JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2014/860279 860279 Research Article Riesz Basicity for General Systems of Functions Sarsenbi A. M. 1 http://orcid.org/0000-0002-7874-9324 Terekhin P. A. 2 Banaś Józef 1 Department of Mathematics M. Auezov South Kazakhstan State University Tauke Han Avenue 5 Shymkent 160012 Kazakhstan ukgu.kz 2 Department of Mechanics and Mathematics N. G. Chernyshevsky Saratov State University Astrakhanskaya 83 Saratov 410012 Russia 2014 1282014 2014 11 05 2014 08 07 2014 12 8 2014 2014 Copyright © 2014 A. M. Sarsenbi and P. A. Terekhin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper we find the general conditions for a complete biorthogonal conjugate system to form a Riesz basis. We show that if a complete biorthogonal conjugate system is uniformly bounded and its coefficient space is solid, then the system forms a Riesz basis. We also construct affine Riesz bases as an application to the main result.

1. Main Result

The aim of this paper is to find the general conditions for a complete biorthogonal conjugate system to form a Riesz basis, following the results obtained by Bari , Christensen , Sarsenbi with coauthors , San Antolin and Zalik , and Guo .

Let {un(x)}n=1 and {vn(x)}n=1 be a complete biorthogonal conjugate system of functions from L2(0,1) space.

By system coefficient space {un(x)}n=1 we denote the space X(u) of all the numeric sequences a={an}n=1 such that the series n=1anun(x) converges in L2(0,1). It is evident that coefficient space X(u) is complete under the norm aX(u)=supnNk=1nakuk(x), and a natural basis ɛi={δij}j=1, iN, where δij is a Kronecker delta, forms a X(u) space basis.

A Banach coordinate space X of numeric sequences a={an}n=1 is said to be solid if bX follows from aX and |bn||an|, nN (the inequality bXaX, as it is put by the precise definition, is not required here).

It is clear that X(u) space is solid if natural basis is an unconditional basis for X(u). The latter follows from unconditional basicity for a system {un(x)}n=1.

Theorem 1.

Let {un(x)}n=1 and {vn(x)}n=1 be a complete biorthogonal conjugate system of functions that is uniformly bounded: (1)01|un(x)|2dxC,01|vn(x)|2dxC,nN. Let there be given coefficient spaces X(u) and X(v) which are both solid. Then {un(x)}n=1 and {vn(x)}n=1 system form a Riesz basis.

Proof.

We consider the series n=1anun(x) for a numeric sequence {an}n=1l2 and show that series converges for almost all choices of signs, that is, series (2)n=1anrn(t)un(x), where {rn(t)}n=1 is the Rademacher system and converges for almost all t[0,1] in L2 metrics by variable x (e.g., [8, Chapter 2]).

We use the results from  claiming that convergence of series n=1fn(x) for almost all choices of signs is equivalent to (3)(n=1|fn(x)|2)1/2L2(0,1). For the series considered n=1anun(x), by Levi’s theorem we have (4)01n=1|anun(x)|2dx=n=1|an|201|un(x)|2dxCn=1|an|2<, meaning (5)(n=1|anun(x)|2)1/2L2(0,1). Convergence of the series n=1anun(x) for almost all choices of signs is shown.

Now take a fixed t0[0,1] such that the series n=1anrn(t0)un(x) converges in L2(0,1) space. By the solidity condition for coefficient space X(u) in L2(0,1), the series n=1anun(x) converges, too.

Thus for any numeric sequence a={an}n=1l2 the series n=1anun(x) converges in L2(0,1). Then the following equivalent inequalities are satisfied: (6)n=1anun2Bn=1|an|2,{an}n=1l2,n=1|f,un|2Bf2,fL2(0,1). This means that {un(x)}n=1 is Bessel system.

Besselian property for a system {vn(x)}n=1 is proved in the same way. It is clear that Besselian property for both biorthogonal conjugate systems {un(x)}n=1 and {vn(x)}n=1 implies the Riesz basicity for these systems.

Remark 2.

Note that in Theorem 1 we can replace the coefficient space X(u) with X(u)l2 and X(v) with X(v)l2.

2. Affine Riesz bases

Let function u:RR have a support suppu[0,1]. Using the representation n=2k+j, k=0,1,, j=0,,2k-1 for nN, we assume (7)un(x)=uk,j(x)=2k/2u(2kx-j). Besides, we suppose u0(x)=1, x[0,1]. System of functions {un(x)}n=0 is called an affine system generated by a function u. Here and elsewhere we assume (8)uL2(0,1),01u(x)dx=0. Note that the classic example of an affine system of functions is the Haar wavelet {hn(x)}n=0 generated by the function (9)h(x)={1,x[0,12),-1,x[12,1),0,x[0,1). We enumerate the functions of Rademacher system {rk}k=0(10)rk=2-k/2j=02k-1hk,j,k=0,1,. We suppose that an affine system {un(x)}n=0 generator u can be represented by Rademacher system (11)u=k=0akrk,k=0|ak|2<. In this case we have the following completeness criterion for a system {un(x)}n=0. Let the function (12)U(z)=k=0akzk,|z|<1, be analytic in the unit disk with coefficients ak from (11).

Theorem 3 (see [<xref ref-type="bibr" rid="B9">9</xref>]).

A necessary and sufficient condition for an affine system {un(x)}n=0 to be complete in L2(0,1) space is that analytic function U(z) is outer function.

The following results are true for function u in the form (11).

Theorem 4.

System {vn(x)}n=0 that is biorthogonal conjugate to the affine system {un(x)}n=0 exists and is complete in L2(0,1) space if a00.

Proof.

Suppose (13)V(z)=1U(z)=k=0bkzk, that is, (14)a0b0=1,ν=0kaνbk-ν=0,k1. Then it follows from the results of  that (15)vn=v(α1,,αk)=ν=0k2-(k-ν)/2bk-νh(α1,,αν), where nN and n=2k+ν=1kαν2k-ν is binary expansion, h(α1,,αν)=hm is the Haar function for m=2ν+μ=1ναμ2ν-μ, and v0(x)=1, x[0,1]. The explicit representation (15) shows that vn is a Haar polynomial of degree n. Hence it follows that the system {vn(x)}n=0 is complete.

Now we can formulate the Riesz basicity test for affine system {un(x)}n=0 with form (11) generator, based on Theorem 1.

Theorem 5.

Let analytic function U(z) have an absolutely convergent Taylor-series expansion (16)k=0|ak|< and U(z) does not vanish in the closed unit disk (|z|1). Then an affine system of functions {un(x)}n=0  forms a Riesz basis.

Proof.

By the conditions of the theorem, U(z) is outer function. By Theorem 3, an affine system {un(x)}n=0 is complete in L2(0,1) space. By Theorem 4, biorthogonal conjugate system {vn(x)}n=0 is complete, too.

Obviously, unmax{1,u}. From representation (15) we get (17)vn2k=02-k|bk|2<,nN. We need to take into account that by Wiener theorem on absolutely convergent Taylor series we have (18)k=0|bk|<. Finally, from results of  it follows that X(u)l2=l2 and X(v)l2=l2, so all the conditions from Theorem 1 including the Remark are satisfied.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

A. M. Sarsenbi was supported by the Ministry of Education and Science of the Republic of Kazakhstan (0264/GF, 0753/GF). P. A. Terekhin was supported by the Russian Foundation for Basic Research (Grant no. 13-01-00102) and the President of the Russian Federation (Grant no. MD-1354.2013.1). The results of Section 2 were obtained within the framework of the state task of Russian Ministry of Education and Science (Project 1.1520.2014K).

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