MATHEMATICS

S. I. Zukhovitskii and G. I. Eskin

ON THE APPROXIMATION OF ABSTRACT CONTINUOUS FUNCTIONS BY UNBOUNDED OPERATOR-FUNCTIONS

(Presented by Academician N. N. Bogolyubov, 26 IV 1957)

1. Let a system of linear differential equations be given

\[ \frac{dx_i}{dt}+\sum_{k=1}^{n}p_{ik}(t)x_k=f_i(t) \qquad (0\leq t\leq 2\pi)\quad (i=1,2,\ldots,m) \tag{1} \]

and suppose that one seeks such a solution of this system which satisfies certain boundary conditions, for example \(x_k(0)=x_k(2\pi)\) \((k=1,\ldots,n)\), and belongs to a prescribed class \(D\) of vector-functions
\(x=(x_1(t),x_2(t),\ldots,x_n(t))\).

If in the class \(D\) there is no such (exact) solution, then one may pose the question of finding in it the best approximate solution of system (1), i.e. such a vector-function
\(x_0=(x_{10}(t),x_{20}(t),\ldots,x_{n0}(t))\) that

\[ \max_i\left\| \frac{dx_{i0}}{dt}+\sum_{k=1}^{n}p_{ik}(t)x_{k0}-f_i(t) \right\| = \inf_{x\in D}\max_i \left\| \frac{dx_i}{dt}+\sum_{k=1}^{n}p_{ik}(t)x_k-f_i(t) \right\|. \]

For each \(i=1,2,\ldots,m\), the left-hand side of the \(i\)-th equation of system (1) may be regarded as the value of an operator \(A_i\), acting from the Hilbert space \(H\) of vector-functions
\(x=(x_1(t),x_2(t),\ldots,x_n(t))\) (where

\[ x_k(t)\in L^2(0,2\pi)\quad (k=1,2,\ldots,n) \quad\text{and}\quad \|x\|_H=\left(\sum_{k=1}^{n}\|x_k\|_{L^2}^{2}\right)^{1/2} \]

) into \(L^2(0,2\pi)\).

The same problem can be posed when one is dealing, in general, with a system of operator equations. More precisely, suppose that on some compact set \(Q\) there is considered an operator-function \(A(q)\), which for each \(q\in Q\) is a linear operator acting from a Hilbert space \(H_1\) into a Hilbert space \(H_2\), with a domain of definition \(D\) common to all \(q\in Q\), and such that for each fixed \(x\in H_1\) the function \(A(q)x\), with values in \(H_2\), is continuous on \(Q\). Let \(f(q)\) be a function continuous on \(Q\) with values in \(H_2\). The problem consists in finding a vector \(x_0\in D\) such that the function \(A(q)x_0\) deviates least on \(Q\) from the function \(f(q)\), i.e. so that*

\[ \max_{q\in D}\|A(q)x_0-f(q)\|_2 = \inf_{x\in D}\max_{q\in Q}\|A(q)x-r(q)\|_2. \]

* We shall denote the norm in \(H_1\) by the subscript 1, and in \(H_2\) by the subscript 2; the zero in \(H_1\) will be denoted by \(\theta_1\), and in \(H_2\) by \(\theta_2\).

Let \(R\) be the subspace of those vectors \(x\in D\) for which \(A(q)x=\theta_2\) for all \(q\in Q\), and let \(S\) be the orthogonal complement of \(R\) in \(H_1\) (for the operator-functions considered below, the linear manifold \(R\) will turn out to be a subspace); then each vector \(x\in D\) is represented in the form \(x=x_R+x_S\) \((x_S\in D\cap S)\), and

\[ \inf_{x\in D}\max_{q\in Q}\|A(q)x-f(q)\|_2 = \inf_{x_S\in D\cap S}\max_{q\in Q}\|A(q)x_S-f(q)\|_2 . \]

Therefore in what follows, without stipulating this, we shall assume that \(R=\theta_1\) and \(S=H_1\).

In (¹) an analogous problem was considered only for the case when the operator-function \(A(q)\), for each \(q\in Q\), was a linear bounded operator acting from the Hilbert space \(H\) into the same space. In the present paper we consider linear unbounded (closed) operators; we prove that property a), given in (¹), is not only a sufficient but also a necessary condition for the existence of a least-deviating function both in the case of bounded and in the case of closed operators; we consider questions of uniqueness of the function of least deviation and extend some of the results obtained to Banach spaces.

2. Theorem 1. Let an operator-function \(A(q)\) be given on the compact set \(Q\), possessing the following two properties: 1) for each \(q\in Q\), \(A(q)\) is a closed linear operator acting from the Hilbert space \(H_1\) into the Hilbert space \(H_2\), with a domain of definition \(D_{A(q)}=D\) dense in \(H_1\) and common to all \(q\in Q\); 2) for each fixed \(x\in D\), \(A(q)x\) is a continuous function on \(Q\) with values in \(H_2\).

Then, in order that for every function \(f(q)\), continuous on \(Q\) and with values in \(H_2\), there exist a vector \(x_0\in D\) such that the function \(A(q)x_0\) deviates least on \(Q\) from the function \(f(q)\):

\[ \inf_{x\in D}\max_{q\in Q}\|A(q)x-f(q)\|_2 = \max_{q\in Q}\|A(q)x_0-f(q)\|_2, \]

it is necessary and sufficient that \(A(q)\) possess the following property:

\[ \max_{q\in Q}\|A(q)x\|_2\ge m\|x\|_1 \quad \text{for all } x\in D, \tag{a} \]

where \(m>0\).

The necessity of property (a) is proved as follows. The functions \(f(q)\), continuous on \(Q\) and with values in \(H_2\), form a Banach space \(C\) with norm \(\|f\|_C=\max_{q\in Q}\|f(q)\|_2\), and, obviously, in order that for every function \(f(q)\) there exist a function \(A(q)x_0\) deviating least from it, it is necessary that the linear manifold of functions \(A(q)x\) \((x\in D)\) be closed in \(C\), i.e. form a subspace. Further, the operator \(T\), acting from \(H_1\) into \(C\) according to the rule \(Tx=A(q)x\) \((x\in D)\), is closed, since the operators \(A(q)\) are closed for each \(q\in D\); moreover, the operator \(T\) has an inverse \(T^{-1}\), since, by assumption, \(R=\theta_1\). But this inverse operator \(T^{-1}\) is bounded, being closed and defined on a subspace (see, for example, (²), p. 47), so that

\[ \|Tx\|_C=\max_{q\in Q}\|A(q)x\|_2\ge \|x\|_1 \quad (x\in D), \]

where \(m>0\).

The sufficiency of property (a) is proved with more difficulty. We only note that it also essentially uses the closedness of the operator \(A(q)\) and one theorem of S. Mazur (see (³), p. 207), on which the proof of Theorem 1 in (¹) was also based.

1. Examples of the realization of the conditions of the preceding theorem, besides the operator-functions I—IV of work (¹), which are bounded operators for every \(q \in Q\), may also be furnished by the following operator-functions:

V.
\[ A(q)x=\sum_{i=1}^{N}\xi_i f_i(q), \]
where \(f_1(q), f_2(q), \ldots, f_N(q)\) are functions continuous on \(Q\) with values in the Hilbert space \(H\), and \(x=(\xi_1,\xi_2,\ldots,\xi_N)\) is a point of a real or complex Euclidean space \(R_N\); the question is that of approximation of abstract functions \(f(q)\) by polynomials
\[ \sum_{i=1}^{N}\xi_i f_i(q), \]
considered in (⁴). Here \(H_1=R_N,\ H_2=H\), and the operators \(A(q)\) are bounded.

In particular, if \(f_1(q), f_2(q), \ldots, f_N(q)\) are real or complex continuous functions on \(Q\), then \(H=R_1\), and the question is that of the classical problem of Chebyshev approximation of a numerical function \(f(q)\), continuous on \(Q\), by a polynomial composed of numerical functions \(f_1(q), f_2(q), \ldots, f_N(q)\) continuous on \(Q\).

VI. \(A(q)=P_1+\lambda(q)E\), where \(P_1=id/dt\) is the differentiation operator on \([0,2\pi]\) with boundary conditions \(x(0)=x(2\pi)\), and \(\lambda(q)\) is a function continuous on \(Q\), and at least for one \(q_0\in Q\) we have \(\lambda(q_0)\ne 0,\ \pm 1,\ \pm 2,\ldots\). Here \(H_1=H_2=L^2(0,2\pi)\), and the operators \(A(q)\) are unbounded.

VII. For every \(q\in Q\), \(A(q)\) is the Sturm—Liouville operator
\[ A(q)x=\frac{d}{dt}\left[P_1(q,t)\frac{dx}{dt}\right]+p_2(q,t)x(t)\qquad (0\leq t\leq 2\pi), \]
depending continuously on \(q\), with boundary conditions \(x(0)=x(2\pi)=0\), and for some \(q_0\in Q\) zero is a regular point of the spectrum of the operator \(A(q_0)\). Here \(H_1=H_2=L^2(0,2\pi)\), and the operators \(A(q)\) are also unbounded.

In general, obviously, property (a) is possessed by every operator-function \(A(q)\) satisfying conditions 1), 2) if, for some \(q_0\in Q\), the operator \(A(q_0)\) has a bounded inverse.

1. Without dwelling on the characteristic property of the function of least deviation, since, obviously, theorem 2 from (¹) carries over without change also to the operator-functions considered in the present work, we pass to the consideration of the question of uniqueness of the least-deviating function.

Theorem 2. Let \(A(q)\) satisfy the conditions of Theorem 1 and \(\dim H_1<\dim H_2\). Then, in order that for every function \(f(q)\), continuous on \(Q\) and with values in \(H_2\), there exist a unique function \(A(q)x_0\) least deviating from it on \(Q\), it is necessary and sufficient that, for \(x\ne\theta_1\), the equation \(A(q)x=\theta_2\) have not a single root on \(Q\).

The following theorem is a transfer to the case of operator-functions of the corresponding theorems from (⁴,⁵).

Theorem 3. Let \(A(q)\) be, for every \(q\in Q\), a linear operator acting from \(H_1\) into \(H_2\), and let, for every fixed \(x\in H_1\), the function \(A(q)x\) be continuous on \(Q\). Let the number \(n\) \((1\leq n<\infty)\) be such that
\[ (n-1)\dim H_2<\dim H_1\leq n\dim H_2. \]
Then, for uniqueness of the function of least deviation, it is necessary and sufficient that the following conditions be fulfilled:

1) for \(x\ne\theta_1\), the equation \(A(q)x=\theta_2\) has on \(Q\) no more than \(n-1\) roots;

2) for any distinct \(q_1,q_2,\ldots,q_{n-1}\) from \(Q\) and any \(y_1,y_2,\ldots,y_{n-1}\) from \(H_2\), there exists a vector \(x\in H_1\) such that
\[ A(q_1)x=y_1,\quad A(q_2)x=y_2,\ldots,\quad A(q_{n-1})x=y_{n-1}. \]

When \(\dim H_1=n\dim H_2\), condition 2) is a consequence of condition 1).

1. In the case of Banach spaces we note the following theorems:

Theorem 4. Let \(A(q)\), for each \(q\in Q\), be a linear discontinuous operator acting from the Banach space \(B_1\), with weakly compact sphere (for example, reflexive), into the Banach space \(B_2\), and suppose that for each fixed \(x\in B_1\) the function \(A(q)x\) is continuous. Then, in order that for every continuous function \(f(q)\) on \(Q\) with values in \(B_2\) there exist a function \(A(q)x_0\) of least deviation from it, it is necessary and sufficient that \(A(q)\) possess property (a)

\[ \max_{q\in Q}\|A(q)x\|_{B_2}\ge m\|x\|_{B_1}\quad \text{for all } x\in B_1 . \]

Theorem 5. Let \(A(q)\) satisfy the conditions of Theorem 4, suppose that \(B_2\) is strictly convex with weakly compact sphere, and, in addition, has the following property: if, for some \(q_1\in Q\), the equation \(A(q_1)x=\theta_2\) has a solution \(x_0\ne\theta_1\), then the adjoint equation \(A^*(q_1)Y=\theta_1^*\) also has a solution \(Y_0\ne\theta_2^*\). Then, for uniqueness of the function of least deviation, it is necessary and sufficient that the equation \(A(q)x=\theta_2\), for \(x\ne\theta_1\), have no roots on \(Q\).

The conditions of the last two theorems are satisfied, for example, by the operator-function \(A(q)=E-T(q)\), where \(T(q)\), for each \(q\in Q\), is a completely continuous operator from \(B_1\) into \(B_2\).

Remarks. \(1^\circ\). Theorem 3 also carries over to the case of Banach spaces \(B_1\) and \(B_2\) under the condition that \(B_2\) is strictly convex, and Theorem 2—if, in addition, the sphere in \(B_2\) is weakly compact.

\(2^\circ\). Let us note that the preceding theorems for the case of Banach spaces were formulated under the assumption that
\[ R=\{x:\ A(q)x=\theta_2 \text{ for all } q\in Q\}=\theta_1 . \]
In the contrary case, instead of \(B_1\) one should consider the factor space \(B_1/R\) of elements \(X\) with norm
\[ \|X\|=\inf_{x\in X}\|x\|_{B_1}. \]

Lutsk State Pedagogical Institute
named after Lesya Ukrainka

Received
24 IV 1957

References

1. S. I. Zukhovitskii, Matem. sborn., 37 (79), issue 1, 3 (1955).
2. E. Hille, Functional Analysis and Semigroups, Moscow, 1951.
3. S. Banach, Course of Functional Analysis, Kiev, 1948.
4. M. I. Zukhovitskii, S. B. Stechkin, DAN, 106, No. 3, 385 (1956).
5. S. I. Zukhovitskii, M. G. Krein, Uspekhi matem. nauk, 3, issue 1 (23), 216 (1948).