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 [1], Christensen [2], Sarsenbi with coauthors [35], San Antolin and Zalik [6], and Guo [7].

Let and be a complete biorthogonal conjugate system of functions from space.

By system coefficient space we denote the space of all the numeric sequences such that the series converges in . It is evident that coefficient space is complete under the norm , and a natural basis , , where is a Kronecker delta, forms a space basis.

A Banach coordinate space of numeric sequences is said to be solid if follows from and , (the inequality , as it is put by the precise definition, is not required here).

It is clear that space is solid if natural basis is an unconditional basis for . The latter follows from unconditional basicity for a system .

Theorem 1. Let and be a complete biorthogonal conjugate system of functions that is uniformly bounded: Let there be given coefficient spaces and which are both solid. Then and system form a Riesz basis.

Proof. We consider the series for a numeric sequence and show that series converges for almost all choices of signs, that is, series where is the Rademacher system and converges for almost all in metrics by variable (e.g., [8, Chapter 2]).
We use the results from [8] claiming that convergence of series for almost all choices of signs is equivalent to For the series considered , by Levi’s theorem we have meaning Convergence of the series for almost all choices of signs is shown.
Now take a fixed such that the series converges in space. By the solidity condition for coefficient space in , the series converges, too.
Thus for any numeric sequence the series converges in . Then the following equivalent inequalities are satisfied: This means that is Bessel system.
Besselian property for a system is proved in the same way. It is clear that Besselian property for both biorthogonal conjugate systems and implies the Riesz basicity for these systems.

Remark 2. Note that in Theorem 1 we can replace the coefficient space with and with .

2. Affine Riesz bases

Let function have a support . Using the representation , , for , we assume Besides, we suppose , . System of functions is called an affine system generated by a function . Here and elsewhere we assume Note that the classic example of an affine system of functions is the Haar wavelet generated by the function We enumerate the functions of Rademacher system We suppose that an affine system generator can be represented by Rademacher system In this case we have the following completeness criterion for a system . Let the function be analytic in the unit disk with coefficients from (11).

Theorem 3 (see [9]). A necessary and sufficient condition for an affine system to be complete in space is that analytic function is outer function.

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

Theorem 4. System that is biorthogonal conjugate to the affine system exists and is complete in space if .

Proof. Suppose that is, Then it follows from the results of [10] that where and is binary expansion, is the Haar function for , and , . The explicit representation (15) shows that is a Haar polynomial of degree . Hence it follows that the system is complete.

Now we can formulate the Riesz basicity test for affine system with form (11) generator, based on Theorem 1.

Theorem 5. Let analytic function have an absolutely convergent Taylor-series expansion and does not vanish in the closed unit disk . Then an affine system of functions   forms a Riesz basis.

Proof. By the conditions of the theorem, is outer function. By Theorem 3, an affine system is complete in space. By Theorem 4, biorthogonal conjugate system is complete, too.
Obviously, . From representation (15) we get We need to take into account that by Wiener theorem on absolutely convergent Taylor series we have Finally, from results of [11] it follows that and , 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.


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).