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Construction and Stability of Riesz Bases.

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 {[[lambda].sub.n]} for which [exp(i[[lambda].sub.n]x]} is a Riesz basis in [L.sup.2] [-[pi], [pi]] [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 [L.sup.2][0, [pi]] 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 {1} [union] {cos(2nx)} [union] {sin(2nx)} and |sin((2n - 1)x)} [union] {cos((2n - 1)x)} and study the stability of them. On the other, we consider the problem of finding a new sequence associated with eigenfunctions of Sturm-Liouville problem

-y" + qy = [lambda]y, on [0, [pi]];

y (0) = y([pi]) = 0, (1)

such that it forms a Riesz basis.

2. Riesz Bases Generated by Sines and Cosines

Let us first recall some basic concepts. Let {[f.sub.n]}, n [member of] N, be a sequence in a Hilbert space H, where N is the set of positive integers. The sequence is called complete if its closed span equals H [5, P. 154]. We say that {[f.sub.n]} is a Bessel sequence if [[summation].sup.[infinity].sub.n=1] [[absolute value of f, [f.sub.n]>].sup.2], < [infinity] to for every element f [member of] H and that the sequence {[f.sub.n]} is a Riesz-Fischer sequence if the moment problem <f, [f.sub.n]> = [c.sub.n] (n = 1, 2, 3, ...) admits at least one solution f [member of] H whenever {[c.sub.n]} [member of] [l.sup.2] [5, P. 154].

A basis {[f.sub.n]} 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 {[f.sub.n]} and {[g.sub.n]} from Hilbert space H are called quadratically close if [[summation].sup.[infinity].sub.n=1] [[parallel] [f.sub.n] - [g.sub.n][parallel].sup.2] < [infinity] [5, P- 45]. A sequence {[[lambda].sub.n]} of real or complex numbers is said to be separated if, for some positive number [epsilon], [absolute value of [[lambda].sub.n] - [[lambda].sub.m]] [greater than or equal to] [member of] whenever n [not equal to] m [5, P. 98]. A sequence {[f.sub.n]} is called [omega]-linearly independent if the equality [[summation].sup.[infinity].sub.n=1] [c.sub.n][f.sub.n] = 0 is possible only for [c.sub.n] = 0 (n [greater than or equal to] 1) [5, P. 40].

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

Lemma 1 ([5, P. 155]).

(i) The sequence {[f.sub.n]} is a Bessel sequence with bound M if and only if the inequality

[[parallel][summation over (n)][c.sub.n] [f.sub.n][parallel].sup.2] [less than or equal to] M [[absolute value of [c.sub.n]].sup.2] (2)

holds for every finite systems {[c.sub.n]} of complex numbers.

(i) The sequence {[f.sub.n]} is a Riesz-Fischer sequence with bound m if and only if the inequality

m[summation over (n)][[absolute value of [c.sub.n]].sup.2] [less than or equal to] [[parallel][summation over (n)][c.sub.n] [f.sub.n][parallel].sup.2] (3)

holds for every finite systems {[c.sub.n]} of complex numbers.

Lemma 2 ([6, P. 95]). Let two sequences {[f.sub.n]} and {[g.sub.n]} be quadratically close and let {[f.sub.n]} be an Riesz basis in H.

(i) If the sequence {[g.sub.n]} is [omega]-linearly independent, then {[g.sub.n]} is a Riesz basis in H.

(ii) If the sequence {[f.sub.n]} is complete in H, then {[f.sub.n]} is [omega]-linearly independent.

Using the above lemmas, we obtain the following lemmas.

Lemma 3. If {cos([[lambda].sub.n]x)} [union] {sin([[??].sub.n]x)} is a Riesz-Fischer sequence in [L.sup.2][0, [pi]] with real [[lambda].sub.n] and [[??].sub.n], then the sequences {[[lambda].sub.n]} and {[[??].sub.n]} are separated, respectively.

Proof. Let m be a lower bound of {cos([[lambda].sub.n]v)} [union] {sin([[??].sub.n]x)}. With [c.sub.m] = 1, [c.sub.k] = -1 and [c.sub.n] = 0, [d.sub.n] = 0, it follows from (3) that

[square root of 2]m [less than or equal to] [parallel]cos ([[lambda].sub.m]x) - cos ([[lambda].sub.k](x)[parallel]. (4)

On the other hand,

[mathematical expression not reproducible]. (5)

Thus {[[lambda].sub.n]} is separated by definition.

Similarly, setting [d.sub.m] = 1, [d.sub.k] = -1 and [c.sub.n] = 0, [d.sub.n] = 0 in (3), we also have that {[[??].sub.n]} is separated.

Lemma 4. Let {[[lambda].sub.n]} [union] {[[??].sub.n]} and {[[mu].sub.n]} [union] {[[??].sub.n]}, n [member of] N, be two sequences of nonnegative real numbers such that [[lambda].sub.n] [not equal to] [[lambda].sub.k], [mathematical expression not reproducible] and

[mathematical expression not reproducible]. (6)

Then {cos([[lambda].sub.n]x)} [union] {sin([[lambda].sub.n]x)} is a Riesz basis in [L.sup.2][0, [pi]] if and only if {cos([[mu].sub.n]x)} [union] {sin([[??].sub.n])} is a Riesz basis in [L.sup.2][0, [pi]].

Proof. Let [mathematical expression not reproducible]. Suppose that {[f.sub.n]} [union] {[[??].sub.n]} is a Riesz basis in [L.sup.2][0, [pi]]. By Lemma 3, we find that the sequences {[[lambda].sub.n]} and {[[??].sub.n]} are separated, respectively. Using (6), we get that the sequences {[[mu].sub.n]} and {[[??].sub.n]} are also separated, respectively. Therefore, we can assume

[mathematical expression not reproducible]. (7)

and there is a positive e such that [[mu].sub.n] [greater than or equal to] n[epsilon] and [[??].sub.n] [greater than or equal to] ne for all n [member of] N. Since

[mathematical expression not reproducible], (8)

we obtain that

[mathematical expression not reproducible]; (8)

thus two sequences {[f.sub.n]} [union] {[[??].sub.n]} and {[g.sub.n]} [union] {[g.sub.n]} are quadratically close. In particular, {[f.sub.n] - [g.sub.n]} [union] {[[??].sub.n] - [[??].sub.n]} is a Bessel sequence. We can define a bounded linear operator

[mathematical expression not reproducible] (10)

on [L.sup.2][0, [pi]], as {[f.sub.n]} [union] {[[??].sub.n]} is a Riesz basis. From (9), we have that T is a Hilbert-Schmidt operator. Furthermore, by Lemma 2, it is sufficient to prove that 1 is a regular point of T in order to prove that {[g.sub.n]} [union] {[[??].sub.n]} is a Riesz basis.

Assume that 1 is not a regular point of T. By the compactness of T, I - T is not one to one; i.e., there exists a sequence {[c.sub.n]} [union] {[d.sub.n]} [member of] [l.sup.2], not identically zero, such that

[[infinity].summation over (n=1)] [c.sub.n][g.sub.n] + [[infinity].summation over (n=1)] [d.sub.n][[??].sub.n] = 0. (11)

Let [lambda] [member of] C such that [lambda] [not equal to] [+ or -][[mu].sub.n], [+ or -][[??].sub.n] for all n [member of] N. Then, the series

[mathematical expression not reproducible] (12)

is convergent uniformly on [0, [pi]]. Similarly,

[mathematical expression not reproducible] (13)

also converges uniformly on [0, n]. Because of

[mathematical expression not reproducible], (14)

we can deduce that

[mathematical expression not reproducible]. (15)

When m [right arrow] [infinity], the sequence on the right-hand side of (15) converges to--[[lambda].sup.2]g(x) in [L.sup.2][0, [pi]]. This shows that g(x) is twice differentiable and g"(x) = -[[lambda].sup.2]g(x) for all v [member of] [0, [pi]]. Due to

[mathematical expression not reproducible], (16)

we obtain that

g(x) = [union] ([lambda]) cos ([lambda]x) + v([lambda]) sin ([lambda]x), (17)

where u([lambda]) = g(0) and v([lambda]) = [[lambda].sup.-1]g'(0). The functions u([lambda]) and v([lambda]) are meromorphic and not identically zero, respectively. Thus it has at most countably many zeros. If u([lambda])v([lambda]) [not equal to] 0, by (12) and (17), we have that {cos([lambda]v)} [union] {sin([lambda]v)} is in the closed linear span of {cos([[mu].sub.n]x)} [union] {sin([[??].sub.n])}. Owing to {cos([lambda]v)} [union] {sin([lambda]v)} which is continuous about (x, [lambda]), we get that {cos([lambda]v)} [union] {sin([lambda]v)} is in the closed linear span of {cos([[mu].sub.n]x)} [union] {sin([[??].sub.n]x)} for all [lambda] [member of] C. It follows that {sin(nx)}, n [member of] N, is in the closed linear span of {[g.sub.n](x)} [union] {[[??].sub.n] (v)}, so {[g.sub.n](x)} [union] {[[??].sub.n](x)} is complete in [L.sup.2][0, [pi]]. Hence the R(I - T) is dense in [L.sup.2][0, [pi]]. Using the fact that T is compact, we have that R(I-T) = [L.sup.2][0, [pi]] and I-T is one to one; this contradicts the assumption.

Similarly, assume that {[g.sub.n](x)} [union] {[[??].sub.n](x)} is a Riesz basis in [L.sup.2][0, [pi]], then {[f.sub.n](x)} [union] {[[?].sub.n](x)} is also a Riesz basis in [L.sup.2][0,7r].

Now we shall introduce our main results.

Theorem 5. (i) The sequence {1} [union] {sin(2nx)} [union] {cos(2nv)} is a orthonormal basis and Riesz basis in [L.sup.2][0, [pi]],

(ii) The sequence {sin((2n - 1)v)} [union] {cos((2n - 1)x)} is a orthonormal basis and Riesz basis in [L.sup.2][0, [pi]],

Proof, (i) Suppose that f(x) [member of] [L.sup.2][0, [pi]] satisfies

[mathematical expression not reproducible]. (10)

Let F(x) = [[integral].sup.x.sub.0] f(t)dt; integration by parts yields that

[mathematical expression not reproducible]. (19)

Thus

[[integral].sup.x.sub.0] F (x) sin (2nx) dx = 0. (20)

Setting t = 2x - [pi], we obtain

[mathematical expression not reproducible]. (21)

[mathematical expression not reproducible]. (22)

Combining (18), (21), and (22), we obtain f(x) [equivalent to] 0. Therefore, {1} [union] {cos(2nv)} [union] {sin(2nx)}, n [member of] N, is complete in [L.sup.2][0, [pi]]. The orthogonality of {1} [union] {cos(2nx)} [union] {sin(2nx)}, n [member of] N, will be proved by establishing that {cos(2nv)} and {sin(2mv)} are orthogonal for all m, n [member of] N, using the fact that {1} [union] {cos(2nv)} and {1} [union] {sin(2nx)}, n [member of] N, are the orthogonal sequences in [L.sup.2][0, [pi]], respectively.

It follows from

<cos (2nx), sin (2mx)> = [[integral].sup.[pi].sub.0] cos (2nx) sin (2mx) dx

= 0 (23)

that cos(2nx) and sin(2mv) are orthogonal for all m, n [member of] N. Clearly, it is also a Riesz basis in [L.sup.2][0, [pi]]. This completes the proof of (i).

(ii) Suppose f(x) [member of] [L.sup.2][0, [pi]], such that

[mathematical expression not reproducible]. (24)

Let F(x) = [[lambda].sub.n] f(t)dt. By partial integration,

[mathematical expression not reproducible]. (25)

Hence

[integral].sup.[pi].sub.0] F(x) cos ((2n - 1) x) dx = 0. (26)

Setting t = 2x - [pi], we obtain

[mathematical expression not reproducible]. (27)

Similarly, using the method in (i), the desired results can be obtained. The proof is completed.

Theorem 6. Let [[delta].sub.n] [member of] [l.sup.2], n [member of] N.

(i) If [mathematical expression not reproducible], where m [not equal to] k, then the sequences {1} [union] {cos([[lambda].sub.n]x)} [union] {sin([[lambda].sub.n]x)} and {1} [union] {cos([[lambda].sub.n]x)} [union] {sin([[??].sub.n]x)} are the Riesz basis in [L.sup.2][0, [pi]], respectively.

(ii) If [mathematical expression not reproducible], where m [not equal to] k, then the sequences [mathematical expression not reproducible] are the Riesz basis in [L.sup.2][0, [pi]], respectively.

Proof. (i) By the assumptions (i) of Theorem 6, we have

[mathematical expression not reproducible]. (28)

Therefore

[[infinity].summation over (n=1)] [([[lambda].sub.n] - 2n).sup.2] + [[infinity].summation over (n=1)] [([[??].sub.n] - 2n).sup.2] < [infinity]. (29)

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 problem

-y" + q(x) y = [[lambda].sub.n] y, x [member of] [0. [pi]],

Y(0) = y([pi]) = 0, (30)

where [lambda] [member of] C and q(x) [member of] [L.sup.2]([0, [pi]], R).

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

[[lambda].sub.n] = n + O(1/n) (31)

and corresponding normalized eigenfunctions are

[y.sub.n] (x) = [square root of 2/[[pi] sin (nx) + O(1/n) (32)

Theorem 7. Let [u.sub.n](x, q) = [g.sub.1](x, [[lambda].sub.n]) [g.su.2] (x, [[lambda].sub.n]), n [member of] N, where [g.sub.i](x, [[lambda].sub.n]), i = 1, 2, are the solutions of (30) satisfying the initial conditions

[g.sub.1] (0, [lambda], q) = [g'.sub.2] (0, [lambda], q) = 1;

[g'.sub.1] (0, [lambda], q) = [g.sub.2] (0, [lambda], q) = 0. (33)

Then, for m, n [member of] N, we have

(i) <[y.sup.2.sub.m], (d/dx) [y.sup.2.sub.n]> = 0;

(ii) <[u.sub.m], (d/dx) [y.sup.2.sub.n]) = ([pi]/2)[[delta].sub.mn];

(iii) <[u.sub.m], (d/dx)[u.sub.n]) = 0.

Proof. (i) Using integration by parts we obtain

[mathematical expression not reproducible]. (34)

This clearly vanishes for m = n. If m [not equal to] n, then [[lambda].sub.n] [not equal to] [[lambda].sub.m], and we can use

[[y.sub.m], [y.sub.n]]' = ([[lambda].sub.m] - [[lambda].sub.n]) [y.sub.m] [y.sub.n] (35)

to obtain

[mathematical expression not reproducible]. (36)

(ii) Again, integration by parts yields

[mathematical expression not reproducible]. (37)

where [g.sub.i] = [g.sub.i]x, [[lambda].sub.m]) (i = 1, 2). m [not equal to] n, we may use

[[g.sub.i], [y.sub.n]]' = ([[lambda].sub.m] - [[lambda].sub.n]) [g.sub.i][y.sub.n] i = 1,2 (38)

to obtain

[mathematical expression not reproducible]. (39)

If m = n, then eigenfunction ym is a multiple of solution [g.sub.2];

hence [[g.sub.2], [y.sub.m]] = 0, and by the Wronskian identity, we get

[mathematical expression not reproducible] (40)

(iii) If m = n, then the conclusion holds clearly. If m = n, using the same procedure in (i) we have

[mathematical expression not reproducible] (41)

Theorem 8. For every q(x) [member of] [L.sup.2]([0, [pi]], R), the sequence {1} [union] {1 - [pi][y.sup.2.sub.n]} [union] {([pi]/2n)(d/dx)[y.sup.2.sub.n]}, n [member of] N, is a Riesz basis in [L.sup.2][0, [pi]].

Proof. It is clear that the element 1 - [pi][y.sup.2.sub.n] is not in the closed linear span of {1, 1 - [pi][y.sup.2.sub.m]}, n [not equal to] m, as

[mathematical expression not reproducible], (42)

but

[mathematical expression not reproducible], (43)

by Theorem 7. Hence {1} [union] {1 - [pi][y.sup.2.sub.n]} is [omega]-linearly independent. Similarly, the sequence {([pi]/2n)(d/dx)[y.sup.2.sub.n]} is [omega]-linearly independent. It follows from Theorem 7 that, for all m, n [member of] N,

[mathematical expression not reproducible]. (44)

Therefore, the two sequences {1} [union] {1 - [pi][y.sup.2.sub.n]} and {([pi]/2n)(d/dx)[y.sup.2.sub.n]} are mutually perpendicular. Hence, the sequence {1} [union] {1 - [pi][y.sup.2.sub.n]} [union] {([pi]/2n)(d/dx)[y.sup.2.sub.n]} is [omega]-linearly independent.

By the expression of [y.sub.n](x), we have

[mathematical expression not reproducible]. (45)

Thus the sequence {1} [union] {1 - [pi][y.sup.2.sub.n]} [union] {([pi]/2n)(d/dx)[y.sup.2.sub.n]} is quadratically close with the Riesz basis {1} [union] {cos(2nx)} [union] {sin(2nx)}. 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 {1} U {cos(2nx)} [union] {sin(2nx)} and {sin((2n - 1)x)} [union] {cos((2n - 1)x)} form a Riesz basis in [L.sup.2][0, [pi]], respectively. Based on this result, we find that a new sequence associated with eigenfunctions of Sturm-Liouville problem forms a Riesz basis in [L.sup.2][0, [pi]].

https://doi.org/10.1155/2018/5063847

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).

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Yulin Bai, (1,2) Wanyi Wang, (1,3) Guixia Wang, (2) and Suqin Ge (iD) (4)

(1) School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

(2) College of Mathematics Science, Inner Mongolia Normal University, Hohhot 010022, China

(3) Inner Mongolia Agricultural University, Hohhot 010018, China

(4) School of Science, Inner Mongolia University of Science and Technology, Baotou 014010, China

Correspondence should be addressed to Suqin Ge; 15647280518@163.com

Received 13 July 2018; Accepted 18 August 2018; Published 2 September 2018

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Author:Bai, Yulin; Wang, Wanyi; Wang, Guixia; Ge, Suqin
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Date:Jan 1, 2018
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