MATHEMATICS

V. N. BUROV

ON THE APPROXIMATION OF FUNCTIONS BY POLYNOMIALS SATISFYING NONLINEAR RELATIONS

(Presented by Academician V. I. Smirnov on 9 I 1961)

1°. Let a continuous real-valued function \(f(q)\) be given on a compact set \(Q\), and let a class \(K\) of real generalized polynomials of the form

\[ y(q)=\sum_{k=1}^{n} a_k \varphi_k(q), \tag{1} \]

be specified, where \(\{\varphi_k(q)\}_1^n\) is a system of \(n\) linearly independent functions continuous on \(Q\). Suppose it is required to approximate uniformly the function \(f(q)\) by polynomials (1) subject to given relations, which will be discussed below in 2°.

For each polynomial \(y(q)\in K\), put

\[ L_f(y)=\max_{q\in Q}|f(q)-y(q)|, \tag{2} \]

and by \(E(y)\) denote the set of all points of maximum deviation of \(y(q)\) from \(f(q)\), i.e. the points \(q\in Q\) at which \(|f(q)-y(q)|=L_f(y)\).

It is known \((^1)\) that the problem of best approximation (in the sense of P. L. Chebyshev) of the function \(f(q)\) by polynomials from \(K\) is always solvable. Therefore we are entitled to set

\[ \min_{y\in K} L_f(y)=\rho^*. \tag{3} \]

Passing to geometric language, we shall identify a polynomial \(y(q)\in K\) with its representing point \((a_1,a_2,\ldots,a_n)\) in the \(n\)-dimensional coefficient space \(R_n\). Then the aggregate of all polynomials of best approximation for \(f(q)\) may be interpreted as a bounded, closed, and convex set \(V^*\equiv V_{\rho^*}\subset R_n\). By \(S_{\rho^*}\) we denote the boundary of \(V_{\rho^*}\).

If \(L>\rho^*\), then the equality \(L_f(y)=L\) is, obviously, realized on a nonempty subset of polynomials from \(K\), which form in \(R_n\) a closed surface \(S_L\) bounding the convex body \(V_L\). It is clear that for \(L'<L''\) we shall have

\[ V_{L'}\subset V_{L''}, \qquad S_{L'}\cap S_{L''}=\Lambda. \tag{4} \]

2°. In order to impose a relation on the polynomials (1), let us single out in \(R_n\) a nonempty closed set \(\Omega\) and define a narrower class of comparison polynomials \(K_\Omega\subset K\), considering \(y(q)\in K_\Omega\) if and only if \((a_1,a_2,\ldots,a_n)\in\Omega\). In concrete cases the set \(\Omega\) may be specified, for example, by inequalities

\[ \alpha_j \leqslant \omega_j(y) \leqslant \beta_j \qquad (j=1,2,\ldots,p), \tag{5} \]

where \(\omega_j(y)\) are continuous functionals, and \(\alpha_j,\beta_j\) are constant numbers ensuring the admissibility of the relations (5), i.e. the nonemptiness of \(\Omega\). In particular, this

may be the linear relations, which have also been studied repeatedly \((^{2-8})\),

\[ \omega_j(y)\equiv \sum_{k=1}^{n}\delta_{jk}a_k=\beta_j\quad (j=1,2,\ldots,p;\ p\leq n). \tag{6} \]

Definition. \(y^*(q)\) is called an extremal polynomial of the function \(f(q)\) in the class \(K_\Omega\), if \(y^*(q)\in K_\Omega\) and for it

\[ L_f(y^*)=\inf_{y\in K_\Omega}L_f(y)\equiv \rho_\Omega . \tag{7} \]

Let the set of the representing points of all extremal polynomials in the space \(R_n\) be denoted by \(V_\Omega^*\).

\(3^\circ\). From relations (4) and the closedness of the set \(\Omega\) there follows the existence of a least value \(L\geq \rho^*\), for which the intersection \(\Omega\cap S_L\) is nonempty. In turn, hence there follows:

Theorem 1. For every continuous function \(f(q)\) on \(Q\) there exists at least one extremal polynomial \(y^*(q)\), and

\[ \rho_\Omega=\min_{\Omega\cap S_L\ne \Lambda}L,\qquad V_\Omega^*=\Omega\cap S_{\rho_\Omega}. \tag{8} \]

Moreover, if \(\Omega\) is convex, then \(V_\Omega^*\) is also convex and lies in a supporting hyperplane for \(V_{\rho_\Omega}\). At the same time, on the compact set \(Q\) there exists a common basis (cf. \((^{4\mathrm{b}})\)) of joint deviation points of all extremal polynomials.

Theorem 2. For uniqueness of the extremal polynomial \(y^*(q)\) it is sufficient that the convex hull of the set \(\Omega\) be strictly convex, contain no points of \(V^*\) in its interior, and that its surface be wholly contained in \(\Omega\).

It is not difficult to extend from the linear case the criterion of B. A. Rymarenko \((^{4\mathrm{a}})\).

Theorem 3. In order that the polynomial \(y^*(q)\in K_\Omega\) be extremal, it is sufficient—and, in the case of convexity of the set \(\Omega\), necessary—that for every polynomial \(y(q)\in K_\Omega\) one have

\[ \min_{q\in E(y^*)}\{[y(q)-y^*(q)][f(q)-y^*(q)]\}\leq 0 . \tag{9} \]

\(4^\circ\). Let \(\Omega\) be convex and \(y_0^*\in V_\Omega^*\). Then, if also \(y^*\in V_\Omega^*\), it is necessary that

\[ \min_{q\in E(y_0^*)}\{[y^*(q)-y_0^*(q)][f(q)-y_0^*(q)]\}=0. \tag{10} \]

Conversely, if a polynomial \(y^*(q)\in K_\Omega\) satisfies relation (10), then for sufficiently small \(\lambda\geq 0\) the comparison polynomial \(y_0^*(q)+\lambda [y^*(q)-y_0^*(q)]\) will be extremal.

The introduction of the parameter \(\lambda\) and the geometric considerations of items \(1^\circ\)—\(3^\circ\) make it possible to apply the theory of the one-parameter problem, set forth in the author’s papers \((^9)\). This helps to clarify questions concerning the general form of extremal polynomials.

Theorem 4. Let \(y_0^*(q)\) be one of the extremal polynomials. In order that \(y^*(q)\in K_\Omega\) also be an extremal polynomial, it is sufficient—and, when \(\Omega\) is convex, necessary—that the conditions

\[ N(y^*-y_0^*)\cap E(y_0^*)\ne \Lambda, \tag{11} \]

\[ \inf_{q\in Q\setminus N(y^*-y_0^*)} \left\{ \frac{[f(q)-y_0^*(q)]\,\operatorname{sign}[y^*(q)-y_0^*(q)]+\rho_\Omega} {|y^*(q)-y_0^*(q)|} \right\}\geq 1, \tag{12} \]

be fulfilled, where \(N(h)\) denotes the set of all zeros of the polynomial \(h(q)\in K\) on the compact set \(Q\).

Theorem 5. If the set \(\Omega\) generates the linear relations (6), then the general form of all extremal polynomials is given by the formula

\[ y^*(q)=y_0^*(q)+\lambda h(q), \tag{13} \]

where \(y_0^*(q)\) is any fixed extremal polynomial; \(h(q)\) is an arbitrary polynomial of the class \(K\) satisfying the conditions

\[ \omega_j(h)=0 \quad (j=1,2,\ldots,p); \tag{14} \]

\[ N(h)\cap E(y_0^*)\ne \Lambda; \tag{15} \]

\[ \max_{q\in Q} |h(q)|=1, \tag{16} \]

and \(\lambda\) is any value from the segment \([\lambda^-,\lambda^+]\), where

\[ \lambda^-=\sup_{q\in Q\setminus N(h)} \left\{ \frac{[f(q)-y_0^*(q)]\operatorname{sign}h(q)-\rho_\Omega}{|h(q)|} \right\}, \]

\[ \lambda^+=\inf_{q\in Q\setminus N(h)} \left\{ \frac{[f(q)-y_0^*(q)]\operatorname{sign}h(q)+\rho_\Omega}{|h(q)|} \right\}. \tag{17} \]

\(5^\circ\). The geometric approach set forth above opens the possibility of far-reaching applications. For example, it facilitates the analysis of approximation problems by polynomials with nonnegative coefficients, with fixed coefficients at the specified \(\varphi_k(q)\), by regularly monotone \({}^{(10)}\) polynomials, etc., when \(\Omega\) is known in advance to be convex. One may also, proceeding from the “other end” (cf. \({}^{(8)}\)), impose relations under which preassigned polynomials with equal \(L_f(y)\) will turn out to be extremal.

In conclusion the author expresses deep gratitude to Prof. B. A. Rymarenko for discussion of the results of this work.

Leningrad State
Pedagogical Institute
named after A. I. Herzen

Received
8 I 1961

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