MATHEMATICS

E. V. VORONOVSKAYA

ON UNIFORMLY BEST APPROXIMATION OF POLYNOMIALS

(Presented by Academician S. L. Sobolev on 28 XII 1956)

Let us consider the special case of the Chebyshev problem on the best approximation of \(f(x)\), continuous on \([0,1]\), by polynomials of a prescribed degree. Let \(f(x)=P_n(x)\); for \(m<n\) one seeks \(P_m(x)\) for which \(\max_{[0,1]} |P_n(x)-P_m(x)|\) has the least value \(L\). As is known, such a polynomial \(P_m(x)\) is unique, and the difference \(P_n(x)-P_m(x)\) attains \(\pm L\) at not fewer than \((m+2)\) points on \([0,1]\), with successive alternation of sign. Setting \(P_n(x)-P_m(x)=Y_n(x)\), we obtain the following formulation of the problem: among polynomials of degree \(n\) with prescribed \(n-m\) leading coefficients, find the one which on \([0,1]\) deviates least from zero, and the deviation \(L\) itself. Thus the number \(s\) of deviation points (nodes) of the polynomial \(Q_n(x)=Y_n(x)/L\) is subject to the condition \(s\ge m+2\).

The problem posed is completely solved if \(Q_n(x)\) is a polynomial of class II, i.e., if the number of its nodes \(s>n/2+1\) for \(\max_{[0,1]} |Q_n|=1\). For this it is sufficient that \(m+2>n/2+1\). Hence:

\[ m<n<2m+2. \tag{1} \]

The results obtained by us for polynomials of class II make it possible to regard these polynomials as known; we shall present them.

The characteristics of polynomials of class II are: \(n\)—the degree, \(s\)—the number of nodes \((\sigma_i)_1^s\) on \([0,1]\), \(p\)—the number of repetitions of the sign of the polynomial at the boundaries of the intervals \((\sigma_i,\sigma_{i+1})\); they form the passport of the polynomial \([n,s,p]\); by \(q\) we denote the number of alternations of signs of the polynomial, so that \(p+q=s-1\) \(\left({}^{1}\right)\).

For example, the passport \([n,n+1,0]\) characterizes the polynomials \(\pm T_n(x)=\pm \cos n\arccos(2x-1)\) and only them; here \(q=n\).

The totality of the nodes of a polynomial, to each of which is assigned the sign taken by the polynomial at this point, is called its distribution: \(\underset{\pm}{(\sigma_i)_1^s}\).

The existence of polynomials of any passport, their properties, and their analytic construction are obtained by means of the simplest linear functionals. Each such \(Q_n(x)\) is an extremal polynomial of a certain segment-functional \(F\), namely: if we prescribe the parameters \((\mu_i)_0^n\) and set \(F(x^i)=\mu_i\) \((i=0,1,\ldots,n)\), \(F(P_n)=\sum_0^n p_i\mu_i\), then there exists an extremal polynomial \(Q_n(x)\) for which \(F(Q_n)=+N\), where \(N\) is the norm of \(F\); under the condition \(s>n/2+1\) the extremal polynomial is unique \(\left({}^{2}\right)\).

Theorem 1. All polynomials of a given passport \([n,s,p]\) form a family depending on \(l\) independent variable parameters, where

\[ l+s=n+1. \tag{2} \]

For the problem posed above, consider the family of polynomials of passport \([n,s,0]\), under the assumption that all polynomials of higher passports (i.e., with a larger number of nodes) have already been studied.

Theorem 2. The segment-functional

\[ (\mu_i)_0^n=0_0,\;0_1,\ldots,0_{n-l+1},\;1_{n-l},\;\vartheta_1,\;\vartheta_2,\ldots,\vartheta_l \tag{*} \]

completely determines all polynomials of passport \([n,s,0]\) (up to sign)
\[ Q_n(x,\vartheta_1,\vartheta_2,\ldots,\vartheta_l), \]
when the parameters \((\vartheta_i)_1^l\) vary in some bounded \(l\)-dimensional domain \(M_l\). At each interior point of \(M_l\) the segment determines one extremal polynomial of passport \([n,s,0]\); conversely, any given polynomial of such a passport is, up to sign, an extremal polynomial \((*)\) at one and only one interior point of \(M_l\).

Theorem 3. To points \((\vartheta_i)_1^l\) of the \(l\)-dimensional space lying outside or on the boundary of \(M_l\), the segment \((*)\) corresponds, as extremal polynomials, in a unique way (but not one-to-one) all (and only) polynomials of higher passports for which the number of alternations \(q>s-1\).

(The last assertion is obvious, since when nodes are added to an existing distribution \((\sigma_i)_1^s\), the number of alternations cannot decrease.)

Theorem 4. In the family of polynomials
\[ Q_n(x,\vartheta_1,\vartheta_2,\ldots,\vartheta_l), \]
defined by the segment \((*)\) in the domain \(M_l\), the deformation parameters \((\vartheta_i)\) may be replaced by other parameters completely equivalent to them, \((\theta_i)\)—the highest coefficients of the polynomial, namely:

\[ Q_n(x,\vartheta_1,\ldots,\vartheta_l) = \theta_l x^n+\theta_{l-1}x^{n-1}+\cdots+\theta_1x^{\,n-l+1} +y_{n-l}(\theta_1,\ldots,\theta_l)x^{\,n-l}+\cdots . \]

The one-to-one correspondence between the old and new parameters is expressed by the formulas

\[ \vartheta_1=-\frac{\partial y_{n-l}}{\partial \theta_1},\ldots, \vartheta_l=-\frac{\partial y_{n-l}}{\partial \theta_l}. \tag{3} \]

Let us note that Theorem 4 extends without change of formulation to polynomials of any other passport \([n,s,p]\); only in the case \(p>0\) will the defining functional, which also contains \(l\) variable parameters, be of a form different from \((*)\), and formulas (3) will likewise be replaced by others (see, for example, (3) for the passport \([n,n,1]\)).

We shall call the resolvent of a polynomial of class II \(Q_n(x)\) with distribution \((\sigma_i)_1^s\) the polynomial
\[ R_s(x)=\prod_1^s (x-\sigma_i)=x^s+\alpha_{s-1}x^{s-1}+\cdots . \]
In the case where the entire family of polynomials of a given passport
\[ Q_n(x,\theta_1,\theta_2,\ldots,\theta_l) \]
is considered (see Theorem 4), we have
\[ \alpha_i=\alpha_i(\theta_1,\theta_2,\ldots,\theta_l). \]
Polynomials of class II, distributed by passports, reveal a deep unity of analytic structure.

Theorem 5. For polynomials of any passport \([n,s,p]\)
\[ Q_n(x_1,\theta_1,\ldots,\theta_l) = \theta_l x^n+\theta_{l-1}x^{n-1}+\cdots+\theta_1x^{\,n-l+1} +y_{n-l}x^{\,n-l}+\cdots \]
the following differential relations hold:

\[ \frac{\partial Q_n}{\partial \theta_i} = \varphi_{i-1}(x)R_s(x), \qquad i=1,2,\ldots,l, \tag{4} \]

i.e., the derivative of a class-II polynomial with respect to each of its deformation parameters \(\theta_i\) is always a multiple of the resolvent of the polynomial.

The coefficients of the polynomial \(\varphi_{i-1}(x)\) are easily expressed in terms of derivatives of the coefficients \((y_i)\) from the condition that certain coefficients of the polynomial \(\partial Q_n/\partial\theta_i\) vanish.

Theorem 5 gives a method for the analytic construction of polynomials of class II as integrals of a system of ordinary differential equations; for this it is quite sufficient to use the first equation (4) with \(i=1\), i.e.

\[ \frac{\partial Q_n}{\partial \vartheta_1}=R_s(x). \tag{4′} \]

In this equation the resolvent is eliminated by means of the easily verified relation

\[ x(x-1)\frac{\partial Q_n}{\partial x} = n\vartheta_1 R_s(x)\psi_l(x). \tag{5} \]

Here \(\psi_l(x)=\prod_1^l (x-\lambda_i)\), where \((\lambda_i)\) are the roots of \(\partial Q_n/\partial x\) which are not among the nodes of \(Q_n\). We note that formula (5) is written for the case when the nodes \((\sigma_i)\) include \(0\) and \(1\); all other cases reduce to this one (for the particular case, see (4)).

After the elimination of \(R_s(x)\), equation (4) decomposes into a system of \(n\) equations, differential with respect to \(s-1\) unknown functions \(y_{s-1},\ldots,y_1\) (\(y_0=\pm 1\)) and algebraic with respect to \(l\) unknown coefficients of \(\psi_l(x)\).

Additional conditions for determining the constants of integration are obtained from the analytic form of the polynomial \(Q_n(x)\) on the boundary of the domain \(M_l\), where it becomes a polynomial of a higher passport. Thus, for example\(^5\), in order to construct polynomials of passport \([n,n-1,0]\) (the polynomials of N. I. Akhiezer), one must regard as already known the polynomials of passport \([n,n,0]\) (i.e. the polynomials of E. I. Zolotarev).

Considering all polynomials of class II \(Q_n(x)\) as known, let us return to the problem posed at the beginning of this note. Suppose that the leading coefficients \(Y_n(x)\), \(l+1\) in number, are given; denote them by \(b_n,b_{n-1},\ldots,b_{n-l}\). The number \(s\) of nodes of the polynomial \(Q_n(x)=Y_n(x)/L\) is subject to the condition \(s\ge n+1-l\); consequently, \(Q_n(x)\) is determined by the functional (*) in the domain \(M_l\), if \(s=n+1-l\), and by the same functional outside and on the boundary of \(M_l\), if \(Q_n(x)\) has a larger number of nodes.

Theorem 6. Among the polynomials determined by the segment (*) over the entire unbounded domain of variation \((\vartheta_i)_1^l\), there is one and only one polynomial \(Q_n(x)\) whose coefficients \(q_n,q_{n-1},\ldots,q_{n-l}\) are proportional respectively to the numbers \(b_n,b_{n-1},\ldots,b_{n-l}\); if \(1/M\) denotes the common value of the ratios \(q_i/b_i\), then \(|M|=L\) and \(Y_n(x)=|M|Q_n(x)\).

To find \(Q_n(x)\) and \(M\), in the family of polynomials \(Q_n(x,\vartheta_1,\ldots,\vartheta_l)\) determined by the segment (*), one must put \(\vartheta_i=\lambda b_i\) \((i=n,n-1,\ldots,n-l+1)\) and

\[ y_{n-l}(\lambda b_{n-l+1},\ldots,\lambda b_n)=\lambda b_{n-l}. \tag{6} \]

From equation (6) the unique (according to Theorem 6) real root \(\lambda=\lambda_0\) is determined. If it turns out that \(\lambda_0<0\), then the found \(Q_n(x)\) should be replaced by \(-Q_n(x)\). The deviation is \(L=1/\lambda_0\).

Leningrad Institute
of Aviation Instrument Engineering

Received
27 XII 1956

CITED LITERATURE

1. E. V. Voronovskaya, DAN, 99, No. 1 (1954).
2. E. V. Voronovskaya, Uspekhi Mat. Nauk, 9, issue 4 (1954).
3. E. V. Voronovskaya, DAN, 99, No. 2 (1954).
4. E. V. Voronovskaya, Uspekhi Mat. Nauk, 11, issue 1 (1956).
5. E. V. Voronovskaya, DAN, 110, No. 5 (1956).