D. L. BERMAN

ON SOME EXTREMAL PROBLEMS IN THE THEORY OF POLYNOMIAL OPERATORS

(Presented by Academician S. N. Bernstein, November 10, 1964)

1°. Let us introduce notation. \(\Pi_n\) is the set of all trigonometric polynomials of order \(\leq n\); \(L_1\) is the set of all summable \(2\pi\)-periodic functions; \(E\) is a linear normed function space possessing the following properties: 1) the elements of \(E\) are functions from \(L_1\); 2) if \(f \in E\), then the shifted function \(f_t(x)=f(x+t)\), for any \(-\infty<t<\infty\), also belongs to \(E\), and \(\|f_t\|=\|f\|\); 3) \(E\) contains the set of all trigonometric polynomials. The most important special cases of the space \(E\) are: the space \(C\) of all continuous \(2\pi\)-periodic functions, and the space \(L_r\) of all \(2\pi\)-periodic functions summable to the \(r\)-th power. Put

\[ \sigma_n(f,x)=\int_0^{2\pi} f(x+t)\Phi(t)\,dt, \]

where

\[ \Phi(t)=\frac{r_0}{2}+\sum_{k=1}^{n} r_k \sin(kt+\alpha_k). \tag{1} \]

Denote by \(\Omega_n^\Phi(E)\) the set of all linear operations \(U_n\) from \(E\) into \(E\) possessing the property that \(U_n(t_n)=\sigma_n(t_n)\) if \(t_n\in\Pi_n\). The set \(\Omega_{n,n}^\Phi(E)\) consists of all linear operations \(U_{n,n}\) from \(E\) into \(E\) for which the conditions are fulfilled: 1) for every \(f\in E\), \(U_{n,n}(f)\in\Pi_n\); 2) if \(t_n\in\Pi_n\), then \(U_{n,n}(t_n)=\sigma_n(t_n)\). It is obvious that \(\Omega_{n,n}^\Phi(E)\subset\Omega_n^\Phi(E)\). There exist operations belonging to \(\Omega_n^\Phi(E)\) but not belonging to \(\Omega_{n,n}^\Phi(E)\). Introduce the numbers

\[ \rho_n(E)=\rho_n^\Phi(E)= \inf_{U_n\in\Omega_n^\Phi(E)} \|U_n\|; \qquad \rho_{n,n}(E)=\rho_{n,n}^\Phi(E)= \inf_{U_{n,n}\in\Omega_{n,n}^\Phi(E)} \|U_{n,n}\|. \]

It is clear that

\[ \rho_{n,n}(E)\geq \rho_n(E). \tag{2} \]

The ratio \(\rho_{n,n}(E):\rho_n(E)\) depends essentially on the space \(E\) (2). In \(L_2\) the following theorem holds.

Theorem 1. The equalities

\[ \rho_{n,n}^\Phi(L_2)=\rho_n^\Phi(L_2)=\pi \max_{j=0,1,\ldots,n} r_j \tag{3} \]

hold.

Let us outline the proof. It is easy to see that

\[ \left\|U_n\left(\frac{\cos kx}{\|\cos kx\|_E}\right)\right\|=\pi r_k,\qquad k=0,1,2,\ldots,n, \]

where \(\|\cos kx\|_E\) is the norm of \(\cos kx\) in the metric of \(E\). Therefore

\[ \rho_n(E)\geq \pi r_{j_0},\qquad r_{j_0}=\max_{j=0,1,2,\ldots,n} r_j . \tag{4} \]

With the aid of Parseval’s equality it is easy to obtain that

\[ \|\sigma_n\|_{L_2}\leq \pi r_{j_0}. \]

Consequently, since \(\sigma_n\in \Omega^\Phi_{n,n}(E)\), we have

\[ \rho_{n,n}(E)\leq \pi r_{j_0}. \tag{5} \]

From (2) and (4), (5), (3) follows.

Of particular interest is the case when

\[ \Phi(t)=\frac{1}{\pi}D_n^{(k)}(t), \tag{6} \]

where \(D_n(t)\) is the Dirichlet kernel and \(D_n^k(t)\) is the derivative of order \(k\). In this case \(r_{j_0}=n^k/\pi\), and therefore equality (3) takes the form

\[ \rho_{n,n}(L_2)=\rho_n(L_2)=n^k . \tag{7} \]

The validity of equality (7) was pointed out to the author by A. N. Kolmogorov.

\(2^\circ\). In the case \(E=\widetilde C\) or \(E_1=\widetilde L_1\), the study of the ratio \(\rho_{n,n}(E):\rho_n(E)\) is considerably more difficult. In [1] it was established that if \(\Phi(t)\) is defined according to (6), then

\[ \lim_{n\to\infty}\left(\frac{\rho_{n,n}(\widetilde C)}{\rho_n(\widetilde C)}:\frac{4}{\pi^2}\ln n\right)=1. \tag{8} \]

(8) also remains valid when \(\widetilde C\) is replaced by \(\widetilde L_1\). The question arises of studying \(\rho_{n,n}(E):\rho_n(E)\) in the case of an arbitrary kernel \(\Phi(t)\) of the form (1).

Theorem 2. Let \(E=\widetilde C\) or \(E=\widetilde L_1\). Suppose that the kernel \(\Phi(t)\) satisfies the conditions \(r_0=0\) and

\[ \Phi_1(t)=r_n+2\sum_{k=1}^{n-1} r_n\cos\bigl[(n-k)t+\alpha_n-\alpha_k\bigr]\geq 0,\qquad -\infty<t<\infty . \tag{9} \]

Then the equality holds

\[ \rho^\Phi_{n,n}(E)/\rho^\Phi_n(E)=\int_0^{2\pi}|\Phi(t)|\,dt\ /\ \pi r_n . \]

In the course of the proof the following lemma plays an important role:

Lemma. Suppose that the kernel \(\Phi(t)\) satisfies the conditions of Theorem 2. Then, for an arbitrary space of type \(E\), the equality holds

\[ \rho^\Phi_n(E)=\pi r_n . \tag{10} \]

We outline the proof. Put

\[ \overline U(f,x)=\overline U(f,g,x)=\int_0^{2\pi} f(x+t)g(nt+\alpha_n)\Phi_1(t)\,dt, \tag{11} \]

where \(\Phi_1(t)\) is defined according to (9), and \(g(t)\) is an arbitrary \(2\pi\)-periodic continuous function whose Fourier expansion begins with \(\sin t\). It is not difficult to verify that \(\overline U\in\Omega_n^\Phi(E)\). In view of (11) we have

\[ \|\overline U(f)\|\leq \|f\|\int_0^{2\pi}|g(nt+\alpha_n)|\,|\Phi_1(t)|\,dt . \tag{12} \]

The legitimacy of passing to the norm under the integral sign in the case \(E=C\) or \(E=\widetilde L_1\) is obvious. In the general case it is easy to justify. Following the arguments of F. Riesz \((^2)\), put

\[ g(t)=\sin t-r^2\sin 3t+r^4\sin 5t-\cdots =\frac{(1-r^2)\sin t}{1+2r^2\cos 2t+r^4},\quad 0<r<1. \]

Since

\[ \int_0^{2\pi}|g(t)|\,dt=\frac{4}{r}\operatorname{arc\,tg} r \]

and \(\Phi_1(t)\geq 0\), it follows from (12) that
\(\|\overline U\|\leq 4r_n r^{-1}\operatorname{arc\,tg} r\). Therefore

\[ \rho_n^\Phi(E)\leq 4r_n\operatorname{arc\,tg} r/r. \]

Letting \(r\to 1\), we obtain

\[ \rho_n^\Phi(E)\leq \pi r_n. \tag{13} \]

On the other hand, by virtue of (4) one always has
\(\rho_n^\Phi(E)\geq \pi r_n\). From this and (13), (10) follows.

Remark. Equality (8) is a special case of Theorem 2.

\(3^\circ\). Let \(L=L_{[a,b]}\) be the set of all summable on the segment \([a,b]\) functions \(f(x)\) with \(\|f\|=\int_a^b |f|\,dx\). Consider some subspace \(\Sigma\) of the space \(L\) and a linear operator \(U\) from \(L\) into \(\Sigma\). Put that \(\{\chi_i\}_{i=1}^p\) are eigenfunctions of the operator \(U\), \(U(\chi_k)=\mu_k\chi_k\), \(k=1,2,\ldots,p\). We shall assume that \(|\chi_k(x)|\leq \Delta\), \(k=1,2,\ldots,p\), where \(\Delta\) does not depend on \(x\) or \(k\).

Theorem 3. If (in \(L\)) there exist a sequence of functionals \(\{\psi_k\}_{k=1}^p\) and a sequence of positive numbers \(\{a_k\}_{k=1}^p\) such that for any \(f\in\Sigma\) the inequality

\[ \sum_{k=1}^p a_k|\psi_k(f)|\leq C\int_a^b |f|\,dx, \]

where \(C\) is a constant, holds, then

\[ \|U\|\geq \sum_{k=1}^p a_k\mu_k\psi_k(\chi_k^2)\,/\,C\Delta(b-a). \]

A special case of this theorem is

Theorem 4. Let \(U_n\) be a linear operator from \(\widetilde L_1\) into \(\Pi_n\), and let the polynomials \(\{t_k^{(n)}\}_{k=1}^m\), \(m\leq n\), be its fixed points, where \(t_k^{(n)}\) is a polynomial of degree \(k\). Then, if the polynomials \(\{t_k^{(n)}\}_{k=1}^n\) are uniformly bounded in the aggregate, then

\[ \|U_n\|\geq C_1\ln\frac{n}{n-m+1}. \]

From this estimate the theorem follows:

Theorem 5. Let \(\overline{\lim}_{n\to\infty}\frac{m_n}{n}=1\), and let \(\{U_n\}_{n=1}^\infty\) be a sequence of linear operators from \(\widetilde L_1\) into \(\widetilde L_1\), where \(U_n\), \(n=1,2,\ldots\), has the properties: a) for any \(f\in\widetilde L_1\), \(U_n(f)\in\Pi_n\); b) there exists a sequence of polynomials \(\{t_k^{(n)}\}_{k=1}^{m_n}\), \(m_n\leq n\), \(n=1,2,\ldots\) \((t_k^{(n)}\) is a polynomial of degree \(k)\), uniformly bounded in the aggregate, such that \(U_n(t_k^{(n)})=t_k^{(n)}\), \(k=1,2,\ldots,m_n\), \(n=1,2\).

Then for some \(f \in \widetilde{L}_{1}\) the equality

\[ \overline{\lim_{n\to\infty}} \,\|U_n(f)-f\|_{\widetilde{L}_{1}}=\infty . \]

This theorem is a strengthening of the Lozinskii–Kharshiladze theorem \((^3)\), since in the latter it is required that any polynomial \(t_n\in \Pi_n\) be a fixed point of the operator \(U_n\). The conditions of Theorem 5 are satisfied, for example, by the polynomials

\[ t_k^{(n)}(x)=\sin\bigl(kx+a_k^{(n)}\bigr), \]

\[ k=1,2,\ldots,m_n;\qquad n=1,2,\ldots\quad (m_n\le n). \]

Leningrad Institute of Soviet Trade
named after F. Engels

Received
2 X 1964

REFERENCES

\(^1\) D. L. Berman, DAN, 138, No. 4 (1961). \(^2\) F. Riesz, C. R., 158 (1914). \(^3\) I. P. Natanson, Constructive Function Theory, 1949.