Riesz system

A concept in the theory of orthogonal systems (cf. Orthonormal system). Let a complete system of functions $\{\psi_n\}$ be fixed in the space $L_2=L_2(a,b)$. It is considered normalized, or, more generally, almost normalized, i.e. there are numbers $m>0$ and $M>0$ for which $m\leq\|\psi_n\|\leq M$ for all $n\in\mathbf N$. Weakening the requirement concerning the orthogonality of the system $\{\psi_n\}$ one assumes that there exists a complete system of functions $\{g_n\}$ in $L_2$ such that $(\psi_n,g_n)=1$, $(\psi_n,g_m)=0$ for all $n\neq m$. In particular, when the system $\{\psi_n\}$ is orthonormal, $g_n=\psi_n$ for all $n\in\mathbf N$. If a series

$$\sum_{n=1}^\infty a_n\psi_n$$

converges to a function $f$ in $L_2$, then $a_n=(f,g_n)$ for all $n\in\mathbf N$. Thus it makes sense to call the number $a_n=(f,g_n)$ the $n$-th Fourier coefficient of the function $f$ with respect to the system $\{\psi_n\}$. In the proofs of a number of theorems in the theory of orthogonal series, the Bessel inequality and the Riesz–Fischer theorem are of great importance. In the general case these theorems are not valid, therefore one has to single out the special class of Riesz systems, i.e. systems $\{\psi_n\}$ satisfying

1) for any function $f$ the series of the squares of the Fourier coefficients is absolutely convergent, i.e.

$$\sum_{n=1}^\infty|(f,g_n)|^2<+\infty;$$

2) for any sequence of numbers $\{a_n\}\in l_2$ there exists a function $f$ for which the $a_n$ are its Fourier coefficients with respect to the system $\{\psi_n\}$, that is, $a_n=(f,g_n)$ for all $n\in\mathbf N$.

The first requirement on the system $\{\psi_n\}$ replaces the Bessel inequality, the second the Riesz–Fischer theorem. N.K. Bari has proved (see [2]) that a system $\{\psi_n\}$ is a Riesz system if and only if there exists a continuous linear operator $A$, invertible in $L_2$, such that the system of functions $\{A\psi_n\}$ is complete and orthonormal. Therefore, a Riesz system is also called a Riesz basis, equivalent to an orthonormal basis. Bari has indicated a convenient criterion for being a Riesz system. A complete system of functions $\{\psi_n\}$ in $L_2$ is a Riesz system if and only if the Gram matrix $\|(\psi_n,\psi_m)\|$ determines a continuous invertible linear operator in $l_2$. Under an arbitrary permutation of the elements of a Riesz system one obtains again a Riesz system. Conversely, if a basis in $L_2$ is still a basis after any permutation of its elements, then by normalizing it one obtains a Riesz system. A natural generalization of a Riesz system is obtained by replacing $L_2$ by the linear closure of a system $\{\psi_n\}$ with respect to the norm of the Hilbert space from which the elements $\psi_n$ are taken (see [4]).

References

[1]	N.K. Bari, "Sur les bases dans l'espace de Hilbert" Dokl. Akad. Nauk SSSR , 54 (1946) pp. 379–382
[2]	N.K. Bari, "Biorthogonal systems and bases in Hilbert space" Uchen. Zap. Moskov. Gos. Univ. , 148 : 4 (1951) pp. 69–107 (In Russian)
[3]	I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Introduction to the theory of linear nonselfadjoint operators" , Transl. Math. Monogr. , 18 , Amer. Math. Soc. (1969) (Translated from Russian)
[4]	V.F. Gaposhkin, "Lacunary series and independent functions" Russian Math. Surveys , 21 : 6 (1966) pp. 1–82 Uspekhi Mat. Nauk , 21 : 6 (1966) pp. 3–82