MATHEMATICS

S. Ya. KHAVINSON

ON A CLASS OF EXTREMAL PROBLEMS FOR POLYNOMIALS

(Presented by Academician S. N. Bernstein on 20 X 1959)

Let \(E\) be a locally convex linear topological space (real or complex); \(p(x)\) a continuous symmetric convex functional in \(E\); \(E_n\) an \(n\)-dimensional linear space with a locally convex topology, consisting of points \((\lambda)=(\lambda_1,\ldots,\lambda_n)\), real or complex; \(p_1(\lambda)=p_1(\lambda_1,\ldots,\lambda_n)\) a continuous symmetric convex functional in \(E_n\). Let \(y, x_1,\ldots,x_n\) be linearly independent elements of \(E\). We shall call linear combinations \(\sum_{\nu=1}^n \lambda_\nu x_\nu\) polynomials. We shall be interested in the following two problems.

Problem I\(_{[p,p_1]}\). Find

\[ \alpha=\inf_{\lambda_1,\ldots,\lambda_n}\left[p\left(y-\sum_{\nu=1}^{n}\lambda_\nu x_\nu\right)+p_1(\lambda_1,\ldots,\lambda_n)\right]. \tag{1} \]

Problem II. Find

\[ \beta=\sup |f(y)| \tag{2} \]

over all continuous linear functionals \(f\in E^*\) satisfying the conditions

\[ |f(x)|\leq p(x),\quad x\in E; \tag{3} \]

\[ \left|\sum_1 \lambda_\nu f(x_\nu)\right|\leq p_1(\lambda_1,\ldots,\lambda_n) \tag{4} \]

for all \((\lambda)\in E_n\).

The close connection between Problems I and II is contained in the following main theorem:

Theorem 1. \(\alpha=\beta\).

The proof is based on duality lemmas of functional analysis, which slightly generalize analogous lemmas previously used in investigations of extremal problems \(({}^{1-4})\), etc. For lack of space we do not formulate these propositions.

Let us give some concrete examples of our problems.

Problem I\(_{[C(Q),p_1]}\). Find

\[ \alpha=\inf_{(\lambda)}\left[\max_{t\in Q}\left|y(t)-\sum_1^n \lambda_\nu x_\nu(t)\right|+p_1(\lambda_1,\ldots,\lambda_n)\right]; \tag{5} \]

\(Q\) is some compact set; \(y(t), x_1(t),\ldots,x_n(t)\) are continuous functions.

The problem dual to it:

Problem II\(_{[C(Q),p_1]}\). Find

\[ \beta=\sup \left|\int_Q y(t)\,dg\right| \tag{6} \]

over all measures \(g\) on \(Q\) satisfying the inequalities

\[ \int_Q |dg| \leqslant 1 \tag{7} \]

and (4) (here \(f(x_i)=\int_Q x_i(t)\,dg\)).

Further:

Problem \(\mathrm{I}_{[L^p,\rho_1]}\), \(p\geqslant 1\). Find

\[ \alpha=\inf_{(\lambda)} \left[ \left\{ \int_a^b \left|\, y(t)-\sum_1^n \lambda_\nu x_\nu(t)\,\right|^p dt \right\}^{1/p} +\rho_1(\lambda_1,\ldots,\lambda_n) \right]. \tag{8} \]

Problem \(\mathrm{II}_{[L^q,\rho_1]}\), \(q>1\). Find

\[ \beta=\sup_{\alpha(t)} \left| \int_a^b \alpha(t)y(t)\,dt \right| \tag{9} \]

over all \(\alpha(t)\) for which

\[ \int_a^b |\alpha(t)|^q dt \leqslant 1 \tag{10} \]

and (4) holds \(\left(f(x_i)=\int_a^b \alpha(t)x_i(t)\,dt\right)\). In the case \(q=\infty\), inequality (10) becomes

\[ |\alpha(t)|\leqslant 1 \quad \text{almost everywhere on } [a,b]. \tag{11} \]

The problems \(\mathrm{I}_{[L^p,\rho_1]}\) and \(\mathrm{II}_{[L^q,\rho_1]}\) are dual if \(\frac1p+\frac1q=1\). Specifying now \(\rho_1\), we obtain, for example, the problem:

Problem \(\mathrm{I}_{[C(Q),\{\varepsilon_\nu\}]}\). Find

\[ \inf_{(\lambda)} \left[ \max_{t\in Q} \left|\, y(t)-\sum_1^n \lambda_\nu x_\nu(t)\,\right| +\sum_1^n \varepsilon_\nu |\lambda_\nu| \right], \]

where \(\varepsilon_\nu\geqslant 0\) \((\nu=1,\ldots,n)\) are given.

Dual to it is

Problem \(\mathrm{II}_{[C(Q),\{\varepsilon_\nu\}]}\). Find (6) under conditions (7) and

\[ \left| \int_Q x_\nu(t)\,dg \right| \leqslant \varepsilon_\nu, \quad \nu=1,\ldots,n. \tag{12} \]

In an analogous way we obtain the problems \(\mathrm{I}_{[L^p,\{\varepsilon_\nu\}]}\) and \(\mathrm{II}_{[L^q,\{\varepsilon_\nu\}]}\). As \(\rho_1\) one may take various other functionals, for example,

\[ \left\{\sum_1^n \varepsilon_\nu |\lambda_\nu|^r\right\}^{1/r}, \quad r\geqslant 1, \]

where \(\varepsilon_\nu\geqslant 0\), \(\nu=1,\ldots,n\), are given. Or

\[ \rho_1(\lambda_1,\ldots,\lambda_n)= \left\{ \int_a^b \left|\sum_1^n \lambda_\nu x_\nu(t)\right|^r dt \right\}^{1/r} \]

and so on.

Theorem 2. There exist extremal polynomials

\[ P^*=\sum_1^n \lambda_\nu^* x_\nu \]

in problem I and extremal functionals in problem II. In order that the polynomial \(P^*\) be extremal in problem I, and the functional \(f^*\), satisfying-

for (3) and (4) to be extremal in Problem II, it is necessary and sufficient that

\[ f^*(y-P^*)=e^{i\theta}p(y-P^*); \tag{13} \]

\[ f^*(P^*)=\sum_{1}^{n}\lambda_\nu^* f^*(x_\nu)=e^{i\theta}p_1(\lambda_1^*,\ldots,\lambda_n^*); \tag{14} \]

\(\theta\) is a real number.

Transferring in a natural way to our case of linear topological spaces the terminology used by M. G. Krein in (¹), we obtain the following criteria for uniqueness of solutions, analogous to those given in (¹) for the problems considered there:

Theorem 3. If among the solutions of Problem I there exists at least one solution \(P^*\) such that \(y-P^*\) is a normal element, then the solution of Problem II is unique up to a constant factor \(K\), \(|K|=1\). If among the solutions of Problem II there is at least one normal one, then the solution of Problem I is unique. In particular, if \(E\) is a strictly normed space with norm \(p(x)\), then the solution of Problem I is unique.

Simple examples show that strict normedness of the functional \(p_1(\lambda)\) does not ensure uniqueness.

We shall denote by \(B\) the convex body in the Euclidean space \(R_n\) formed by the points \((f(x_1),\ldots,f(x_n))\), where \(f(x)\) satisfies (3). The following deserves to be noted.

Theorem 4. If \(E\) is a strictly normed space with norm \(p(x)\), and \(f^*(x)\) is a functional extremal in Problem II, then the point \((f^*(x_2),\ldots,\ldots,f^*(x_n))\) is necessarily an interior point for \(B\). (If \(E\) is not strictly normed, then the point \((f^*(x_1,\ldots,f^*(x_n))\) may also lie on the boundary of \(B\).)

Consider the problems \(\mathrm{I}_{[C(Q),p_1]}\) and \(\mathrm{II}_{[C(Q),p_1]}\).

Theorem 5. There exist subsets \(Q_r\subset Q\), consisting of \(r\) points (\(r\le n+1\) in the real case and \(r\le 2n+1\) in the complex case), such that the solutions of the problems \(\mathrm{I}_{[C(Q_r),p_1]}\), \(\mathrm{II}_{[C(Q_r),p_1]}\) coincide with the solutions of the problems \(\mathrm{I}_{[C(Q),p_1]}\) and \(\mathrm{II}_{[C(Q),p_1]}\), respectively.

In the case \(p_1\equiv 0\) we obtain a well-known theorem of approximation theory ((⁵–⁷), see also (⁸)).

Theorem 6. In order that the polynomial

\[ P^*(t)=\sum_{1}^{n}\lambda_\nu^*x_\nu(t) \]

be extremal in the problem \(\mathrm{I}_{[C(Q),p_1]}\), it is necessary and sufficient that there exist \(r\) points \(t_1,\ldots,t_r\subset Q\) (\(r\le n+1\) in the real case and \(r\le 2n+1\) in the complex case), positive numbers \(\mu_1,\ldots,\mu_r\), and real numbers \(\theta_1,\ldots,\theta_r\) such that the relations

\[ y(t_j)-P^*(t_j)=Me^{-i\theta_j},\qquad M=\max_{t\in Q}|y(t)-P^*(t)|,\qquad j=1,\ldots,r; \tag{15} \]

\[ \sum_{j=1}^{r}\mu_j e^{i\theta_j}P^*(t_j)=p_1(\lambda_1^*,\ldots,\lambda_n^*), \tag{16} \]

hold, while for any polynomial

\[ P(t)=\sum_{1}^{n}\lambda_\nu x_\nu(t) \]

\[ \left|\sum_{j=1}^{n}\mu_j e^{i\theta_j}P(t_j)\right|\le p_1(\lambda_1,\ldots,\lambda_n). \tag{17} \]

Finally,

\[ \sum_{1}^{n}\mu_j=1. \tag{18} \]

For \(p_1(\lambda_1,\ldots,\lambda_n)\equiv 0\), Theorem 6 gives, in the complex case, the theorem of E. Ya. Remez \(({}^{9,10})\) (see also \(({}^{11})\)), which is another form of the earlier theorem found by A. N. Kolmogorov \(({}^{12})\), and in the real case the classical theorem of P. L. Chebyshev follows from it.

Theorem 7. If \(x_1(t),\ldots,x_n(t)\) is a P. L. Chebyshev system on \(Q\), then for an arbitrary functional \(p_1(\lambda_1,\ldots,\lambda_n)\) there exist numbers \(0<\delta_1\leqslant\delta_0\) such that for all \(\delta\leqslant\delta_0\) in problems I and \(\mathrm{II}_{[C(Q),\delta p_1]}\), for an arbitrary continuous \(y(t)\) on \(Q\), for the number of points \(r\) in Theorems 5 and 6 we have: \(r=n+1\) in the real case, \(n+1\leqslant r\leqslant 2n+1\) in the complex case; for all \(\delta\leqslant\delta_1\), for arbitrary \(y(t)\) continuous on \(Q\), the solution of problem \(\mathrm{I}_{[C(Q),p_1]}\) is unique.

We note that, if the smallness of \(\delta\) is not required, both assertions of the theorem cease to be valid.

Theorem 8. Let \(x_1(t),\ldots,x_n(t)\) be a real P. L. Chebyshev system. There exists a number \(\delta_2\), \(0<\delta_2\leqslant\delta_1\), such that for all \(\delta\leqslant\delta_2\) the unique extremal polynomial in the problem \(\mathrm{I}_{[C(Q),\delta p_1]}\) with arbitrary continuous \(y(t)\) will be the polynomial \(P^*\) of least deviation from \(y(t)\) on \(Q\).

For the complex case Theorem 8 is valid under one additional condition on the system \(x_1(t),\ldots,x_n(t)\).

Theorem 8 makes it possible to solve effectively the problem \(\mathrm{II}_{[C(Q),\{\varepsilon_\nu\}]}\) for small \(\varepsilon_\nu\) (this smallness is expressed by the requirement that certain determinants be positive). For example, the following is solved effectively for small \(\varepsilon_\nu\).

Problem. Find

\[ \sup_g \left| \int_{-1}^{+1} t^n\,dg \right|, \]

if

\[ \int_{-1}^{+1} |dg|\leqslant 1,\qquad \left|\int_{-1}^{+1} t^\nu\,dg\right|\leqslant \varepsilon_\nu,\qquad \nu=1,\ldots,n-1. \]

We do not write out the explicit formulas for lack of space.

For problems I and \(\mathrm{II}_{[L^p,p_1]}\), Theorem 2 leads to criteria for an extremal polynomial which generalize the known criteria for the polynomial of best approximation in the \(L^p\) metrics. In the case \(p>1\), uniqueness of the solution follows from Theorem 3. In the case \(p=1\), uniqueness does not always hold.

Theorem 9. If \(x_1(t),\ldots,x_n(t)\) is a P. L. Chebyshev system on \((a,b)\), the function \(y(t)\) is continuous on \((a,b)\), and \(p_1(\lambda_1,\ldots,\lambda_n)\) is such that from the fulfillment of (11) and (4) (here

\[ f(x_\nu)=\int_a^b \alpha(t)x_\nu(t)\,dt \]

) it follows that the point \((f(x_1),\ldots,f(x_n))\) belongs to the interior of the body \(B\) (see Theorem 4), then the extremal polynomial \(P^*\) for the problem \(\mathrm{I}_{[L^1,p_1]}\) is unique. In this case the difference \(y(t)-P^*(t)\) changes sign at no fewer than \(n\) points.

Received
29 IX 1959

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