UDC 517.512.6

MATHEMATICS

M. S. ALIEV

ON THE BEST APPROXIMATION OF CONTINUOUS FUNCTIONS IN THE METRIC \(C[a,b]\) BY GENERALIZED POLYNOMIALS

(Presented by Academician V. I. Smirnov on 13 V 1967)

In the present paper we indicate necessary and sufficient conditions that must be imposed on a system of functions \(\{\varphi_i(x)\}_1^{n+1}\) so that every polynomial

\[ P(c,x)=\sum_{i=1}^{n+1} c_i\varphi_i(x) \]

least deviating in the metric \(C[a,b]\) from a function \(f(x)\in C[a,b]\) possess no fewer than \(k+1\) \((0\le k\le n+1)\) points of alternance.

Definition. A system of continuous and linearly independent functions \(\{\varphi_i(x)\}_1^{n+1}\) on \([a,b]\) will be called a system of type \(k\) \((0\le k\le n)\) if: 1) for any points \(a\le x_1<x_2<\cdots<x_{n+1}\le b\) such that

\[ \left| \begin{array}{ccc} \varphi_1(x_1) & \ldots & \varphi_{n+1}(x_1)\\ \cdots & \cdots & \cdots\\ \varphi_1(x_{n+1}) & \ldots & \varphi_{n+1}(x_{n+1}) \end{array} \right|=0, \]

there exist \(i+1\) rows \((k\le i\le n)\) for which all minors of order \(i+1\) are equal to zero (at least for one \(i\)); 2) in at least one such determinant of order \(i+1\) all complementary minors \(D_j^i\) \((j=1,\ldots,i+1)\) of the elements of at least one column are different from zero, and in the sequence \(D_1^i,\ldots,D_{i+1}^i\) there are no more than \(i-k\) sign changes; moreover, the set of points \(\{x_i\}_1^{n+1}\) for which in this sequence (for some \(i\)) there will be exactly \(i-k\) sign changes.

Examples of systems of type \(k\):

1. Define on the segment \([-1,2]\) continuous functions in the following way:

\[ \varphi_1(x)=1,\quad \varphi_2(x)=x,\ldots,\quad \varphi_{k-1}(x)=x^{k-2},\quad \varphi_k(x)=x^k, \]

\[ \varphi_{k+1}(x)=x^{k+1}+a_0x^k+\cdots+a_k, \]

\[ \varphi_i(x)= \begin{cases} 0 & \text{for } x\in[-1,1],\\ x^i-1 & \text{for } x\in(1,2], \end{cases} \qquad (i=k+2,\ldots,n) \]

\[ \varphi_{n+1}(x)= \begin{cases} 0 & \text{for } x\in[-1,1],\\ x^{k-1}-1 & \text{for } x\in(1,2]. \end{cases} \]

It is possible to choose the parameters \(a_0,a_1,\ldots,a_k\) so that no \(k\) functions from \(\{\varphi_i(x)\}_1^{n+1}\) form a Chebyshev system (\(T\)-system) on \([-1,2]\), but the whole system will be of type \(k\).

1. The system of functions \(1,x^2,x^4,\ldots,x^{2n}\) will be a system of type 1 on the segment \([-a,a]\), where \(a>0\).

Remark. If \(\{\varphi_i(x)\}_1^{n+1}\) form a Chebyshev system (\(T\)-system) on \([a,b]\), then we shall regard its type as equal to \(n+1\).

Theorem. In order that, for every function \(f(x)\) continuous on \([a,b]\), every polynomial least deviating from it

\[ p(c,x)=\sum_{i=1}^{n+1} c_i\varphi_i(x) \]

possess no fewer than \(k+1\) point alternations, and, in order that for every function \(f(x)\) at least one of the polynomials of least deviation from it have exactly \(k+1\) point alternations, it is necessary and sufficient that the system \(\{\varphi_i(x)\}_1^{n+1}\) be of type \(k\) \((0\leq k\leq n+1)\) on \([a,b]\).

In what follows we shall use the criterion for polynomials of least deviation when, on \([a,b]\), the functions \(\{\varphi_i(x)\}_1^{n+1}\) do not form a Chebyshev system \((^{1,2})\), and the following lemmas.

Lemma 1. The equality is valid
\[ \sum_{i=k}^{k+l-1}(-1)^{k+l-1}D \begin{pmatrix} \varphi_1,\ldots,\varphi_k\\ x_1,\ldots,x_{k-1},x_i \end{pmatrix} D \begin{pmatrix} \varphi_1,\ldots,\varphi_k\\ x_l,\ldots,x_{i-1},x_{i+1},\ldots,x_{k+l-1},x \end{pmatrix} = \]
\[ = D \begin{pmatrix} \varphi_1,\ldots,\varphi_k\\ x_1,\ldots,x_{k-1},x \end{pmatrix} D \begin{pmatrix} \varphi_1,\ldots,\varphi_k\\ x_l,\ldots,x_{k+l-1} \end{pmatrix} \qquad (k\geq 1,\ 1\leq l\leq k), \]
where
\[ D \begin{pmatrix} \varphi_1,\ldots,\varphi_k\\ x_{t_1},\ldots,x_{t_k} \end{pmatrix} = \begin{vmatrix} \varphi_1(x_{t_1}),\ldots,\varphi_k(x_{t_1})\\ \ldots\\ \varphi_1(x_{t_k}),\ldots,\varphi_k(x_{t_k}) \end{vmatrix}. \]

Lemma 2. If
\[ D \begin{pmatrix} \varphi_{t_1},\ldots,\varphi_{t_i}\\ x_1,\ldots,x_i \end{pmatrix} \neq 0, \]
then
\[ (-1)^{\,n-i+1}D \begin{pmatrix} \varphi_1,\ldots,\varphi_{n+1}\\ x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_{n+1},x \end{pmatrix} + \]
\[ + \sum_{r=i+1}^{n+1}(-1)^{\,n-r+1} \frac{ D \begin{pmatrix} \varphi_{t_1},\ldots,\varphi_{t_i}\\ x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_i,x_r \end{pmatrix} }{ D \begin{pmatrix} \varphi_{t_1},\ldots,\varphi_{t_i}\\ x_1,\ldots,x_i \end{pmatrix} } \times \]
\[ \times D \begin{pmatrix} \varphi_1,\ldots,\varphi_{n+1}\\ x_1,\ldots,x_{r-1},x_{r+1},\ldots,x_{n+1},x \end{pmatrix} = \]
\[ = \frac{ D \begin{pmatrix} \varphi_{t_1},\ldots,\varphi_{t_i}\\ x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_i,x \end{pmatrix} }{ D \begin{pmatrix} \varphi_{t_1},\ldots,\varphi_{t_i}\\ x_1,\ldots,x_i \end{pmatrix} } D \begin{pmatrix} \varphi_1,\ldots,\varphi_{n+1}\\ x_1,\ldots,x_{n+1} \end{pmatrix} \]
\[ (j=1,\ldots,i). \]

Lemma 3. If the system \(\{\varphi_i(x)\}_1^{n+1}\) is of type \(k\), then in the sequence \(D_1^{n+1},D_2^{n+1},\ldots,D_{n+2}^{n+1}\) \((D_j^{n+1}\neq 0;\ j=1,\ldots,n+2)\) there will be no more than \(n+1-k\) changes of sign, where \(D_j^{n+1}\) \((j=1,\ldots,n+2)\) are obtained by deleting the \(j\)-th row from the matrix
\[ \begin{Vmatrix} \varphi_1(x_1)\ldots\varphi_{n+1}(x_1)\\ \ldots\\ \varphi_1(x_{n+2})\ldots\varphi_{n+1}(x_{n+2}) \end{Vmatrix}. \]

Lemma 4. If in the sequence of complementary minors \(D_1^i,\ldots,D_{i+1}^i\) \((D_j^i\neq 0,\ j=1,\ldots,i+1)\) of the elements of one column of some determinant of order \((i+1)\) of the matrix
\[ \begin{Vmatrix} \varphi_1(x_{\mu_1})\ldots\varphi_{n+1}(x_{\mu_1})\\ \ldots\\ \varphi_1(x_{\mu_{i+1}})\ldots\varphi_{n+1}(x_{\mu_{i+1}}) \end{Vmatrix} \tag{*} \]

\((\mu_1,\mu_2,\ldots,\mu_{i+1}\) are the numbers of the rows appearing in condition 1) of the definition) has \(p\) changes of sign, then in the row of nonzero complementary minors \(D_j^i\) \((j=1,\ldots,i+1)\) of the elements of any other column of the same matrix () there will also be \(p\) changes of sign, and if the complementary minor of at least one element of some determinant of order \((i+1)\) of the matrix () is equal to zero, the complementary minors of all the remaining elements of the corresponding column are also equal to zero.

We outline the proof of the theorem.

Sufficiency. Let a system of type \(k\) and \(p(c,x)\) deviate least from \(f(x)\) on \([a,b]\). From (1) it follows that among the points of the minimal subset there will be \(i+1\) \((k\le i\le n+1)\) points \(x_1,\ldots,x_{i+1}\) for which: a) the matrix (*) has rank \(i\); b) there exist \(D_j^i\ne0\) \((j=1,\ldots,i+1)\); c)
\(\operatorname{sign}[p(c,x_j)-f(x_j)]=\delta\,\operatorname{sign}(-1)^jD_j^i\), where \(\delta=\pm1\).

Since in the row \(D_1^i,\ldots,D_{i+1}^i\) \((k\le i\le n+1)\) there are no more than \(i-k\) changes of sign, in the row \(-D_1^i,D_2^i,\ldots,(-1)^{i+1}D_{i+1}^i\) there will be no fewer than \(k\) changes of sign. Consequently, at least \(k+1\) points from \(\{x_i\}_1^{i+1}\) will be alternation points.

Necessity. Let every polynomial \(p(c,x)\) that deviates least from \(f(x)\) possess at least \(k+1\) pointwise alternations.

Suppose that the system \(\{\varphi_i(x)\}_1^{n+1}\) is of type \(r\) (for definiteness, \(r<k\)). Then, by definition, there exists a system of points \(\{x_j\}_1^{i+1}\) for which, in the corresponding row \(D_1^i,\ldots,D_{i+1}^i\), there will be exactly \(i-r\) changes of sign.

Put \(f(x_j)=\operatorname{sign}[(-1)^jD_j^i]\) \((j=1,\ldots,i+1)\), \(f(a)=\operatorname{sign}(-D_1^i)\), \(f(b)=\operatorname{sign}(-1)^{i+1}D_{i+1}^i\), and at the remaining points of \([a,b]\) let \(f(x)\) be linear. From condition 1) of the definition we have

\[ \sum_{j=1}^{i+1}(-1)^jD_j^i p(c,x_j)=0. \]

Consequently, the least deviation

\[ h(c)=|p(c,x_j)-f(x_j)|=|p(c,x_j)-[\operatorname{sign}(-1)^jD_j^i]| \]

is not less than 1. \(p(c,x)\equiv0\) deviates least from \(f(x)\). For it

\[ \operatorname{sign}[p(c,x_j)-f(x_j)]=-\operatorname{sign}[(-1)^jD_j^i] \]

and, consequently, the number of alternation points will be \(r+1<k+1\), which contradicts the assumption.

It is easy to show that any system of functions \(\{\varphi_i(x)\}_1^{n+1}\) linearly independent on \([a,b]\) will be a system of some definite type \(k\) \((0\le k\le n+1)\).

We consider it our duty to express gratitude to I. V. Tsenov for posing the problem and for supervising the work.

Dagestan State University
named after V. I. Lenin

Received
10 V 1967

CITED LITERATURE

1. S. I. Zukhovitskii, Uspekhi Mat. Nauk, 2, no. 2 (1956).
2. E. Ya. Remez, On Methods for the Best, in the Sense of P. L. Chebyshev, Approximate Representation of Functions, Kyiv, 1935.