D. L. BERMAN

LINEAR POLYNOMIAL OPERATIONS

(Presented by Academician S. N. Bernstein on 20 XI 1961)

1.

Let \(\widetilde C\) be the space of all continuous \(2\pi\)-periodic functions \(f(x)\) with norm
\[ \|f\|_{\widetilde C}=\max_{0\le x<2\pi}|f(x)|; \]
let \(p\) and \(q\) be integers satisfying the inequalities \(0\le q\le p\). By \(\Delta_{q,p}\) we shall denote the set of all linear operators \(U_{q,p}\) from \(\widetilde C\) into \(\widetilde C\) satisfying the following conditions: 1) for any \(f\in\widetilde C\), \(U_{q,p}(f,x)\) is a trigonometric polynomial of order \(\le p\); 2) if \(f(x)\) is a trigonometric polynomial of order \(\le q\), then \(U_{q,p}(f,x)=f(x)\). Put
\[ \rho_{q,p}=\inf_{U_{q,p}\in\Delta_{q,p}}\|U_{q,p}\|. \]
Since \(\Delta_{0,p}\supset\Delta_{1,p}\supset\cdots\supset\Delta_{p,p}\), we have
\[ \rho_{0,p}\le \rho_{1,p}\le\cdots\le\rho_{p-1,p}\le\rho_{p,p}. \]
There arises the natural question of finding, in each class of operators \(\Delta_{q,p}\), \(q=0,1,2,\ldots,p\), an operator of least norm and of computing \(\rho_{q,p}\). The solution of this question is given by Theorem 1.

Theorem 1. In the class of operators \(\Delta_{q,p}\), the operator of least norm is
\[ \overline U_{q,p}(f,x)=\frac1\pi\int_0^{2\pi} f(x+t)\left[D_q(t)+\sum_{i=1}^{p-q}\widetilde\alpha_i\cos(q+i)t\right]\,dt, \]
where \(D_q(t)\) is the Dirichlet kernel of order \(q\), and the numbers \(\{\widetilde\alpha_i\}_{i=1}^{p-q}\) are determined from the condition that the integral
\[ I(\alpha_1,\alpha_2,\ldots,\alpha_{p-q}) = \int_0^\pi\left|D_q(t)+\sum_{i=1}^{p-q}\alpha_i\cos(q+i)t\right|\,dt \]
be minimal. Moreover,
\[ \rho_{q,p}=\frac2\pi\int_0^\pi\left|D_q(t)+\sum_{i=1}^{p-q}\widetilde\alpha_i\cos(q+i)t\right|\,dt. \]

This theorem is a consequence of the general Theorem 2 from \((^1)\). With the aid of Theorem 1 it is easy to obtain the following results.

A. In the class of operators \(\Delta_{0,p}\), the partial Fejér sum of order \(p\) has least norm, and
\[ \rho_{0,p}=1. \]

B. In the class of operators \(\Delta_{p-1,p}\), the operator
\[ \overline U_{p-1,p}=\frac1{2\pi}\int_0^{2\pi} f(x+t)\sin pt\,\cot\frac t2\,dt \tag{1} \]
has least norm, and
\[ \rho_{p-1,p}=\frac1{2\pi}\int_0^{2\pi}\left|\sin pt\,\cot\frac t2\right|\,dt\simeq \frac4{\pi^2}\ln p. \]

The operator (1) often occurs in the theory of Fourier series (see, for example, \((^{2,3})\)), but its extremal property apparently has not been noted until now.

C. In the class \(\Delta_{p,p}\), the partial sum of the series has least norm ...

Fourier sum \(S_p(f,x)\) and \(\rho_{p,p}=L_p\), where \(L_p\) is the Lebesgue constant of order \(p\). This result is known \((^4)\).

Let us note that assertion A can also easily be obtained without Theorem 1.

2. For given \(q\) and \(p\) one can construct the de la Vallée-Poussin partial sum

\[ \sigma_{q,p}=\sigma_{q,p}(f,x)= \frac{S_q(f,x)+S_{q+1}(f,x)+\ldots+S_p(f,x)}{p-q+1}. \]

It is obvious that \(\sigma_{q,p}\in\Delta_{q,p}\). The question arises: when is the partial sum \(\sigma_{q,p}\) extremal in the class of operators \(\Delta_{q,p}\)?

Theorem 1 makes it possible to answer this question as well.

Theorem 2. In order that \(\sigma_{q,p}\) be an extremal operator in the class \(\Delta_{q,p}\), it is necessary and sufficient that \(2q\) be divisible by \((p-q+1)\).

Since \(2q\) is divisible by \((p-q+1)\) for \(q\) respectively equal to \(0,(p-1),p\), it follows, according to Theorem 2, that \(\sigma_{0,p}, \sigma_{p-1,p}, \sigma_{p,p}\) are extremal operators respectively for the classes \(\Delta_{0,p}, \Delta_{p-1,p}, \Delta_{p,p}\). It is not difficult to see that we have again obtained assertions A, B, C, since \(\sigma_{0,p}, \sigma_{p-1,p}, \sigma_{p,p}\) coincide respectively with the Fejér partial sum of order \(p\), with the operator (1), and with \(S_p(f,x)\). In this connection the result of Theorem 3 is of interest.

Theorem 3. If \((p+1)\) is a prime number, then among the partial sums \(\sigma_{0,p}, \sigma_{1,p},\ldots,\sigma_{p-1,p},\sigma_{p,p}\), only the partial sums \(\sigma_{0,p}, \sigma_{p-1,p}, \sigma_{p,p}\) are extremal operators respectively in their classes \(\Delta_{0,p}, \Delta_{p-1,p}, \Delta_{p,p}\). Every other de la Vallée-Poussin partial sum is not an extremal operator in its class.

3. Let us consider one more problem which can be solved with the help of Theorem 1. Let \(\overline{\Delta}_{q,2p}\) be the set of all linear operators \(W_{q,2p}(f,x)\) from \(\widetilde C\) into \(\widetilde C\) possessing the following properties: 1) for every \(f\in\widetilde C\), \(W_{q,2p}(f,x)\) is a trigonometric polynomial of order \(\le p\); 2) if \(f\in\widetilde C\) is a polynomial of order \(\le q\), then \(W_{q,2p}(f,x)=\overline f(x)\), where \(\overline f(x)\) is the polynomial conjugate to \(f(x)\). Put

\[ \overline\rho_{q,2p}=\inf_{W\in\overline\Delta}\|W_{q,1p}\|. \]

Theorem 4. In the class \(\overline{\Delta}_{2p-1,2p}\) the operator

\[ W_{2p-1,2p}(f,x)=\frac1\pi\int_0^{2\pi} f(x+t)\Phi(t)\,dt, \]

has the smallest norm, where

\[ \Phi(t)=-\left(\frac{\sin 2pt}{2}+\sum_{k=1}^{2p-1}\sin kt\right); \]

\[ \overline\rho_{2p-1,2p}=\frac2\pi\left(\frac11+\frac13+\ldots+\frac1{2p-1}\right). \]

4. As is known \((^{5,6})\), the general definition of a linear trigonometric polynomial operation of type \(\Phi_n\) is as follows: 1) \(U_n(f,x)\) is a linear operator from \(\widetilde C\) into \(\widetilde C\); 2) for every \(f\in\widetilde C\), \(U_n(f,x)\) is a trigonometric polynomial of order not exceeding \(n\); 3) for every trigonometric polynomial \(f(x)\) of order not exceeding \(n\) the equality

\[ U_n(f,x)=\sigma_n(f,x), \]

holds, where

\[ \sigma_n(f,x)=\int_0^{2\pi} f(x,t)\Phi_n(t)\,dt \tag{*} \]

and \(\Phi_n(t)\) is a given trigonometric polynomial of order \(n\).

Suppose that for every \(f \in \widetilde C\) the relation

\[ \sigma_n(f,x) \to f(x), \qquad n \to \infty . \tag{2} \]

holds uniformly.

The question is whether, starting from an arbitrary operation \(U_n(f,x)\) of type \(\Phi_n\), one can recover the function \(f(x)\) as \(n \to \infty\). The answer to this question is positive in principle, for Theorem 5 holds.

Theorem 5. Let, for every \(f \in \widetilde C\), the relation (2) hold uniformly. Then for every \(f(x)\), every \(x\), and every operation \(U_n(f,x)\) of type \(\Phi_n\), there exists a real number \(\tau_n\), depending on \(f, x, n\), such that

\[ \lim_{n\to\infty} U_n(f_{\tau_n}, x-\tau_n)=f(x), \]

where \(f_t(x)=f(x+t)\).

The proof of the theorem follows from the equality (5) *

\[ \frac{1}{2\pi}\int_0^{2\pi} U_n(f_t, x-t)\,dt=\sigma_n(f,x), \tag{3} \]

if the mean-value theorem is applied to its left-hand side. Here one must take into account that \(U_n(f_t,x-t)\) is a continuous function of \(t\).

1. At present there exist two methods for studying polynomial operations. The starting point of one method is Faber’s work \((^7)\), and the starting point of the other method is Fejér’s work \((^8)\). The second method is well known, since it is presented in many textbooks \((^{9-12})\). In this connection it is curious that the second method is, in a certain sense, a consequence of the first method. The basis of the first method is formula (3) and its special case

\[ \frac{1}{2\pi}\int_0^{2\pi} U_n(f_t, x-t)\,dt=S_n(f,x), \tag{4} \]

which holds if, for every polynomial \(T\) of order \(n\), \(U_n(T,x)=T(x)\). Applying the mean-value theorem to the left-hand side of (4), we obtain that there exists a real \(\alpha\) such that

\[ U_n(f_\alpha, x-\alpha)=S_n(f,x). \]

In particular, there exists an \(\alpha\) such that

\[ U_n(f_\alpha, -\alpha)=S_n(f,0). \]

This equality and its various modifications constitute the basis of the second method.

\[ \text{*} \]

• Incidentally, this equality generalizes to the case where the convolution (*) is replaced by the integral

\[ \sigma(f,x)=\int_0^{2\pi} f(t)\Phi_{n,m}(x,t)\,dt, \]

where \(\Phi_{n,m}(x,t)\) is a trigonometric polynomial of order \(n\) with respect to \(x\) and of order \(m\) with respect to \(t\), with \(m \leq n\). In this case equality (3) takes the form:

\[ \frac{1}{2\pi}\int_0^{2\pi} U_n(f_t,x-t)\,dt = \frac{1}{2\pi}\int_0^{2\pi} f(z)\,dz \int_0^{2\pi} \Phi_{n,m}(x-t,z-t)\,dt . \]

6. Let

\[ 0 \leqslant x_0^{(n)} < x_1^{(n)} < \cdots < x_{2n}^{(n)} \leqslant 2\pi, \tag{\(m_1\)} \]

\[ 0 \leqslant y_0^{(n)} < y_1^{(n)} < \cdots < y_{2n}^{(n)} \leqslant 2\pi \tag{\(m_2\)} \]

be two arbitrary systems of nodes of trigonometric interpolation. By \(L_n(f,x,m_i)\), \(i=1,2\), denote the corresponding Lagrange interpolation polynomials of order \(n\). Marcinkiewicz \({}^{13}\) proved the equality

\[ \frac{1}{2\pi}\int_0^{2\pi} L_n(f_t,x-t,m_i)\,dt=S_n(f,x);\qquad i=1,2 \]

(which is a special case of (4)). Therefore

\[ \frac{1}{2\pi}\int_0^{2\pi}\bigl[L_n(f_t,x-t,m_1)-L_n(f_t,x-t,m_2)\bigr]\,dt=0. \]

Since the integrand depends continuously on \(t\), by the mean-value theorem there exists a real \(a\), \(a=a(f,x,m_1,m_2)\), such that

\[ L_n(f_a,x-a,m_1)=L_n(f_a,x-a,m_2). \tag{5} \]

It is not difficult to verify that

\[ L_n(f_a,x-a,m_i)=L_n(f,x,m_i+a),\qquad i=1,2, \]

where by \(m_i+a\) is denoted the system of nodes obtained from the system \(m_i\) by a shift by \(a\). Therefore equality (5) takes the form

\[ L_n(f,x,m_1+a)=L_n(f,x,m_2+a). \]

Thus the following theorem has been proved:

Theorem 6. Let \(m_i\), \(i=1,2\), be arbitrary systems of interpolation nodes and \(f\in \widetilde C\). Then for any point \(x\) one can specify a shift of the nodes \(m_1\) and \(m_2\) such that the values at the point \(x\) of the Lagrange interpolation polynomials constructed for the shifted nodes are equal to each other.

Received
28 X 1961

References

\({}^{1}\) D. L. Berman, DAN, 138, No. 4 (1961).
\({}^{2}\) N. K. Bari, Trigonometric Series, 1961.
\({}^{3}\) A. Zygmund, Trigonometric Series, 1939.
\({}^{4}\) S. M. Lozinskii, DAN, 61, No. 2 (1948).
\({}^{5}\) D. L. Berman, DAN, 85, No. 1 (1952).
\({}^{6}\) S. M. Lozinskii, DAN, 89, No. 5 (1953).
\({}^{7}\) G. Faber, Jahresber. DMV, 23, 192 (1914).
\({}^{8}\) L. Fejér, Math. Zs., 32, 426 (1930).
\({}^{9}\) V. L. Goncharov, Theory of Interpolation and Approximation of Functions, 1934.
\({}^{10}\) I. P. Natanson, Constructive Theory of Functions, 1949.
\({}^{11}\) P. P. Korovkin, Linear Operators and Approximation Theory, 1959.
\({}^{12}\) A. F. Timan, Theory of Approximation of Functions, 1960.
\({}^{13}\) I. Marcinkiewicz, Acta Litt. Sci. Szeged, 8, 127 (1937).