Vert ^{2}+\Vert S_{J}f\Vert ^{2}\\ \ge \frac{3}{4}A^{2}\langle f,f\rangle . \end{array} \end{aligned}$$
In addition, if \(\{(W_{i},w_{i})\}_{i\in I}\) is a Parseval fusion frame for \(\mathcal {H}\) , then we have
$$\begin{aligned} \begin{array}{l} \sum \limits _{i\in I}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}+\Vert S_{J^{c}}f\Vert ^{2}\\ =\sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}(f)\Vert ^{2}+\Vert S_{J}f\Vert ^{2}\\ \ge \frac{3}{4}\langle f,f\rangle . \end{array} \end{aligned}$$
(10)

Proof

Since \(\{(W_{i},w_{i})\}_{i\in I}\) be a tight fusion frame for \(\mathcal {H}\) with the fusion frame bound A, then for any \(f\in \mathcal {H}\),
$$\begin{aligned} \sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}=\langle Sf,f\rangle =A\Vert f\Vert ^{2},\,\pi _{W_{i}}S^{-1}=\frac{1}{A}\pi _{W_{i}}, \end{aligned}$$
and
$$\begin{aligned} \sum \limits _{i\in I}w_{i}^{2}\Vert \pi _{W_{i}}S^{-1}S_{J}f\Vert ^{2}=\frac{1}{A}\Vert S_{J}f\Vert ^{2}. \end{aligned}$$
It follows from Theorem 6 that, for any \(f\in \mathcal {H}\),
$$\begin{aligned} \begin{array}{l} A\sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}+\Vert S_{J^{c}}f\Vert ^{2} \\=A\sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}(f)\Vert ^{2}+\Vert S_{J}f\Vert ^{2} \\\ge \frac{3}{4}A^{2}\langle f, f\rangle . \end{array} \end{aligned}$$

Theorem 7

Let \(\{(W_{i},w_{i})\}_{i\in I}\) be a tight fusion frame for \(\mathcal {H}\) with the fusion frame bound A. Then, for any \(J,\,E\subset I\) with \(J\cap E=\emptyset\), and any \(f\in \mathcal {H}\), we have
$$\begin{aligned} \begin{array}{l} \Vert S_{J\cup E}f\Vert ^{2}-\Vert S_{J^{c}\setminus E}f\Vert ^{2} \\=\Vert S_{J}f\Vert ^{2}-\Vert S_{J^{c}}f\Vert ^{2}+ 2A\sum \limits _{i\in E}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}. \end{array} \end{aligned}$$
(11)

Proof

Applying Corollary 2 yields that
$$\begin{aligned} \begin{array}{l} \Vert S_{J\cup E}f\Vert ^{2}-\Vert S_{J^{c}\setminus E}f\Vert ^{2}\\ \quad =A\sum \limits _{i\in J\cup E}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}- A\sum \limits _{i\in J^{c}\setminus E}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}\\ \quad =A\sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}- A\sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}\\ \qquad +\,2A\sum \limits _{i\in E}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}\\ \quad =\Vert S_{J}f\Vert ^{2}-\Vert S_{J^{c}}f\Vert ^{2}+2A\sum \limits _{i\in E}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}. \end{array} \end{aligned}$$

Similarly Corollary 3.6 in Xiao and Zeng (2010), obtain

Corollary 3

Let \(\{(W_{i},w_{i})\}_{i\in I}\) be a tight fusion frame for \(\mathcal {H}\) with the fusion frame bound A. Then, for any \(J_{i}\subset I,\,(i\in N)\), where \(N\ge 2\) is a positive integer, with \(J_{i}\cap J_{j}=\emptyset\), for \(i\ne j\), \(I=\cup _{i=1}^{N}J_{i}\). Then for any \(f\in \mathcal {H}\), we have
$$\begin{aligned} \begin{array}{l} \Vert S_{(\cup _{i=N_{1}}^{N_{4}}J_{i})}f\Vert ^{2}-\Vert S_{(\cup _{i=1}^{N_{1}-1}J_{i}+\cup _{i=N_{4}-1}^{N}J_{i})}f\Vert ^{2}\\ =\Vert S_{(\cup _{i=N_{2}}^{N_{3}}J_{i})}f\Vert ^{2}-\Vert S_{(\cup _{i=N_{2}}^{N_{3}}J_{i})^{c}}f\Vert ^{2}\\ +\,2A\sum _{(\cup _{i=N_{1}}^{N_{2}-1}+\cup _{i=N_{3}+1}^{N_{4}})}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}, \end{array} \end{aligned}$$
where \(N_{i},\,(1\le i\le 4)\) are positive integers satisfying \(1\le N_{1}\le N_{2}<N_{3}<N_{4}\le N-1\).

Proof

Applying (11), replace J and E by \(\cup _{i=N_{2}}^{N_{3}}J_{i}\) and \(\cup _{i=N_{1}}^{N_{2}-1}J_{i}+\cup _{i=N_{3}+1}^{N_{4}}J_{i}\), the above result hold.□

The inequality (10) in Corollary 2 leads us to introduce some notations \(v_{-}(\mathbf {W},J)\) and \(v_{+}(\mathbf {W},J)\). Let \({\mathbf {W}}=\{(W_{i},w_{i})\}_{i\in I}\) be a Parseval fusion frame. For any \(J\subset I\) and \(f\in \mathcal {H}\), define
$$\begin{aligned} v_{+}({\mathbf {W}},J)=\sup _{f\ne 0}\frac{\sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}+\Vert \sum \limits _{i\in J}w_{i}^{2}\pi _{w_{i}}f\Vert ^{2}}{\Vert f\Vert ^{2}}, \end{aligned}$$
and
$$\begin{aligned} v_{-}({\mathbf {W}},J)=\inf _{f\ne 0}\frac{\sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}+\Vert \sum \limits _{i\in J}w_{i}^{2}\pi _{w_{i}}f\Vert ^{2}}{\Vert f\Vert ^{2}}. \end{aligned}$$

Theorem 8

\(v_{-}({\mathbf {W}},J)\) and \(v_{+}({\mathbf {W}},J)\) have the following properties:
  1. 1.

    \(\frac{3}{4}\le v_{-}({\mathbf {W}},J)\le v_{+}({\mathbf {W}},J)\le 1\);

     
  2. 2.

    \(v_{-}({\mathbf {W}},J^{c})=v_{-}({\mathbf {W}},J)\), \(v_{+} ({\mathbf {W}},J^{c})=v_{+}({\mathbf {W}},J)\);

     
  3. 3.

    \(v_{-}({\mathbf {W}},J)=v_{+}({\mathbf {W}},J)\), \(v_{-}({\mathbf {W}},\emptyset )=v_{+}({\mathbf {W}},\emptyset )\).

     

Proof

By inequality (10), \(\frac{3}{4}\le v_{-}({\mathbf {W}},J)\) holds trivially.

For any \(f, g\in \mathcal {H}\) and any \(J\subset I\), we have
$$\begin{aligned} \begin{array}{lll} \Vert \sum \limits _{i\in J}w_{i}^{2}\pi _{w_{i}}f\Vert ^{2}&{}=&{} \sup \limits _{\Vert g\Vert =1}|\langle \sum \limits _{i\in J}w_{i}^{2}\pi _{w_{i}}f,g\rangle |^{2}\\ &{}=&{}\sup \limits _{\Vert g\Vert =1}|\sum \limits _{i\in J}w_{i}^{2}\langle \pi _{w_{i}}f,\pi _{w_{i}}g\rangle |^{2}\\ &{}\le &{}\sup \limits _{\Vert g\Vert =1}\sum \limits _{i\in J}w_{i}^{2} \Vert \pi _{w_{i}}f\Vert ^{2}\sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{w_{i}}g\Vert ^{2}\\ &{}=&{}\sup \limits _{\Vert g\Vert =1}\Vert g\Vert ^{2}\sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{w_{i}}f\Vert ^{2}\\ &{}=&{}\sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{w_{i}}f\Vert ^{2}. \end{array} \end{aligned}$$
Hence,
$$\begin{aligned} \begin{array}{l} \sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}+ \Vert \sum \limits _{i\in J}w_{i}^{2}\pi _{w_{i}}f\Vert ^{2}\\ \le \sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}+ \sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{w_{i}}f\Vert ^{2}\le \Vert f\Vert ^{2}, \end{array} \end{aligned}$$
This implies that \(\Vert \sum \limits _{i\in J}w_{i}^{2}\pi _{w_{i}}f\Vert ^{2}\le \sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{w_{i}}f\Vert ^{2}\). That is \(v_{+}({\mathbf {W}},J)\le 1\).

(2) and (3) follow directly by inequality (10) in Corollary 2.□

Some results for the Parseval fusion frame were established in Xiyan et al. (2009). For the reader’s convenience and our results equivalence, we not only recall its formulation but also provide its proof as follows.

Theorem 9

Let \({\mathbf {W}}=\{(W_{i},w_{i})\}_{i\in I}\) be a Parseval fusion frame for \(\mathcal {H}\). Then, for any \(J\subset I\) and any \(f\in \mathcal {H}\), the following statements are equivalent:
  1. 1.

    \(v_{-}({\mathbf {W}},J)=v_{+}({\mathbf {W}},J)=1\);

     
  2. 2.

    \(\sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{w_{i}}f\Vert ^{2}= \Vert \sum \limits _{i\in J}w_{i}^{2}\pi _{w_{i}}f\Vert ^{2}\);

     
  3. 3.

    \(\sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{w_{i}}f\Vert ^{2}= \Vert \sum \limits _{i\in J^{c}}w_{i}^{2}\pi _{w_{i}}f\Vert ^{2}\);

     
  4. 4.

    \(S_{J}S_{J^{c}}f=0\).

     

Proof

(1)\(\Rightarrow\) (2). Since \(\mathbf {W}\) is a Parseval fusion frame, then for any \(f\in \mathcal {H}\), we have \(\sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}+\sum \limits _{i\in J}w_{i}^{2} \Vert \pi _{w_{i}}f\Vert ^{2}=\Vert f\Vert ^{2}\). This implies that
$$\begin{aligned} \begin{array}{lll} \sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{w_{i}}f\Vert ^{2}&{}=&{}\Vert f\Vert ^{2}- \sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}\\ &{}=&{}\sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}+ \Vert \sum \limits _{i\in J}w_{i}^{2}\pi _{w_{i}}f\Vert ^{2}\\ &{}&{}-\sum \limits _{i\in J^{c}}w_{i}^{2}\Vert \pi _{W_{i}}f\Vert ^{2}\\ &{}=&{}\Vert \sum \limits _{i\in J}w_{i}^{2}\pi _{w_{i}}f\Vert ^{2}. \end{array} \end{aligned}$$
Applying (10), (3) \(\Leftarrow\) (2) \(\Rightarrow\) (1) hold trivially.
(2)\(\Leftrightarrow\) (4) follows from
$$\begin{aligned} \begin{array}{lll} \sum \limits _{i\in J}w_{i}^{2}\Vert \pi _{w_{i}}f\Vert ^{2}-\Vert \sum \limits _{i\in J}w_{i}^{2} \pi _{w_{i}}f\Vert ^{2}\\ =\langle S_{J}f,f\rangle -\langle S_{J}f,S_{J}f\rangle =\langle (S_{J}-S_{J}^{2})f,f\rangle \\ =\langle S_{J}(I-S_{J})f,f\rangle =\langle S_{J}S_{J^{c}}f,f\rangle . \end{array} \end{aligned}$$

Conclusions

In frame theory, fusion frames have some properties similar to those of frames in Hilbert spaces, but not all of their properties are similar. Many excellent results of frames have been achieved and applied successfully, which properties of the frames may be extended to the fusion frames, which requires a lot of efforts to deal with. In this paper, we extend some equalities and inequalities of the frame to the fusion frames, which generalize and improve the remarkable results which have been obtained.

Declarations

Acknowledgements

This work was supported by National Natural Science foundation of China (11271001,61370147,11101071), 973 Program (2013CB329404, 2008CB317110), and Sichuan Province Science and Technology Research Project (12ZC1802).

Authors’ contributions

This work was carried out in collaboration among the authors. All authors have a good contribution to design the study, and to perform the analysis of this research work. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical Sciences, University of Electronic Science and Technology of China

References

  1. Asgari M, Khosravi A (2005) Frames and bases of subspaces in hilbert spaces. J Math Anal Appl 308(2):541–553View ArticleGoogle Scholar
  2. Balan R, Casazza P, Edidin D (2006) On signal reconstruction without phase. Appl Comput Harmonic Anal 20(3):345–356View ArticleGoogle Scholar
  3. Balan R, Casazza P, Edidin D, Kutyniok G (2007) A new identity for parseval frames. Proc Am Math Soc 135(4):1007–1015View ArticleGoogle Scholar
  4. Benedetto JJ, Colella D (1995) Wavelet analysis of spectrogram seizure chirps. In: SPIE’s 1995 international symposium on optical science, engineering, and instrumentation. International Society for Optics and Photonics, pp. 512–521Google Scholar
  5. Benedetto JJ, Pfander GE (1998) Wavelet periodicity detection algorithms. International Society for Optics and Photonics, pp 48–55Google Scholar
  6. Casazza PG, Kutyniok G (2004) Frames of subspaces. Contemp Math 345:87–114View ArticleGoogle Scholar
  7. Christensen O (2008) Frames and bases: an introductory course. Springer, BerlinView ArticleGoogle Scholar
  8. Daubechies I (1992) Ten lectures on wavelets[M]. Society for Industrial and Applied Mathematics, PhiladelphiaView ArticleGoogle Scholar
  9. Daubechies I, Grossmann A, Meyer Y (1986) Painless nonorthogonal expansions. J Math Phys 27(5):1271–1283View ArticleGoogle Scholar
  10. Duffin RJ, Schaeffer AC (1952) A class of nonharmonic fourier series. Trans Am Math Soc 341–366Google Scholar
  11. Fornasier M (2002) Decompositions of hilbert spaces: local construction of global frames. In: Proceedings of international conference on constructive function theory, Varna, pp 275–281Google Scholar
  12. Goyal VK, Vetterli M, Thao NT (1998) Quantized overcomplete expansions in IRN: analysis, synthesis, and algorithms. IEEE Trans Inf Theory 44(1):16–31Google Scholar
  13. Gǎvruţa P (2006) On some identities and inequalities for frames in hilbert spaces. J Math Anal Appl 321(1):469–478View ArticleGoogle Scholar
  14. Gǎvruţa P (2007) On the duality of fusion frames. J Math Anal Appl 333(2):871–879View ArticleGoogle Scholar
  15. Unser M (1995) Texture classification and segmentation using wavelet frames. IEEE Trans Image Process 4(11):1549–1560View ArticleGoogle Scholar
  16. Xiao X, Zeng X (2010) Some equalities and inequalities of g-continuous frames. Sci China Math 53(10):2621–2632View ArticleGoogle Scholar
  17. Xiao X, Zhu Y, Ding M (2014) Erasures and equalities for fusion frames in Hilbert spaces. Bull Malays Math Sci Soc 1–11Google Scholar
  18. Xiyan Y, Guibao G, Ali M (2009) On the equality of fusion frames1. In: International mathematical forum, vol 4, pp 1059–1066Google Scholar
  19. Zhu X, Wu G (2010) A note on some equalities for frames in hilbert spaces. Appl Math Lett 23(7):788–790View ArticleGoogle Scholar

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