Construction and Stability of Riesz Bases

Abstract

We construct some new Riesz bases and consider the stability of them. The investigation is based on the stability of Riesz bases of cosines and sines in the Hilbert space .

1. Introduction

As is well known Riesz basis is not only a base but also a special frame. The research of frame and Riesz basis plays important role in theoretical research of wavelet analysis [1]; because of the redundancy of frame and Riesz basis, they have been extensively applied in signal denoising, feature extraction, robust signal processing, and so on. Therefore, construction of Riesz basis has attracted much attention of the researchers due to their wide applications.

In 1934, Paley and Wiener studied the problem of finding sequences for which is a Riesz basis in [2]. Since then many results on the Riesz basis have been obtained [3–5]. Also the Riesz basis of the systems of sines and cosines in and Riesz basis associated with Sturm-Liouville problems have been studied in many papers [6–12]; moreover, on the problems of expansion of eigenfunctions, we refer to [13–18] and references cited therein.

Motivated by these works, on the one hand, we construct two groups of Riesz bases and and study the stability of them. On the other, we consider the problem of finding a new sequence associated with eigenfunctions of Sturm-Liouville problemsuch that it forms a Riesz basis.

2. Riesz Bases Generated by Sines and Cosines

Let us first recall some basic concepts. Let , , be a sequence in a Hilbert space , where is the set of positive integers. The sequence is called complete if its closed span equals [5, P. 154]. We say that is a Bessel sequence if for every element and that the sequence is a Riesz-Fischer sequence if the moment problem admits at least one solution whenever [5, P. 154].

A basis of Hilbert space is called a Riesz basis if it is obtained from an orthonormal basis by means of a bounded linear invertible operator. Two sequences of elements and from Hilbert space are called quadratically close if [5, P. 45]. A sequence of real or complex numbers is said to be separated if, for some positive number , whenever [5, P. 98]. A sequence is called -linearly independent if the equality is possible only for [5, P. 40].

Next we need the following lemmas to get our main results.

Lemma 1 ([5, P. 155]).  (i)The sequence is a Bessel sequence with bound if and only if the inequality holds for every finite systems of complex numbers.(i)The sequence is a Riesz-Fischer sequence with bound if and only if the inequality holds for every finite systems of complex numbers.

Lemma 2 ([6, P. 95]). Let two sequences and be quadratically close and let be an Riesz basis in . (i)If the sequence is -linearly independent, then is a Riesz basis in .(ii)If the sequence is complete in , then is -linearly independent.

Using the above lemmas, we obtain the following lemmas.

Lemma 3. If is a Riesz-Fischer sequence in with real and , then the sequences and are separated, respectively.

Proof. Let be a lower bound of . With , and , , it follows from (3) that On the other hand, Thus is separated by definition.
Similarly, setting , and , in (3), we also have that is separated.

Lemma 4. Let and , , be two sequences of nonnegative real numbers such that , , , and for all andThen is a Riesz basis in if and only if is a Riesz basis in .

Proof. Let , and , . Suppose that is a Riesz basis in . By Lemma 3, we find that the sequences and are separated, respectively. Using (6), we get that the sequences and are also separated, respectively. Therefore, we can assume and there is a positive such that and for all . Since we obtain thatthus two sequences and are quadratically close. In particular, is a Bessel sequence.
We can define a bounded linear operator on , as is a Riesz basis. From (9), we have that is a Hilbert-Schmidt operator. Furthermore, by Lemma 2, it is sufficient to prove that is a regular point of in order to prove that is a Riesz basis.
Assume that is not a regular point of . By the compactness of , is not one to one; i.e., there exists a sequence , not identically zero, such that Let such that for all . Then, the seriesis convergent uniformly on . Similarly, also converges uniformly on . Because of we can deduce thatWhen , the sequence on the right-hand side of (15) converges to in . This shows that is twice differentiable and for all . Due to we obtain thatwhere and . The functions and are meromorphic and not identically zero, respectively. Thus it has at most countably many zeros. If , by (12) and (17), we have that is in the closed linear span of . Owing to which is continuous about , we get that is in the closed linear span of for all . It follows that , , is in the closed linear span of , so is complete in . Hence the is dense in . Using the fact that is compact, we have that and is one to one; this contradicts the assumption.
Similarly, assume that is a Riesz basis in , then is also a Riesz basis in .

Now we shall introduce our main results.

Theorem 5. (i) The sequence is a orthonormal basis and Riesz basis in .
(ii) The sequence is a orthonormal basis and Riesz basis in .

Proof. (i) Suppose that satisfiesLet ; integration by parts yields that Thus Setting , we obtainCombining (18), (21), and (22), we obtain . Therefore, , , is complete in . The orthogonality of , , will be proved by establishing that and are orthogonal for all , using the fact that and , , are the orthogonal sequences in , respectively.
It follows from that and are orthogonal for all . Clearly, it is also a Riesz basis in . This completes the proof of (i).
(ii) Suppose , such that Let . By partial integration, Hence Setting , we obtain Similarly, using the method in (i), the desired results can be obtained. The proof is completed.

Theorem 6. Let , . (i)If , and , , where , then the sequences and are the Riesz basis in , respectively.(ii)If , and , , where , then the sequences and are the Riesz basis in , respectively.

Proof. (i) By the assumptions (i) of Theorem 6, we have Therefore Hence the result follows from Theorem 5 and Lemma 4.
The proof of the second part of this theorem follows in a similar manner.

3. Riesz Bases Associated with the Eigenfunctions of Strum-Liouville Problems

We consider the Strum-Liouville problemwhere and .

It is well known that (see, for example, [19]) the eigenvalues of problem (30) are and corresponding normalized eigenfunctions are

Theorem 7. Let , , where , , are the solutions of (30) satisfying the initial conditions Then, for , we have (i);(ii);(iii).

Proof. (i) Using integration by parts we obtain This clearly vanishes for . If , then , and we can use to obtain (ii) Again, integration by parts yields where . If , we may use to obtain If , then eigenfunction is a multiple of solution ; hence , and by the Wronskian identity, we get (iii) If , then the conclusion holds clearly. If , using the same procedure in (i) we have

Theorem 8. For every , the sequence , , is a Riesz basis in .

Proof. It is clear that the element is not in the closed linear span of , , as but by Theorem 7. Hence is -linearly independent. Similarly, the sequence is -linearly independent. It follows from Theorem 7 that, for all , Therefore, the two sequences and are mutually perpendicular. Hence, the sequence is -linearly independent.
By the expression of , we have Thus the sequence is quadratically close with the Riesz basis . The statement follows directly from Lemma 2.

4. Conclusion

Riesz bases have been extensively applied in signal denoising, feature extraction, robust signal processing, and also the corresponding inverse problems. This paper gives that and form a Riesz basis in , respectively. Based on this result, we find that a new sequence associated with eigenfunctions of Sturm-Liouville problem forms a Riesz basis in

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.

Acknowledgments

The work of the authors is supported by the National Nature Science Foundation of China (no. 11361039), the Inner Mongolia Natural Science Foundation (nos. 2017MS0124, 2017MS0125, and 2017MS(LH)0105), and the Inner Mongolia Autonomous Region University Scientific Research Project (nos. NJZY17045 and NJZC16165).