D. L. BERMAN

MAJORANTS FOR THE DERIVATIVE OF A POLYNOMIAL

(Presented by Academician S. N. Bernstein on 5 VI 1964)

1°. Let \(\Pi_n\) denote the set of all polynomials of degree \(\leq n\). Let \(k\) be a fixed integer \(\geq 1\). For a given value of \(x\), \(-\infty < x < \infty\), put

\[ M_k(x)=M_{k,n}(x)=\sup |P_n^{(k)}(x)|, \]

where \(P_n^{(k)}(x)\) denotes, as usual, the \(k\)-th derivative, and the supremum is taken over all \(P_n\in\Pi_n\) satisfying on the segment \([-1,1]\) the inequality

\[ |P_n(x)|\leq 1. \]

We shall call the function \(M_k(x)\) the majorant of V. A. Markov, since for arbitrary \(k\) it was first studied by V. A. Markov \((^1)\).

It is known that the computation of \(M_k(x)\) is connected with great difficulties, which arise at those points \(x\) at which the extremal polynomials are the polynomials of E. I. Zolotarev \((^2)\). Therefore there arises the important question of finding the asymptotic value of \(M_k(x)\). This question was solved comparatively long ago by S. N. Bernstein.

S. N. Bernstein \((^3)\) proved that at all interior points of the interval \([-1,1]\) the asymptotic equality

\[ M_{k,n}(x)\sim \left(\frac{n^2}{1-x^2}\right)^{k/2}. \]

holds.

A simple proof of the complete theorem of V. A. Markov, which concerns all points of the interval \([-1,1]\), was given by S. N. Bernstein in \((^4)\).

2°. Let a sequence of points be given

\[ -1\leq x_0 < x_1 < \cdots < x_n \leq 1. \tag{\(\mathfrak{M}_n\)} \]

Put, for given \(k\) and \(x\), \(-\infty < x < \infty\),

\[ N_k(x)=N_{k,n}(x,\mathfrak{M}_n)=\sup |P_n^{(k)}(x)|, \]

where the supremum is taken over all \(P_n\in\Pi_n\) satisfying the inequalities

\[ |P_n(x_j)|\leq 1,\qquad j=0,1,2,\ldots,n. \tag{1} \]

The present note is devoted mainly to the study of the connection between the functions \(N_k(x)\) and \(M_k(x)\).

3°. Concerning \(N_k(x)\) we shall prove the theorem:

Theorem 1. For any point \(x\) and any sequence \((\mathfrak{M}_n)\) the equality

\[ N_k(x)=\sum_{i=0}^{n}|l_i^{(k)}(x)|, \]

holds, where \(\{l_i(x)\}_{i=0}^{n}\) are the fundamental Lagrange polynomials constructed for the sequence of numbers \((\mathfrak{M}_n)\). The extremal polynomial \(Q_n(x)\), for which

\[ N_k(x)=Q_n^{(k)}(x), \]

is uniquely determined from the conditions

\[ Q_n(x_i)=\operatorname{sign} l_i^{(k)}(x),\qquad i=0,1,2,\ldots,n . \tag{2} \]

Proof. Let \(R_n\in \Pi_n\) and let it satisfy inequality (1). By the Lagrange interpolation formula we have

\[ R_n(x)=\sum_{i=0}^{n} R_n(x_i)l_i(x). \]

Consequently,

\[ R_n^{(k)}(x)=\sum_{i=0}^{n} R_n(x_i)l_i^{(k)}(x). \]

Therefore, by virtue of (1), we obtain that

\[ |R_n^{(k)}(x)|\leq \sum_{i=0}^{n} |l_i^{(k)}(x)|. \tag{3} \]

Let us now recall the definition of \(N_k(x)\). In view of (3), one may conclude that

\[ N_k(x)\leq \sum_{i=0}^{n} |l_i^{(k)}(x)|. \tag{4} \]

On the other hand, the polynomial \(Q_n(x)\), which is uniquely determined by condition (2), satisfies the equalities

\[ Q_n^{(k)}(x)=\sum_{i=0}^{n} Q_n(x_i)l_i^{(k)}(x) =\sum_{i=0}^{n} |l_i^{(k)}(x)|. \]

Thus, by the definition of \(N_k(x)\), we have:

\[ \sum_{i=0}^{n} |l_i^{(k)}(x)|\leq N_k(x). \tag{5} \]

It follows from inequalities (4) and (5) that

\[ N_k(x)=\sum_{i=0}^{n} |l_i^{(k)}(x)|=Q_n^{(k)}(x). \]

Theorem 2. The function \(N_k(x)\) has the following properties:

1) If the system of points \((\mathfrak{M}_n)\) is located symmetrically with respect to the origin, then \(N_k(x)\) is an even function, \(N_k(-x)=N_k(x)\).

2) \(N_k(x)\) is continuous on the entire number axis.

3) The derivative \(N_k'(x)\) exists at all points of the number axis, with the exception of the roots of the polynomials \(\{l_i^{(k)}(x)\}_{i=0}^n\), where \(N_k'(x)\) has discontinuities of the first kind. If \(x_{j,i}^{(k)}\) is a root of the polynomial \(l_j^{(k)}(x)\), then

\[ |N_k'(x_{j,i}^{(k)}+0)-N_k'(x_{j,i}^{(k)}-0)| =2|l_j^{(k+1)}(x_{j,i}^{(k)})|. \]

4°. It is obvious that at each point

\[ M_k(x)\leq N_k(x). \tag{6} \]

It is natural to ask under what conditions the majorants coincide, i.e., \(M_k(x)=N_k(x)\).

For the study of this question the following theorem is useful:

Theorem 3. Suppose that, for a given point \(x\), there exist a polynomial \(R\in \Pi_n\) and a sequence of points \(\mathfrak{M}_{n,x}\) of the form \((\mathfrak{M}_n)\) such that:

1) \(|R(x)|\leq 1,\quad -1\leq x\leq 1\).

2) \(R(x_i^{(n)})=\pm(-1)^i,\quad i=0,1,2,\ldots,n.\)

3) \(\operatorname{sign} l_i^{(k)}(x)=-\operatorname{sign} l_{i+1}^{(k)}(x);\quad i=0,1,2,\ldots,n.\)

Then at this point the equalities hold

\[ M_k(x)=N_k(x)=|R^{(k)}(x)|. \tag{7} \]

Proof. According to Theorem 1,

\[ N_k(x)=\sum_{i=0}^{n}\left|l_i^{(k)}(x)\right|. \]

We now use property 3) of the system of points \(\mathfrak M_{n,x}\); then we obtain that

\[ N_k(x)=\pm \sum_{i=0}^{n}(-1)^i l_i^{(k)}(x). \tag{8} \]

Let us now take into account that the polynomial \(R\) satisfies condition 2). Therefore equality (8) can be written in the form

\[ N_k(x)=\pm \sum_{i=0}^{n} R(x_i)l_i^{(k)}(x). \]

Consequently,

\[ N_k(x)=\left|R^{(k)}(x)\right|. \tag{9} \]

On the other hand, since \(R\in \Pi_n\) and \(|R(x)|\leq 1,\ -1\leq x\leq 1\), it follows from the definition of the function \(M_k(x)\) that

\[ M_k(x)\geq \left|R^{(k)}(x)\right|. \tag{10} \]

Inequalities (10), (6), and (9) lead to equality (7).

\(5^\circ\). Let us now suppose that the system of points \((\mathfrak M_n)\) consists of the numbers

\[ x_j=\cos\frac{n-j}{n}\pi,\qquad j=0,1,2,\ldots,n. \tag{\(\mathfrak M_n^{(0)}\)} \]

Denote the roots of the equations

\[ \bigl((x+1)T_n'(x)\bigr)^{(k)}=0,\qquad T_n(x)=\cos n\arccos x,\qquad \bigl((x-1)T_n'(x)\bigr)^{(k)}=0 \]

respectively by

\[ \xi_1<\xi_2<\cdots<\xi_{n-k} \quad\text{and}\quad \eta_1<\eta_2<\cdots<\eta_{n-k}. \]

V. A. Markov \((^1)\) proved that the inequalities

\[ \xi_1<\eta_1<\xi_2<\eta_2<\cdots<\xi_{n-k}<\eta_{n-k} \]

hold.

Put

\[ E(M)=(-\infty,\xi_1]+[\eta_1,\xi_2]+\cdots+[\eta_{n-k-1},\xi_{n-k}]+[\eta_{n-k},\infty). \]

From V. A. Markov’s considerations \((^1)\) it follows that if, as the system of points \((\mathfrak M_n)\), one takes the system \((\mathfrak M_n^{(0)})\), and as \(R(x)\) the polynomial \(T_n(x)\), then at every point of the set \(E(M)\) all the conditions of Theorem 3 are satisfied. Therefore, from Theorem 3 there follows

Theorem 4. At every point of the set \(E(M)\) the equalities

\[ M_k(x)=N_k(x)=\left|T_n^{(k)}(x)\right| \]

hold.

In connection with this theorem there arises the question of the relation between the majorants \(M_k(x)\) and \(N_k(x)\) on the set \(CE(M)=Z\setminus E(M)\), where \(Z\) is the whole number axis. This question is answered by

Theorem 5. For every point \(x\in Z\setminus E(M)\) there exists a system of points \(\mathfrak M_{n-1}\), depending on \(x\) and \(k\), such that

\[ M_{k,n}(x)=N_{k,n-1}(x,\mathfrak M_{n-1}). \]

This theorem is obtained from the results of V. A. Markov and Theorem 3.

Received
4 VI 1964

CITED LITERATURE

\(^1\) V. A. Markov, On functions least deviating from zero on a given interval, St. Petersburg, 1892.
\(^2\) E. I. Zolotarev, Collected Works, 2, Publishing House of the Academy of Sciences of the USSR, 1932.
\(^3\) S. N. Bernstein, Collected Works, 1, Publishing House of the Academy of Sciences of the USSR, 1952, p. 153.
\(^4\) S. N. Bernstein, Collected Works, 2, Publishing House of the Academy of Sciences of the USSR, 1954, p. 281.