MATHEMATICS

D. L. BERMAN

EXTREMAL PROBLEMS IN THE THEORY OF POLYNOMIAL OPERATORS (THE NONPERIODIC CASE)

(Presented by Academician S. N. Bernstein on 8 V 1961)

1°. Let \(\bar C\) be the space of functions \(f(x)\) continuous on the interval \([-1,1]\), with norm
\[ \|f(x)\|=\max_{-1\le x\le 1}|f(x)|; \]
let \(k\ge 0\) be an integer, and let \(f^{(k)}(x)\) be the \(k\)-th derivative of \(f(x)\); let \(B_\sigma\) be the set of all entire transcendental functions \(f(z)\) of exponential type of degree \(\le \sigma\), bounded on the real axis.

By \(\bar\Omega_n^{(k)}\) we denote the set of all linear operators \(V(f,x)\) from \(\bar C\) into \(\bar C\) having the property that
\[ V(f,x)=f^{(k)}(x), \]
if \(f(x)\) is an algebraic polynomial of degree \(\le n\). Along with the set \(\bar\Omega_n^{(k)}\), consider the set \(\bar\Omega_{n,n}^{(k)}\) of all linear operators \(V_n(f,x)\) from \(\bar C\) into \(\bar C\) possessing the properties: 1) for any \(f\in\bar C\), \(V_n(f,x)\) is a polynomial of degree \(\le n\); 2) if \(f(x)\) is a polynomial of degree \(\le n\), then \(V_n(f,x)=f^{(k)}(x)\). Obviously, \(\bar\Omega_{n,n}^{(k)}\subset\bar\Omega_n^{(k)}\). It is easy to construct such an operator which belongs to \(\bar\Omega_n^{(k)}\) but does not belong to \(\bar\Omega_{n,n}^{(k)}\). Therefore \(\bar\Omega_{n,n}^{(k)}\ne\bar\Omega_n^{(k)}\). Put
\[ \bar\rho_n^{(k)}=\inf_{V\in\bar\Omega_n^{(k)}}\|V\|;\qquad \bar\rho_{n,n}^{(k)}=\inf_{V_n\in\bar\Omega_{n,n}^{(k)}}\|V_n\|. \]
In (1), analogous definitions and notation were given for the periodic case*. As is seen from Theorems 1 and 2, the nonperiodic case differs essentially from the periodic case.

2°. Theorem 1. The equalities
\[ \bar\rho_n^{(k)}=T_n^{(k)}(1),\qquad k=0,1,2,\ldots,n, \]
hold, where \(T_n(x)=\cos n\arccos x\). For every \(0\le k\le n\) one can specify an operation \(V\in\bar\Omega_n^{(k)}\) such that \(\|\bar V\|=\bar\rho_n^{(k)}\).

Theorem 2. The equalities
\[ \bar\rho_{n,n}^{(k)}=T_n^{(k)}(1),\qquad k=1,2,\ldots,n, \]
hold.

For every \(1\le k\le n\) one can specify an operation \(\bar V_{n,n}\) such that
\[ \|\bar V_{n,n}\|=\bar\rho_{n,n}^{(k)}. \]
It follows from Theorems 1 and 2 that, for any \(k\ge 1\),
\[ \bar\rho_{n,n}^{(k)}/\bar\rho_n^{(k)}=1, \]
whereas in the periodic case
\[ \lim_{n\to\infty}\left(\frac{\bar\rho_{n,n}^{(k)}}{\bar\rho_n^{(k)}}:\frac{4}{\pi^2}\ln n\right)=1. \]

* Only the overbar was absent in the notation.

In the \(2\pi\)-periodic case, among all linear operators \(U_n(f,\theta)\) from \(C\) to \(C\) that take functions from \(C\) into trigonometric polynomials of order \(n\) and have the property that \(U_n(f,\theta)=f^{(k)}(\theta)\) if \(f(\theta)\) is a trigonometric polynomial of order \(n\), the operator \(S_n^{(k)}(f,\theta)\) has the smallest norm, where \(S_n(f,\theta)\) is the partial sum of order \(n\) of the Fourier series of the function \(f(\theta)\). Therefore it might seem that in the nonperiodic case, in the class of operators \(\overline{\Omega}_{n,n}^{(k)}\), the operator \(\bar{\sigma}_n^{(k)}(f,x)\) has the smallest norm, where \(\bar{\sigma}_n(f,x)\) is the partial sum of order \(n\) of the Fourier series of the function \(f(x)\) in the P. L. Chebyshev polynomials \(\{T_i(x)\}_{i=0}^{\infty}\). In fact this is not so. A simple calculation shows that the norm of the operator \(\bar{\sigma}_n'(f,x)\) satisfies the inequality

\[ \|\bar{\sigma}_n'\|\geq c n^2\ln n. \tag{1} \]

Since the norm of the extremal operator of the class \(\overline{\Omega}_{n,n}^{(1)}\) is equal to \(n^2\), it follows from inequality (1) that the operator \(\bar{\sigma}_n'(f,x)\) is not extremal in the class \(\overline{\Omega}_{n,n}^{(1)}\). From \((2)\) one can obtain that, for any \(k\geq 1\), the operator

\[ \overline{V}_{n,n}(f,x)=\sum_{j=0}^{n} f(x_j)l_j^{(k)}(x), \]

has the smallest norm in each of the classes \(\overline{\Omega}_n^{(k)}\) and \(\overline{\Omega}_{n,n}^{(k)}\), where \(\{l_j(x)\}_{j=0}^{n}\) are the fundamental Lagrange polynomials constructed at the nodes

\[ x_j=\cos \frac{j\pi}{n},\qquad j=0,1,2,\ldots,n. \tag{\(m_0\)} \]

Thus, for \(k\geq 1\) there is an analogy between the operators \(S_n^{(k)}(f,\theta)\) and \(\overline{V}_{n,n}(f,x)\). It is curious that the problem of computing \(\bar{\rho}_{n,n}^{(k)}\) and of finding an extremal operation in the class \(\overline{\Omega}_{n,n}^{(k)}\) is closely connected with the problem of finding the best system of nodes for parabolic interpolation. Let \((\mathfrak{M})\) be the set of all possible sequences of numbers

\[ -1\leq x_n^{(n)}<x_{n-1}^{(n)}<\cdots<x_0^{(n)}\leq 1. \tag{\(m\)} \]

By \(\{l_j(x,m)\}_{j=0}^{n}\) we denote the fundamental Lagrange polynomials corresponding to the points \((m)\). Put

\[ M_n^{(k)}(m)=\max_{-1\leq x\leq 1}\sum_{j=0}^{n}|l_j^{(k)}(x)|. \]

From \((2)\) and Theorem 2 the following theorem follows:

Theorem 3. For any \(1\leq k\leq n\), the equalities

\[ \inf_{V_n\in \overline{\Omega}_{n,n}^{(k)}} \|V_n\| = \inf_{m\in(\mathfrak{M})} M_n^{(k)}(m) = T_n^{(k)}(1) \]

hold.

For \(k=0\), Theorem 3 ceases to be true, since one can prove that

\[ \lim_{n\to\infty}\frac{\lambda_n^{(0)}}{\bar{\rho}_{n,n}^{(0)}}\geq \frac{\pi}{2}, \qquad \text{where } \lambda_n^{(0)}=\inf_{m\in(\mathfrak{M})}M_n^{(0)}(m). \]

3°. Some results from \((1)\) admit an extension to the case of entire transcendental functions of exponential type. Let \(E\) be a complete linear normed functional space consisting of functions \(f(x)\) defined on the whole real axis \(-\infty<x<\infty\). We shall assume that \(E\) has the following properties: 1) if \(f(x)\in E\), then \(f(z)\) is an entire transcendental function of exponential type; 2) if \(f(x)\in E\) and \(y\) is an arbitrary real number, then \(f(x+y)\in E\) and \(\|f(x+y)\|=\|f(x)\|\). Such spaces were introduced in \((3)\). It is known \((3)\) that if \(f(x)\in E\), then \(f^{(k)}(x)\in E\). Let \(D\) be the differentiation operator.

By \(U(f,x)\) we shall denote an arbitrary linear operator from \(E\) into \(E\) having the property that, if \(f(x)\in B_\sigma\), where \(\sigma>0\) is a fixed number, then

\[ U(f,x)=(D\sin\alpha-\sigma\cos\alpha)^k f(x), \]

where \(\alpha\) is an arbitrary real number. The set of all such operators will be denoted by \(\Omega_{\sigma,\alpha}^{(k)}\). Put
\[ \rho_{\sigma,\alpha}^{(k)}=\inf_{U\in\Omega_{\sigma,\alpha}^{(k)}}\|U\|. \]
The question arises of computing \(\rho_{\sigma,\alpha}^{(k)}\) and of finding in the class \(\Omega_{\sigma,\alpha}^{(k)}\) an extremal operation \(\overline U\) for which \(\|\overline U\|=\rho_{\sigma,\alpha}^{(k)}\). The solution of this question is given by the theorem:

Theorem 4. For any real \(\alpha\) the equalities
\[ \rho_{\sigma,\alpha}^{(k)}=\sigma^k,\qquad k=0,1,2,\ldots \]
hold.

The operation
\[ \overline U(f,x)=\sigma^k\sum_{j_1,j_2,\ldots,j_k} f\left(x+\sum_{s=1}^k\beta_{j_s}\right)\prod_{s=1}^k\rho_{j_s}, \tag{2} \]
\[ \beta_{j_s}=\frac{j_s\pi-\alpha}{\sigma},\qquad \rho_{j_s}=(-1)^{j_s-1}\frac{\sin^2\alpha}{(\alpha-j_s\pi)^2},\qquad -\infty<j_s<\infty,\quad s=1,2,\ldots,k, \]
belongs to the class \(\Omega_{\sigma,\alpha}^{(k)}\) and satisfies the equalities \(\|\overline U\|=\rho_{\sigma,\alpha}^{(k)}=\sigma^k\).

Corollary. Let \(U(f,x)\) be an arbitrary linear operator from \(E\) into \(E\) having the property that for every \(f\in B_\sigma\),
\(U(f,x)=f^{(k)}(x)\). Then \(\|U\|\ge \sigma^k\). Equality is attained for the operation

\[ \overline U(f,x)=\sigma^k\sum_{j_1,j_2,\ldots,j_k} f\left(x+\sum_{s=1}^k\beta_{j_s}\right)\prod_{s=1}^k\rho_{j_s}, \]
\[ \beta_{j_s}=\frac{2j_s-1}{2\sigma}\pi,\qquad \rho_{j_s}=(-1)^{j_s-1}\frac{4}{(2j_s-1)^2\pi^2},\qquad -\infty<j_s<\infty, \]
\[ s=1,2,\ldots,k. \]

It is obvious that the corollary follows directly from Theorem 4 when \(\alpha=\pi/2\).

Remark. It can be proved that for every \(f\in B_\sigma\) the equality
\[ (D\sin\alpha-\sigma\cos\alpha)^k f(x)=\overline U(f,x), \]
holds, where \(\overline U(f,x)\) is defined by formula (2). Hence we obtain that for every \(f(x)\in B_\sigma\) the inequality

\[ \|(D\sin\alpha-\sigma\cos\alpha)^k f\|\le \sigma^k\|f\| \tag{3} \]
is valid.

For \(\alpha=\pi/2\), inequality (3) becomes the well-known inequality of S. N. Bernstein (4). For \(k=1\), inequality (3) is also well known (5).

Received
5 IV 1961

REFERENCES

1. D. L. Berman, DAN, 138, No. 4 (1961).
2. D. L. Berman, DAN, 87, 167 (1952).
3. N. I. Akhiezer, Lectures on the Theory of Approximation, Moscow—Leningrad, 1947.
4. S. N. Bernstein, Collected Works, vol. 1, Publishing House of the Academy of Sciences of the USSR, 1952, p. 269.
5. G. Pólya, G. Szegő, Problems and Theorems in Analysis, part 1, Moscow—Leningrad, 1937, p. 143, problem 165.