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)}_n'(f,x)) satisfies the inequality}^{\infty}). In fact this is not so. A simple calculation shows that the norm of the operator (\bar{\sigma

[
|\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}). From ((2)) one can obtain that, for any (k\geq 1), the operator}^{(1)

[
\overline{V}{n,n}(f,x)=\sum(x),}^{n} f(x_j)l_j^{(k)
]

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

[
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})) be the set of all possible sequences of numbers}^{(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

[
-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)})} M_n^{(k)
=
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).})}M_n^{(0)
]

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