MATHEMATICS

Yu. A. KAZ’MIN

ON COMPLETE SYSTEMS AND BASES IN \(L_2\)

(Presented by Academician S. L. Sobolev on 22 XI 1956)

The article considers questions connected with the problem of completeness in \(L_2\) of systems of functions that are “close” in a certain sense (\(L_2\), as usual, is the Hilbert space of functions square-summable on the segment \([a,b]\)). First, a theorem of a general character is proved, which is subsequently used as a criterion in solving the question of the completeness of “close” systems.

§ 1. Let the set \(G \subset L_2\) be everywhere dense in \(L_2\). We shall assume that \(G\) forms a space of type (B), and moreover that from
\[ \lim_{n\to\infty}\|f-f_n\|_G=0 \]
it always follows that
\[ \lim_{n\to\infty}\|f-f_n\|_{L_2}=0 \]
(the norm in \(G\) in the case \(G\ne L_2\) is distinct from the norm in \(L_2\)).

The following almost obvious assertion holds:

Theorem 1. Let a bounded linear operator \(T\) be defined on \(G\), mapping \(G\) into itself and possessing a unique inverse \(T^{-1}\). If \(\{g_n\}\), \(n=1,2,\ldots\), is a complete system or a basis in \(G\), then the system of functions
\[ f_n=Tg_n,\qquad n=1,2,\ldots, \]
also forms, respectively, a complete system or a basis in \(G\).

Let us note that if \(\{g_n\}\) is a basis in \(G\), then the spaces of coefficients of expansions both with respect to the functions \(\{g_n\}\) and with respect to the functions \(\{f_n=Tg_n\}\) coincide. In particular, if \(G=L_2\), then a basis \(\{g_n\}\) in \(L_2\) also generates in \(L_2\) a basis \(\{f_n=Tg_n\}\) with the same space of expansion coefficients.

Let us also note that from the very definition of the set \(G\) it follows that a system of functions \(\{f_n\}\) complete in \(G\) is complete in \(L_2\).

§ 2. \(G=L_2\). A system of functions \(\{g_n\}\subset L_2\) is called minimal if none of them belongs to the closed linear span of the others. It is easy to show that the system \(\{g_n\}\) is minimal if and only if from
\[ \lim_{n\to\infty}\left|c_1^{(n)}g_1+c_2^{(n)}g_2+\cdots+c_n^{(n)}g_n\right|=0 \]
it follows that
\[ \lim_{n\to\infty}c_i^{(n)}=0,\qquad i=1,2,\ldots. \]
S. S. Levin proved \((^1)\) that if the system \(\{g_n\}\) is minimal, then there exists a sequence of functions \(\{h_n\}\) forming with \(\{g_n\}\) a biorthogonal system, i.e.
\[ (g_n,h_k)=\int_a^b g_nh_k\,dx=\delta_{nk},\qquad \delta_{nk}= \begin{cases} 1 & \text{if } n=k,\\ 0 & \text{if } n\ne k. \end{cases} \]

Let \(\{g_n\}\) be a complete minimal system of functions, and let the sequence of functions \(\{R_n\}\), \(R_n\in L_2\), \(n=1,2,\ldots\), be such that the double series converges ...

series

\[ \sum_{i=1}^{\infty}\sum_{k=1}^{\infty} (R_i,R_k)(h_i,h_k), \tag{1} \]

\((g_n,h_k)=\delta_{nk}\); then the system of functions \(\{f_n=g_n+\lambda R_n\}\) is complete in \(L_2\) for all regular values \(\lambda\) of the integral equation

\[ f(x)=g(x)+\lambda\int_a^b K(x,s)\,g(s)\,ds \tag{2} \]

with kernel

\[ K(x,s)=\sum_{i=1}^{\infty} R_i(x)h_i(s). \tag{3} \]

Indeed, in this case

\[ \int_a^b\int_a^b [K(x,s)]^2\,dx\,ds = \sum_{i=1}^{\infty}\sum_{k=1}^{\infty}(R_i,R_k)(h_i,h_k)<\infty, \]

and therefore the integral equation (2) defines a linear operator \(T_\lambda=J+\lambda K\), mapping \(L_2\) into itself. It remains to note that \(f_n=T_\lambda g_n\), and, if \(\lambda\) is a regular value of \(T_\lambda\), to apply Theorem 1.

In particular, the system of functions \(\{f_n=g_n+R_n\}\) is complete in \(L_2\) if

\[ \sum_{i=1}^{\infty}\sum_{k=1}^{\infty}(R_i,R_k)(h_i,h_k)<1. \]

A system of functions \(\{f_n\}\), \(n=1,2,\ldots\), is called linearly independent in \(L_2\) (l.i. in \(L_2\)) if from \(\sum_{n=1}^{\infty} c_n f_n=0\) it always follows that \(c_n=0\), \(n=1,2,\ldots\). Every minimal system is l.i. in \(L_2\); the converse, generally speaking, is false.

Lemma 1. If \(\{g_n\}\) is a basis (or a complete minimal system) in \(L_2\), and the functions \(\{f_n=g_n+\lambda R_n\}\) are l.i. in \(L_2\) (respectively, also form a minimal system) and are such that the double series (1) converges, then \(\lambda\) is a regular value of the integral equation (2) with kernel (8).

Otherwise the integral equation

\[ (J+\lambda K)g=0 \tag{4} \]

has in \(L_2\) a nontrivial solution \(g\), \(\|g\|\ne0\). Representing \(g\) in the form

\[ g=\sum_{n=1}^{\infty} c_n g_n \]

(respectively,

\[ g=\lim_{n\to\infty}\left[c_1^{(n)}g_1+c_2^{(n)}g_2+\cdots+c_n^{(n)}g_n\right] \]

) and substituting this expansion into (4), we find that the system \(\{f_n=g_n+\lambda R_n\}\) is not l.i. (respectively, minimal) in \(L_2\), which contradicts what was assumed above.

From Lemma 1 it follows:

Theorem 2. Let the system of functions \(\{f_n\}\) be l.i. in \(L_2\) (or minimal), and let \(\{g_n\}\), \(n=1,2,\ldots\), be a basis (respectively, a complete minimal system) in \(L_2\); if, moreover, the double series

\[ \sum_{i=1}^{\infty}\sum_{k=1}^{\infty}(R_i,R_k)(h_i,h_k), \]

converges, where \(R_n=f_n-g_n\), \((g_n,h_k)=\delta_{nk}\), then the system \(\{f_n\}\) is a basis (respectively, complete) in \(L_2\). The spaces of coefficients of expansions with respect to the systems \(\{g_n\}\) and \(\{f_n\}\) coincide.

The systems of functions \(\{g_n\}\) and \(\{f_n\}\) are called quadratically close if the series
\[ \sum_{n=1}^{\infty}\|R_n\|^2,\qquad R_n=f_n-g_n \]
converges \(((1), p. 51)\), and \(B\)-close if
\[ \sum_{i,k=1}^{\infty}\left|(R_i;R_k)\right|<\infty . \]

From what has been set forth, the following theorems on complete systems, obtained earlier by another method by N. K. Bari, follow as consequences.

\(1^\circ\). If \(\{g_n\}\) is a complete orthonormal system, and the system \(\{f_n\}\) is such that
\[ \sum_{n=1}^{\infty}\|R_n\|^2<1,\qquad R_n=f_n-g_n, \]
then \(\{f_n\}\) is a basis in \(L_2\) with a space of expansion coefficients coinciding with \(l_2\) \(((2), p. 68)\).

\(2^\circ\). If \(\{g_n\}\) is a complete orthonormal system, and the system of functions \(\{f_n\}\), quadratically close to \(\{g_n\}\), is l.i. in \(L_2\), then \(\{f_n\}\) is a basis in \(L_2\) with a space of expansion coefficients coinciding with \(l_2\) \(((2), p. 72)\)*.

We shall call the system \(\{g_n\}\) strictly minimal if there exists a constant \(\delta>0\) such that the distance from any \(g_i\) to the closed linear span of the remaining functions \(\{g_n\}\), \(n\ne i\), is greater than or equal to \(\delta\) for all \(i=1,2,\ldots\). It is said that \(\{g_n\}, \{h_n\}\) form a regular biorthogonal system if \(\{g_n\}\) is a complete system and, in addition, the conditions
\[ \sup_n\|g_n\|<\infty,\qquad \sup_n\|h_n\|<\infty \]
are simultaneously satisfied. A biorthogonal system \(\{g_n\}, \{h_n\}\) is regular if and only if \(\{g_n\}\) is a complete bounded (i.e.
\[ \sup_n\|g_n\|<\infty \]
), strictly minimal system \((^4)\).

Using the results cited and Theorem 2, it is easy to obtain:

Theorem 3. If \(\{g_n\}\) is a complete, bounded, strictly minimal system, and \(\{f_n\}\) is a minimal system \(B\)-close to \(\{g_n\}\), then \(\{f_n\}\) is complete in \(L_2\).

Corollary. Two \(B\)-close, strictly minimal systems are complete or incomplete simultaneously.

Theorem 4. If \(\{g_n\}\) is a complete, bounded, strictly minimal system and
\[ h=\sup_n\|h_n\|, \]
where
\[ (g_n,h_k)=\delta_{nk}, \]
and the system of functions \(\{f_n\}\) is such that
\[ \sum_{i,k=1}^{\infty}|(R_i;R_k)|<\frac{1}{h^2},\qquad R_n=f_n-g_n, \]
then \(\{f_n\}\) is complete in \(L_2\).

A basis is a complete strictly minimal system. Therefore, if \(\{g_n\}\) is a basis in \(L_2\) and
\[ \sup_n\|g_n\|<\infty, \]
and the system of functions \(\{f_n\}\) is l.i. in \(L_2\) and \(B\)-close to \(\{g_n\}\), then \(\{f_n\}\) is also a basis in \(L_2\), with the same space of expansion coefficients as that of \(\{g_n\}\).

It is easy to prove that when the systems of functions \(\{g_n\}\) and \(\{h_n\}\) form a complete biorthogonal sequence, i.e. both systems \(\{g_n\}\) and \(\{h_n\}\) are complete in \(L_2\), in all the cases considered above there exists a system \(\{F_n\}\) which forms, together with \(\{f_n\}\), also a complete biorthogonal system. Indeed, in these cases the operator \(T\), \(f_n=Tg_n\), has a unique inverse \(T^{-1}\). It remains to note that
\[ F_n=(T^{-1})^*h_n, \]
where \((T^{-1})^*\) is the operator adjoint to \(T^{-1}\).

* In a later work, result \(2^\circ\) was extended by N. K. Bari to the so-called Riesz bases \((^3)\); however, it was not possible to obtain these generalizations from our theorems.

§ 3. Let us now consider the case where \(G \ne L_2\). Let \(\{g_n\}\), \(n=1,2,\ldots\), be a complete orthonormal system of functions. Put \(G=A_g\), where the set \(A_g \subset L_2\) is such that if \(f \in A_g\) and

\[ f=\sum_{k=1}^{\infty} c_k g_k, \]

then the series

\[ \sum_{k=1}^{\infty} |c_k| \]

converges. It is obvious that \(A_g\), everywhere dense in \(L_2\), is a space of type \((B)\) with norm, for \(f \in A_g\),

\[ \|f\|_{A_g}=\sum_{k=1}^{\infty} |c_k|, \]

and satisfies all the conditions imposed on the set \(G\) in § 1.

Lemma 2. Let a system of functions \(\{R_n\}\subset A_g\), \(n=1,2,\ldots\), be such that

\[ \sup_n |(R_n,g_k)|=\alpha_k, \]

and the series

\[ \sum_{k=1}^{\infty}\alpha_k \]

converges.

The integral equation

\[ f(x)=g(x)+\lambda\int_a^b K(x,s)g(s)\,ds \]

with kernel

\[ K(x,s)=\sum_{i=1}^{\infty} R_i(x)g_i(s) \]

defines on \(A_g\) a linear operator

\[ T_\lambda=I+\lambda K, \]

which maps \(A_g\) into itself and has in \(A_g\) a unique inverse for all values of \(\lambda\), except those which coincide with the zeros of the entire function

\[ \Delta(\lambda)= \left| \begin{array}{cccc} 1+\lambda a_{11} & \lambda a_{12} & \cdots \\ \lambda a_{21} & 1+\lambda a_{22} & \cdots \\ \cdots & \cdots & \cdots \end{array} \right|, \]

where \(a_{ik}=(g_i,R_k)\).

(It can be shown, relying on Koch’s criterion, that \(\Delta(\lambda)\) is an absolutely convergent determinant\(^5\).)

Without particular difficulty one proves the following modifications of Theorem 1.

Theorem 5. If \(\{g_n\}\) is a complete orthonormal system in \(L_2\), and the system \(\{f_n\}\) is such that

\[ \sup_n |(R_n,g_k)|=\alpha_k,\qquad R_n=f_n-g_n,\qquad \sum_{k=1}^{\infty}\alpha_k<1, \]

then \(\{f_n\}\) is a basis in \(A_g\) with the space of coefficients of expansions coinciding with \(l_1\).

Theorem 6. Let \(\{g_n\}\) be a complete orthonormal system in \(L_2\). Then, in order that the system of functions \(\{f_n\}\),

\[ \sup_n |(R_n,g_k)|=\alpha_k,\qquad R_n=f_n-g_n,\qquad \sum \alpha_k<\infty, \]

be a basis in \(A_g\) with the space of coefficients of expansions coinciding with \(l_1\), it is necessary and sufficient that from

\[ \sum_{n=1}^{\infty} c_n g_n=0,\qquad \sum_{k=1}^{\infty}|c_k|<\infty \]

it always follow that \(c_n=0\) for all \(n\).

In conclusion, we note that a system of functions \(\{f_n\}\) complete in \(A_g\) is complete in \(L_2\).

Azov–Black Sea Institute
of Agricultural Mechanization

Received
21 III 1956

CITED LITERATURE

\(^1\) S. Lewin, Math. Zs., 32, H. 4, 503 (1930).
\(^2\) N. K. Bari, Matem. sborn., 14 (56), 1–2, 51 (1944).
\(^3\) N. K. Bari, DAN, 54, No. 5, 385 (1946); Uch. zap. MGU, issue 148, mathematics, 4, 69 (1951).
\(^4\) M. M. Grinblyum, DAN, 47, No. 2, 79 (1945).
\(^5\) Fr. Riesz, Les systèmes d’équations linéaires à une infinité d’inconnues, Paris, 1913.