MATHEMATICS

Yu. A. SHASHKIN

ON THE CONVERGENCE OF LINEAR POSITIVE OPERATORS IN THE SPACE OF CONTINUOUS FUNCTIONS

(Presented by Academician N. N. Bogolyubov, 27 XI 1959)

Consider, on some compact set \(Q\), the space \(C(Q)\) of real continuous functions \(f(x)\), \(x \in Q\), with norm
\[ \|f\|_C=\max_{x\in Q}|f(x)| \]
and the space \(B(Q)\) of real bounded functions \(\varphi(x)\), \(x \in Q\), with norm
\[ \|\varphi\|_B=\sup_{x\in Q}|\varphi(x)|. \]
It is obvious that \(C(Q)\) is a subspace of \(B(Q)\). Let \(L_n\) \((n=1,2,\ldots)\) be a sequence of linear positive operators mapping \(C(Q)\) into \(B(Q)\). (An operator \(L\) is called positive if, for every nonnegative function \(f(x)\in C(Q)\), the function \(\varphi(x)=L(f,x)\in B(Q)\) is also nonnegative.)

Definition 1. A system of continuous functions
\[ S_m=\{f_0(x), f_1(x),\ldots, f_m(x)\} \]
will be called a \(K\)-system on the compact set \(Q\) if, for any sequence of linear positive operators \(L_n\), from
\[ \|L_n(f_i,x)-f_i(x)\|_B\to 0 \qquad (i=0,1,2,\ldots,m) \]
as \(n\to\infty\), it follows that
\[ \|L_n(f,x)-f(x)\|_B\to 0 \]
for every \(f(x)\in C(Q)\). The number \(m\) is called the order of the system \(S_m\).

The properties of \(K\)-systems have been studied by P. P. Korovkin \((^{1,2})\) (see also \((^3)\)) in the case when the compact set \(Q\) is an interval or a circle; by V. I. Volkov \((^{4,5})\) in the case of a closed domain in the plane; and by E. N. Morozov \((^6)\) in the case of the two-dimensional torus.

In this article we shall consider only such \(K\)-systems for which \(f_0(x)\equiv 1\).

Definition 2. It is said that a system of continuous functions
\[ S_m=\{f_0(x), f_1(x),\ldots, f_m(x)\} \]
has Chebyshev rank equal to \(r\) \((0\le r\le m)\) on the compact set \(Q\) if, for any \(m-r+1\) distinct points \(x_1,\ldots,x_{m-r+1}\) of the compact set \(Q\), the rank of the matrix
\[ \begin{pmatrix} f_0(x_1) & f_1(x_1) & \ldots & f_m(x_1)\\ f_0(x_2) & f_1(x_2) & \ldots & f_m(x_2)\\ \cdots & \cdots & \cdots & \cdots\\ f_0(x_{m-r+1}) & f_1(x_{m-r+1}) & \ldots & f_m(x_{m-r+1}) \end{pmatrix} \]
is equal to \(m-r+1\), and there exist \(m-r+2\) distinct points of the compact set for which the rank of the corresponding matrix is less than \(m-r+2\). In particular, if \(r=0\), then \(S_m\) is called a Chebyshev system. This definition of Chebyshev rank is equivalent to the original one (see, for example, \((^{7,8})\)).

Theorem 1. In order that on a compactum \(Q\) there exist at least one \(K\)-system of finite order, it is necessary and sufficient that the compactum \(Q\) have finite dimension.

Theorem 2. In order that the system \(S_m=\{f_0(x)=1,\ f_1(x),\ldots,\) \(\ldots, f_m(x)\}\) be a \(K\)-system on the compactum \(Q\), it is necessary and sufficient that, for every point \(x_0\in Q\), a linear positive functional \(\Phi\) in the space \(C(Q)\) be uniquely determined by the conditions \(\Phi(f_i)=f_i(x_0)\), \(i=0,1,\ldots,m\).

This theorem can be formulated in other terms:

Theorem 3. In order that the system \(S_m\) be a \(K\)-system on the compactum \(Q\), it is necessary and sufficient that the mapping \(F\) of the compactum \(Q\) onto a subset \(M\) of \(m\)-dimensional Euclidean space \(E^m\), defined by this system,

\[ F(x)=(f_1(x), f_2(x),\ldots, f_m(x))\in E^m,\qquad x\in Q, \]

be homeomorphic and that one of the following mutually equivalent conditions be satisfied:

each point of the set \(M\) is an extreme point of its convex hull;

no point of the set \(M\) belongs to the convex hull of the remaining part of this set;

whatever simplex of positive dimension with vertices from the set \(M\) may be taken, all its interior points do not belong to this set.

From Theorem 3 there follows

Corollary \((^5)\). The order \(m\) and the Chebyshev rank \(r\) of a \(K\)-system on any compactum are related by the relation \(m\ge r+2\).

Denote by \(m(Q)\) the minimal order of \(K\)-systems on the compactum \(Q\). Naturally the question arises of computing \(m(Q)\) for any finite-dimensional compactum. In this direction one can prove the following theorem.

Theorem 4. If the compactum \(Q\) is homeomorphic to an \(n\)-dimensional sphere or to a subset of it, then \(m(Q)\le n+1\). If there exists a nonempty subset \(U\subset Q\), open relative to \(Q\), such that the difference \(Q\setminus U\) is not homeomorphically embedded in \((n-1)\)-dimensional Euclidean space \(E^{n-1}\), then \(m(Q)\ge n+1\).

Proof. The first assertion of the theorem follows from the fact that on the \(n\)-dimensional sphere there exist systems of functions of order \(n+1\) which are \(K\)-systems on the entire sphere and on any of its closed subsets, for example:

\[ \begin{aligned} f_0&=1,\\ f_1&=\cos x_1,\\ f_2&=\sin x_1\cos x_2,\\ &\ldots\ldots\ldots\ldots\\ f_n&=\sin x_1\sin x_2\cdots \sin x_{n-1}\cos x_n,\\ f_{n+1}&=\sin x_1\sin x_2\cdots \sin x_{n-1}\sin x_n\\ (0\le x_i\le \pi,\quad i&=1,2,\ldots,n-1;\qquad 0\le x_n\le 2\pi). \end{aligned} \]

The second assertion follows from a theorem of K. Borsuk \((^9)\).

Corollary. If \(Q\) is the closure of an open subset of the \(n\)-dimensional sphere or of \(n\)-dimensional Euclidean space, then \(m(Q)=n+1\).

Theorem 5. Let \(Q\) be homeomorphic to the closure of an open and connected subset (i.e., a domain) of the \(n\)-dimensional sphere. In order that the system \(S_{n+1}=\{f_0(x)=1,\ f_1(x),\ldots, f_{n+1}(x)\}\) be a \(K\)-system on the compactum \(Q\),

necessary and sufficient that it have Chebyshev rank equal to \(m(Q)-2=n-1\).

In the proof of this theorem the following is used.

Lemma. Let the compact set \(Q\) satisfy the conditions of Theorem 5, and let the system \(S_{n+1}\) have Chebyshev rank \(n-1\) on \(Q\). Then for every interior point \(x_0\in Q\) there exists a polynomial

\[ p(x,x_0)=\sum_{i=0}^{n+1} c_i f_i(x) \]

with respect to the system \(S_{n+1}\) such that

\[ p(x_0,x_0)=0,\qquad p(x,x_0)>0,\qquad x\in Q,\quad x\ne x_0 . \]

One can verify from examples that the conditions imposed on the compact set are essential in Theorem 5; moreover, the following theorem is valid.

Theorem 6. For every disconnected compact set \(Q\) containing more than three points, there exists a system of functions of order \(m=m(Q)\) and Chebyshev rank \(r=m(Q)-2\) which is not a \(K\)-system.

For the set consisting of the intervals \([0,\pi/2]\) and \([\pi,3\pi/2]\), such a system will be, for example, the following Chebyshev system of second order:

\[ f_0(x)=1; \]

\[ f_1(x)= \begin{cases} \cos x, & x\in[0,\pi/2],\\ 2+\cos x, & x\in[\pi,3\pi/2]; \end{cases} \qquad f_2(x)= \begin{cases} \sin x, & x\in[0,\pi/2],\\ 2+\sin x, & x\in[\pi,3\pi/2]. \end{cases} \]

Let us now consider \(K\)-systems in the space of periodic functions of \(n\) variables or, in other words, in the space \(C(T^n)\), where \(T^n\) is the \(n\)-dimensional torus. It is not difficult to see that \(m(T^n)=n+2\) (for \(n\ge 2\)). It is also clear that any system of the form

\[ f_0(x)=1,\qquad f_1(x),\ldots,f_{n+1}(x), \]

\[ f_{n+2}(x)=f_1^2(x)+f_2^2(x)+\cdots+f_{n+1}^2(x),\qquad x\in T^n, \]

where the functions \(f_1(x),\ldots,f_{n+1}(x)\) realize a homeomorphic embedding of the torus \(T^n\) into \((n+1)\)-dimensional Euclidean space, for example:

\[ f_1=a_1\cos x_1, \]

\[ f_2=(a_2+a_1\sin x_1)\cos x_2, \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ f_n=\bigl(a_n+(a_{n-1}+\cdots+(a_2+a_1\sin x_1)\sin x_2\cdots)\sin x_{n-1}\bigr)\cos x_n, \]

\[ f_{n+1}=\bigl(a_n+(a_{n-1}+\cdots+(a_2+a_1\sin x_1)\sin x_2\cdots)\sin x_{n-1}\bigr)\sin x_n, \]

\[ \left(0\le x_i\le 2\pi,\quad i=1,2,\ldots,n;\qquad a_1>0,\quad a_k>\sum_{i=1}^{k-1}a_i,\quad k=2,3,\ldots,n\right). \]

Received
26 XI 1959

CITED LITERATURE

1. P. P. Korovkin, DAN, 90, No. 6, 961 (1953).
2. P. P. Korovkin, Scientific Notes of the Kalinin Pedagogical Institute named after M. I. Kalinin, 26, 95 (1958).
3. P. P. Korovkin, Linear Operators and Approximation Theory, 1959.
4. V. I. Volkov, DAN, 115, No. 1, 17 (1957).
5. V. I. Volkov, Scientific Notes of the Kalinin Pedagogical Institute named after M. I. Kalinin, 26, 27 (1958).
6. E. N. Morozov, ibid., 26, 129 (1958).
7. G. Sh. Rubinshtein, DAN, 102, No. 3, 451 (1955).
8. A. L. Garkavi, Izv. AN SSSR, Ser. Mat., 23, No. 1 (1959).
9. K. Borsuk, Bull. Acad. Sci. Pol., 5, No. 4, 351 (1957).