MATHEMATICS

I. Yu. RYZHAKOV

ON A PROBLEM OF S. N. BERNSTEIN

(Presented by Academician V. I. Smirnov, 12 VI 1963)

Denote by \(C_{2\pi}\) the space of continuous functions \(x(t)\) defined on \([0,2\pi]\) and satisfying the condition \(x(0)=x(2\pi)\); by \(\mathcal P_{n,h}(\alpha,\beta)\) the class of trigonometric polynomials \(P_n(t)\),

\[ P_n(t)=\sum_{k=0}^{n}\left(a_k\cos kt+b_k\sin kt\right) \]

of order not exceeding \(n\), with real coefficients \(a_k\) and \(b_k\) and fixed \(a_h=\alpha\) and \(b_h=\beta\), \(0<h\le n\); and by \(\mathcal P_n\) the class of trigonometric polynomials \(P_n(t)\) subject to the condition

\[ \sup_{[0,2\pi]} |P_n(t)|=1 . \]

In \((^1)\), pp. 28–31, the problem was posed of finding

\[ M_{n,h}(\alpha,\beta)= \inf_{P_n(t)\in\mathcal P_{n,h}(\alpha,\beta)} \sup_{[0,2\pi]} |P_n(t)| . \]

There S. N. Bernstein obtained an estimate of this quantity and investigated its asymptotics.

In the present article the exact value of \(M_{n,h}(\alpha,\beta)\) is found and a polynomial \(\pi_{n,h}(\alpha,\beta,t)\in\mathcal P_{n,h}(\alpha,\beta)\) is constructed for which

\[ \sup_{[0,2\pi]} |\pi_{n,h}(\alpha,\beta,t)|=M_{n,h}(\alpha,\beta). \]

Since \(M_{n,h}(\alpha,\beta)=M_{m,1}(\alpha,\beta)\), where \(m=E(n/h)\), it is possible to restrict oneself to considering the class \(\mathcal P_{n,1}(\alpha,\beta)\). Moreover, if the polynomial \(\pi_{n,1}(\alpha,\beta,t)\) has the least deviation from zero on \([0,2\pi]\) in \(\mathcal P_{n,1}(\alpha,\beta)\), then the polynomial \(A\pi_{n,1}(\alpha,\beta,t+\varphi)\), where

\[ A^2=\frac{1}{\alpha^2+\beta^2}, \qquad \varphi=\operatorname{arc\,tg}\frac{\beta}{\alpha}, \]

belongs to \(\mathcal P_{n,1}(1,0)\), and its deviation from zero is the least in this class. Consequently, it suffices to indicate in \(\mathcal P_{n,1}(1,0)\) a polynomial with least deviation from zero and to find \(M_{n,1}(1,0)\). This last problem may be replaced by the following: in \(\mathcal P_n\), find a polynomial whose coefficient \(a_1\) is largest, and find

\[ \sup_{\mathcal P_n} a_1 . \]

Below precisely this formulation of the problem is considered.

The article uses the method of functionals proposed by E. V. Voronovskaya \((^2,^3)\).

A linear functional \(F\) on \(C_{2\pi}\) may be specified by a moment sequence \((\lambda_k)_{-\infty}^{\infty}\),

\[ \lambda_k=\int_0^{2\pi} e^{ikt}\,dg(t), \qquad k=0,\pm1,\ldots, \]

where \(g(t)\) is a real-valued function of bounded variation on \([0,2\pi]\), with

\[ \operatorname{Var}_{[0,2\pi]} g(t)=\|F\|. \]

A function \(x(t)\in C_{2\pi}\), \(\sup_{[0,2\pi]}|x(t)|=1\), will be called extremal for \(F\) if \(F(x)=\|F\|\).

The segment \((\lambda_k)_{-n}^{n}\), \(\lambda_k=\overline{\lambda}_{-k}\), defines a functional \(F_n\) on the set \(\mathcal P_n\),

\[ F_n(P_n)=\sum_{k=-n}^{n} c_k\lambda_k, \qquad P_n(t)=\sum_{k=-n}^{n} c_k e^{ikt}, \qquad P_n(t)\in\mathcal P_n . \]

The condition necessary and sufficient for \(P_n(t)\), \(P_n(t)\in\mathcal P_n\), \(P_n(t)\ne 1\), to be extremal for \(F_n\), consists in the fulfillment of the equalities

\[ P_n(t_j)=\operatorname{sign}\delta_j. \tag{1} \]

for all nonzero \(\delta_j\). Here \(0\le t_1<t_2<\cdots<t_s<2\pi\) are the deviation points of \(P_n(t)\) on \([0,2\pi)\), and \((\delta_j)_s\) are found from the system

\[ \lambda_k=\sum_{j=1}^{s}\theta_j^k\delta_j,\qquad \theta_j=e^{it_j},\qquad k=0,\pm1,\ldots,\pm n. \tag{2} \]

In the case when (1) is fulfilled, \(\|F_n\|=\sum_{j=1}^{s}|\delta_j|\), and the extension \(F_n=(\lambda_k)_{-n}^{n}\) to the sets \(\mathcal P_{n+p}\), \(p=1,2,\ldots\), with preservation of norms is unique and is realized by the numbers
\[ \lambda_{n+p}=\sum_{j=1}^{s}\theta_j^{\,n+p}\delta_j,\qquad p=1,2,\ldots . \]

Consider the functional \(F_{n,h}=(\lambda_k)_{-n}^{n}\), where \(\lambda_k=0\), \(k=0,1,\ldots,h-1,h+1,\ldots,n\); \(\lambda_n=1\); \(\lambda_k=\lambda_{-k}\). Obviously, \(F_{n,h}(P_n)=c_{-h}+c_h=a_h\). Thus, the problem is reduced to finding an extremal polynomial and the norm of the functional \(F_{n,1}\).

Lemma. Let

\[ \sigma_k= \begin{cases} \displaystyle \cos\frac{2k-1}{2(l+2)}\pi, & k=1,2,\ldots,\frac12(l+1),\\[6pt] \displaystyle \cos\frac{2k+1}{2(l+2)}\pi, & k=\frac12(l+3),\ldots,l+1, \end{cases} \qquad \text{for } l \text{ odd;} \]

\[ \sigma_k= \begin{cases} \displaystyle \cos\frac{k-1}{l+2}\pi, & k=1,2,\ldots,\frac12 l+1,\\[6pt] \displaystyle \cos\frac{k}{l+2}\pi, & k=\frac12 l+2,\ldots,l+2, \end{cases} \qquad \text{for } l \text{ even.} \]

There exists an algebraic polynomial \(H_l(x)\) of degree \(2l+1\) such that

\[ \sup_{[-1,1]}|H_l(x)|=1; \]

\[ H_l(\sigma_k)= \begin{cases} +1, & k=1,2,\ldots,\frac12(l+1),\\ -1, & k=\frac12(l+3),\ldots,l+1, \end{cases} \qquad \text{for } l \text{ odd;} \]

\[ H_l(\sigma_k)= \begin{cases} +1, & k=1,2,\ldots,\frac12 l+1,\\ -1, & k=\frac12 l+2,\ldots,l+2, \end{cases} \qquad \text{for } l \text{ even.} \]

Proof. The polynomial \(H_l(x)\) is the Hermite interpolation polynomial constructed at the nodes \(\sigma_k\):

\[ H_l(x)=2\,\frac{T_{l+2}^2(x)}{(l+2)^2x} \sum_{k=1}^{\frac12(l+1)} \sigma_k\frac{\sigma_k^4-(3\sigma_k^2-2)x^2}{(x^2-\sigma_k^2)^2} \qquad \text{for } l \text{ odd,} \]

\[ H_l(x)=\frac{T_{l+2}^2(x)}{(l+2)^4x} \left[ 1-2(x^2-1) \sum_{k=2}^{\frac12 l+1} \sigma_k\frac{\sigma_k^4-(3\sigma_k^2-2)x^2}{(x^2-\sigma_k^2)^2} \right] \qquad \text{for } l \text{ even.} \]

Here \(T_{l+2}(x)=\cos (l+2)\arccos x\). The condition \(\sup_{[-1,1]} |H_l(x)|=1\) is easily verified if one observes that the expression \(\sigma_k^4-(3\sigma_k^2-2)x^2\) is nonnegative on \([-1,1]\) for all \(k\).

Theorem. Let \(2l+1\le n<2l+3,\ l=0,1,2,\ldots\). The extremal polynomial for \(F_{n,1}\) is \(H_l(\mathrm{const})\).

Proof. We write the system (2) for the functional \(F_{2l+1,1}\) and the polynomial \(H_l(\cos t)\). In our case \(s=2(l+1)\), and the points \((\theta_j)_1^{2(l+1)}\) are the roots of the polynomial
\[ \frac{z^{2(l+2)}-(-1)^l}{z^2+1}. \]
From (2) we find
\[ \delta_k=(-1)^k \frac{\theta_k^{2(l+1)}} {\displaystyle\prod_{j=1}^{k-1}(\theta_k-\theta_j) \prod_{j=k+1}^{2(l+1)}(\theta_j-\theta_k)} \quad k=1,2,\ldots,2(l+1); \]
\[ \arg\delta_k=(k+1/2)\pi+(l+2)t_k \qquad \text{for \(l\) odd,} \]
\[ \arg\delta_k=(k+1)\pi+(l+2)t_k \qquad \text{for \(l\) even.} \]

It is now easy to establish the equalities (1) for the found \(\delta_k\). Thus the theorem is proved for \(n=2l+1\). Extending \(F_{2l+1,1}\) to the set \(\mathscr P_{2l+2}\) with preservation of the norm, we obtain
\[ \lambda_{2l+2}=\sum_{j=1}^{2(l+1)} \theta_j^{2l+2}\delta_j=0, \]
i.e. the indicated extension of the functional \(F_{2l+1,1}\) is the functional \(F_{2l+2,1}\). Since the polynomial \(H_l(\cos t)\) is extremal for \(F_{2l+1,1}\), it is also extremal for \(F_{2l+2,1}\).

Corollary 1. If \(2l+1\le n<2l+3,\ l=0,1,2,\ldots\), then
\[ \sup_{\mathscr P_n} a_1=\frac{2}{l+2}\ctg\frac{\pi}{2(l+2)}. \]

Indeed, for the indicated \(n\),
\[ \sup_{\mathscr P_n} a_1=\sup_{\mathscr P_{2l+1}} a_1 =\|F_{2l+1,1}\| =\sum_{k=1}^{2(l+1)}|\delta_k| =\sum_{k=1}^{2(l+1)} \frac{1}{\displaystyle\prod_{\substack{j=1\\ j\ne k}}^{2(l+1)}|\theta_j-\theta_k|} =\frac{2}{l+2}\ctg\frac{1}{2(l+2)}. \]

Corollary 2. If \(2l+1\le n<2l+3,\ l=0,1,2,\ldots\), then
\[ M_{n,1}(1,0)=\frac12(l+2)\tg\frac{\pi}{2(l+2)},\qquad \pi_{n,1}(1,0,t)=M_{n,1}(1,0)H_l(\cos t). \]

The polynomial \(H_l(\cos ht)\) is, obviously, extremal for \(F_{n,h}\) when
\((2l+1)h\le n<(2l+3)h,\ l=0,1,2,\ldots,\ 0<h\le n\). Any other extremal polynomial of this functional among its points of deviation on \([0,2\pi)\) must necessarily contain all points of deviation \((t_i)_1^{2h(l+1)}\) of the polynomial \(H_l(\cos ht)\). Hence it follows that

Corollary 3. If \((2l+1)h\le n<(2l+2)h,\ l=0,1,2,\ldots,\ 0<h\le n\), then the polynomial \(H_l(\cos ht)\) is the unique extremal polynomial \(F_{n,h}\). For \((2l+2)h\le n<(2l+3)h\) any extremal poly-

the polynomial \(P_{n,h}(t)\) of the functional \(F_{n,h}\) can be written in the form

\[ P_{n,h}(t)=H_l(\cos ht)+\psi(t)\prod_{j=1}^{2h(l+1)}\sin^2\frac{t-t_j}{2}, \]

where \(\psi(t)\) is a polynomial whose choice is restricted only by the condition \(P_{n,h}(t)\in \mathscr P_n\).

Corollary 4. If \((2l+1)h\leqslant n<(2l+3)h,\ l=0,1,2,\ldots,\ 0<h\leqslant n\), then

\[ M_{n,h}(\alpha,\beta)=\frac12\sqrt{\alpha^2+\beta^2}\,(l+2)\tg\frac{\pi}{2(l+2)}; \]

\[ \pi_{n,h}(x,\beta,t)=M_{n,h}(\alpha,\beta)\,H_l[\cos(ht-\varphi)],\qquad \varphi=\operatorname{arc\,tg}\frac{\beta}{\alpha}, \]

where \(\pi_{n,h}(\alpha,\beta,t)\) is the unique polynomial least deviating from zero on \([0,2\pi]\) in the class \(\mathscr P_{n,h}(\alpha,\beta)\), when \((2k+1)h\leqslant n<(2l+2)h\).

The general form of the polynomial of least deviation for the remaining \(n\) is clear from the preceding corollary.

Leningrad Institute
of Aviation Instrument Engineering

Received
12 VI 1963

REFERENCES

¹ S. N. Bernstein, Extremal Properties of Polynomials, 1937. ² E. V. Voronovskaya, Extremal Polynomials of Finite Functionals, L., 1955. ³ E. V. Voronovskaya. Proceedings of the Third All-Union Mathematical Congress, 3, Publishing House of the Academy of Sciences of the USSR, 1958.