Abstract
We obtain a new inequality for frames in Hilbert spaces associated with a scalar and a bounded linear operator induced by two Bessel sequences. It turns out that the corresponding results due to Balan et al. and Găvruţa can be deduced from our result.
1. Introduction
A frame for a Hilbert space firstly emerged in the work on nonharmonic Fourier series owing to Duffin and Schaeffer [1], which has made great contributions to various fields because of its nice properties; the reader can examine the papers [2–12] for background and details of frames.
Balan et al. in [13] showed us a surprising inequality when they further investigated the Parseval frame identity derived from their study on efficient algorithms for signal reconstruction, which was then extended to general frames and alternate dual frames by Găvruţa [14]. In this paper, we establish a new inequality for frames in Hilbert spaces, where a scalar and a bounded linear operator with respect to two Bessel sequences are involved, and it is shown that our result can lead to the corresponding results of Balan et al. and Găvruţa.
The notations , , and are reserved, respectively, for a complex Hilbert space, the identity operator on , and an index set which is finite or countable. The algebra of all bounded linear operators on is designated as .
One says that a family of vectors in is a frame, if there are two positive constants satisfyingThe frame is said to be Parseval if . If satisfies the inequality to the right in (1), we call that is a Bessel sequence for .
For a given frame , the frame operator , a positive, self-adjoint, and invertible operator on , is defined by from which we see that where the involved frame is said to be the canonical dual of .
For any , denote . A positive, bounded linear, and self-adjoint operator induced by and the frame is given below
Suppose that and are two Bessel sequences for . An application of the Cauchy-Schwartz inequality can show that the operatoris well-defined and further . Particularly, if , then both and are frames for . In this case we say that is an alternate dual frame of , and the pair is called an alternate dual frame pair.
2. The Main Results
We need the following simple result on operators to present our main result.
Lemma 1. Suppose that and that . Then for each we have
Proof. A direct calculation gives From this fact and taking into account that we arrive at the relation stated in the lemma.
We can immediately get the following result obtained by Poria in [15], when putting in Lemma 1.
Corollary 2. Suppose that and that . Then for every we have
Theorem 3. Suppose that is a frame for , that and are two Bessel sequences for , and that the operator is defined by (5). Then for each and each , we haveMoreover, if is self-adjoint, then for any and any ,
Proof. We take and for any . Then and further By Lemma 1 we have Therefore, It follows that We now prove the inequality in (10). Again by Lemma 1, for every . Hence, from which we conclude that The proof of (11) is similar to the proof of (10); we leave the details to the reader.
Corollary 4. Suppose that is a frame for with frame operator and that for any . Then for all , for any and any , we have
Proof. Setting for each , then . Taking then and are both Bessel sequences for . For any we have A similar discussion yields We also have Thus the result follows from Theorem 3.
Let be a Parseval frame for ; then . Thus for any , Similarly we have This together with Corollary 4 leads to a result as follows.
Corollary 5. Suppose that is a Parseval frame for . Then for each , for any and any , we have
Corollary 6. Suppose that is an alternate dual frame pair for . Then for each , for any and any , we have
Proof. Since is an alternate dual frame of , . For any , let On the one hand we have On the other hand we have By Theorem 3 the conclusion follows.
Remark 7. Theorems 2.2 and 3.2 in [14] and Proposition 4.1 in [13] can be obtained when taking , respectively, in Corollaries 4, 6, and 5.
As a matter of fact, we can establish a more general inequality for alternate dual frames than that shown in Corollary 6.
Theorem 8. Suppose that is an alternate dual frame pair for . Then for every bounded sequence , for all and all , we have
Proof. We define the operators and by Then both series converge unconditionally and . Since , by Corollary 2 we obtain for each . Hence Therefore, It follows that This completes the proof.
Remark 9. If we take in Theorem 8, then we can obtain Theorem 3.3 in [14].
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that he has no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11761057 and 11561057).