UDC 513.88:513.83+517.948.35+517.948.5:518

MATHEMATICS

D. L. BERMAN

EXTREMAL PROBLEMS IN THE THEORY OF POLYNOMIAL OPERATIONS

(Presented by Academician S. N. Bernstein on December 9, 1965)

1°. Let us introduce notation. \(\Pi_n\) is the set of all trigonometric polynomials of order \(\leq n\). \(\widetilde L\) is the set of all summable \(2\pi\)-periodic functions. \(E\) is a linear normed functional space having the following properties: 1) the elements of \(E\) are functions from \(\widetilde L\); 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 moreover \(\|f_t\|\leq \|f\|\); 3) \(E\) contains the set of all trigonometric polynomials. The most important special cases of the space \(E\) are: the space \(\widetilde C\) of all continuous \(2\pi\)-periodic functions, and the space \(\widetilde L_r\) of all \(2\pi\)-periodic functions summable to the \(r\)-th power. To the polynomial

\[ \Phi(t)=\sum_{k=0}^{n} r_k \sin(kt+\alpha_k) \]

we assign the polynomial

\[ \widetilde{\Phi}(t)=r_n+2\sum_{k=0}^{n-1} r_k \cos[(n-k)t+\alpha_n-\alpha_k], \]

which we shall call associated (after F. Riesz (1)) with the polynomial \(\Phi\). Put

\[ \sigma(f,x)=\int_{0}^{2\pi} f(x+t)\Phi(t)\,dt . \]

Denote by \(\Omega_n^\Phi(E)\) the set of all linear operations \(U\) from \(E\) into \(E\) having the property that \(U(t_n)=\sigma(t_n)\), if \(t_n \in \Pi_n\). We also introduce the set \(\Omega_{n,n+m}^\Phi(E)\), \(m\geq 0\), consisting of all linear operations \(U\) from \(E\) into \(E\) for which the following conditions are satisfied: 1) for every \(f \in E\), \(U(f)\in \Pi_{n+m}\); 2) if \(t_n \in \Pi_n\), then \(U(t_n)=\sigma(t_n)\). Introduce the quantities

\[ \rho_n=\rho_n^\Phi(E)=\inf_{U\in\Omega_n^\Phi(E)} \|U\|;\qquad \rho_{n,n+m}=\rho_{n,n+m}^\Phi(E)=\inf_{U\in\Omega_{n,n+m}^\Phi(E)} \|U\|. \tag{1} \]

Since \(\Omega_{n,n+m}^\Phi \subset \Omega_n^\Phi\), it follows that \(\rho_{n,n+m}\geq \rho_n\), \(m\geq 0\). The present note is devoted to the study of the quantities (1).

2°. Theorem 1. If

\[ \widetilde{\Phi}\left(\varphi_r-\frac{\alpha_n}{n}\right)\geq 0,\qquad \varphi_r=\frac{2r-1}{2n}\pi,\qquad r=1,2,\ldots,2n, \tag{2} \]

then \(\rho_n=\pi r_n\).

In this case the extremal operation is given by the equality

\[ \overline{U}(f,x)=\sum_{r=1}^{2n}\lambda_r f\left(x+\varphi_r-\frac{\alpha_n}{n}\right); \qquad \lambda_r=(-1)^{r-1}\frac{\pi}{2n}\,\widetilde{\Phi}\left(\varphi_r-\frac{\alpha_n}{n}\right), \tag{3} \]

i.e. \(\|\overline{U}\|=\rho_n\).

We outline the proof. It is known from (2) that for any \(t\in \Pi_n\), \(\overline{U}=\sigma(t)\).

Consequently, \(\overline{U}\in \Omega_n^{\Phi}\). Since the inequalities (2) hold, by virtue of (3) we have

\[ \|\overline{U}(f)\|\leq \|f\|\frac{\pi}{2n}\sum_{r=1}^{2n} \widetilde{\Phi}\left(\varphi_r-\frac{\alpha_n}{n}\right). \tag{4} \]

But it is easy to verify that

\[ \frac{1}{2n}\sum_{r=1}^{2n} \widetilde{\Phi}\left(\varphi_r-\frac{\alpha_n}{n}\right)=r_n. \]

Therefore (4) takes the form

\[ \|\overline{U}(f)\|\leq \pi r_n\|f\| \quad\text{or}\quad \|\overline{U}\|\leq \pi r_n. \tag{5} \]

Hence

\[ \rho_n\leq \pi r_n. \tag{6} \]

On the other hand, in [3] it was proved that

\[ \rho_n\geq \pi r_n. \tag{7} \]

From (6) and (7) it follows that

\[ \rho_n=\pi r_n. \tag{8} \]

Since \(\|\overline{U}\|\geq \rho_n\), by virtue of (8) we have

\[ \|\overline{U}\|\geq \pi r_n. \tag{9} \]

The inequalities (5) and (9) lead to the conclusion that

\[ \|\overline{U}\|=\pi r_n. \tag{10} \]

Finally, from (8) and (10) it follows that \(\|\overline{U}\|=\rho_n=\pi r_n\).
Theorem 1 strengthens the lemma from [3].

\(3^\circ\). We now consider the quantity \(\rho_{n,2n}^{\Phi}\) for the case when \(E=\widetilde{C}\) or \(E=L\). In [4, 5] the following theorem was proved.

Theorem. In the space \(\widetilde{C}\) or \(L\), \(\rho_{n,n+m}^{\Phi}\) satisfies the equality

\[ \rho_{n,n+m}^{\Phi}=\inf_{\alpha_k,\beta_k} =I(\alpha_1,\alpha_2,\ldots,\alpha_m;\beta_1,\beta_2,\ldots,\beta_m), \]

where

\[ I(\alpha_1,\alpha_2,\ldots,\alpha_m;\beta_1,\beta_2,\ldots,\beta_m)= \]

\[ =\int_{0}^{2\pi} \left|\Phi(t)+ \sum_{j=n+1}^{m+n} (\alpha_{j-n}\cos jt+\beta_{j-n}\sin jt) \right|\,dt. \tag{11} \]

If the integral (11) attains its minimum for
\(\alpha_j=\alpha_j^{(0)}\), \(\beta_j=\beta_j^{(0)}\), \(j=1,2,\ldots,m\), then the extremal operation is computed by the formula

\[ \overline{U}(f,x)= \int_{0}^{2\pi} f(x+t)\left[ \Phi(t)+ \sum_{j=n+1}^{m+n} (\alpha_{j-n}^{(0)}\cos jt+\beta_{j-n}^{(0)}\sin jt) \right]\,dt, \tag{12} \]

i.e. \(\|\overline{U}\|=\rho_{n,n+m}^{\Phi}\).

This theorem may be formulated in the following equivalent form:

Theorem 2. In order that the operator (12) have the smallest norm in the class of operators \(\Omega_{n,n+m}^{\Phi}\), it is necessary and sufficient that the equalities

\[ \int_{0}^{2\pi} \operatorname{sign}\left[\Phi(t)+\sum_{j=n+1}^{n+m}\left(\alpha_{j-n}^{(0)}\cos jt+\beta_{j-n}^{(0)}\sin jt\right)\right] e^{i(n+k)t}\,dt=0,\quad 1\leq k\leq m. \tag{13} \]

hold. In this case

\[ \left\|\Omega_{n,1n+m}^{\Phi}\right\| = \int_{0}^{2\pi} \left| \Phi(t)+\sum_{j=n+1}^{n+m}\left(\alpha_{j-n}^{(0)}\cos jt+\beta_{j-n}^{(0)}\sin jt\right) \right|\,dt. \]

Theorem 2 makes it possible, for one important case, to compute the quantity \(\rho_{n,2n}\).

Theorem 3. If the kernel \(\Phi\) is such that

\[ \Phi(t)\geq 0,\quad -\infty<t<\infty, \tag{14} \]

then

\[ \rho_{n,2n}=4r_n. \tag{15} \]

Proof. Put

\[ \psi(t)=\sum_{k=0}^{n-1} r_k \sin\bigl[(2n-k)t+2\alpha_n-\alpha_k\bigr]. \]

It is not hard to see that

\[ \Phi(t)+\psi(t)=\sin(nt+\alpha_n)\widetilde{\Phi}(t). \]

Since (14) holds, we have

\[ \int_{0}^{2\pi}\operatorname{sign}(\Phi(t)+\psi(t))e^{i(n+k)t}\,dt = \int_{0}^{2\pi}\operatorname{sign}\sin(nt+\alpha_n)e^{i(n+k)t}\,dt. \tag{16} \]

It is known that

\[ \operatorname{sign}\sin x=\frac{4}{\pi}\sum_{k=0}^{\infty}\frac{\sin(2k+1)x}{2k+1}. \]

Therefore from (16) we conclude that

\[ \int_{0}^{2\pi}\operatorname{sign}(\Phi(t)+\psi(t))e^{i(n+k)t}\,dt=0,\quad k=1,2,\ldots,n. \]

Thus, the equalities (13) are satisfied. Consequently, by Theorem 2, the operator

\[ \overline{U}(f,x)=\int_{0}^{2\pi} f(x+t)[\Phi(t)+\psi(t)]\,dt \]

has the smallest norm in the class \(\Omega_{n,2n}^{\Phi}\), and

\[ \rho_{n,2n}^{\Phi} = \int_{0}^{2\pi}|\sin(nt+\alpha_n)|\widetilde{\Phi}(t)\,dt. \]

From this, by a simple calculation, we obtain (15).

Corollary. If the associated kernel is nonnegative, then in the spaces \(\widetilde{C}\) and \(\widetilde{L}\) the equality

\[ \rho_{n,2n}^{\Phi}:\rho_n^{\Phi}=4:\pi \]

holds.

This assertion follows directly from Theorems 1 and 3.

Theorem 4. Let

\[ \Phi(t)=\sum_{k=1}^{n} b_{n-k}\cos kt, \]

for

\[ b_\nu-2b_{\nu+1}+b_{\nu+2}\geqslant 0,\quad \nu=0,1,2,\ldots,(n-3), \]
\[ b_{n-2}-2b_{n-1}\geqslant 0,\quad b_{n-1}\geqslant 0. \tag{17} \]

Then the extremal operator \(\overline{U}\) from the class \(\Omega^{\Phi}_{n,2n-1}\) is determined by the equality

\[ \overline{U}(f,x)=\int_{0}^{2\pi} f(x+t)\cos nt\,\widetilde{\Phi}(t)\,dt,\qquad \widetilde{\Phi}(t)=b_0+\sum_{j=1}^{n-1}2b_j\cos jt, \tag{18} \]

where \(\rho^{\Phi}_{n,2n-1}=4b_0\).

This theorem follows from Theorem 3 and L. Fejér’s theorem \({}^{6}\), according to which the polynomial (18) is nonnegative if its coefficients satisfy inequalities (17).

Remark. Theorem 4 remains valid also in the case when

\[ \Phi(t)=\sum_{k=1}^{n} b_{n-k}\sin kt. \]

Let us apply Theorem 4 in the case when

\[ \Phi(t)=\frac{1}{\pi}\left(\frac{\sin (n+{}^{1}/_{2})t}{2\sin t/2}\right)' . \tag{19} \]

Then

\[ \widetilde{\Phi}(t)=\frac{\sin^2 nt/2}{\pi\sin^2t/2}. \]

Consequently, \(\rho^{\Phi}_{n,2n-1}=4n:\pi\). It is known \({}^{1}\) that the operator (12), when \(\Phi(t)\) is defined according to (19), has the property that for any \(t\in\Pi_n\), \(\sigma(t)=t'\). Therefore we have

Theorem 5. Among all linear operations from \(\widetilde{C}\) to \(\widetilde{C}\) (from \(\widetilde{L}\) to \(\widetilde{L}\)) that take all functions into trigonometric polynomials of order \(\leqslant (2n-1)\) and have the property that each polynomial \(t\in\Pi_n\) is taken into its derivative, the operator

\[ \overline{U}(f,x)=\frac{1}{\pi}\int_{0}^{2\pi} f(x+t)\sin nt\,\widetilde{\Phi}(t)\,dt \]

has the smallest norm.

Here \(\|U\|=\rho^{\Phi}_{n,2n-1}=4n:\pi\).

Hence, in particular, it is clear that with the aid of operators from \(\Phi^{\Phi}_{n,2n-1}\) it is impossible to prove S. N. Bernstein’s classical theorem on the modulus of the derivative of a trigonometric polynomial with the sharp constant equal to 1. The best constant that can be obtained by operators from \(\Omega^{\Phi}_{n,2n-1}\), as Theorem 5 shows, is \(4:\pi\).

In conclusion we formulate one more problem, which is solved with the aid of Theorem 3. Let \(\mathfrak{M}_n\) be the set of all polynomials of the form \(\Phi(t)=\cos t+a_2\cos 2t+\cdots+a_n\cos nt\), for which the associated polynomials are nonnegative on the number axis. It is required to compute the quantity

\[ \tau_n=\inf_{\Phi=\mathfrak{M}_n}\rho^{\Phi}_{n,2n-1}(E), \]

where \(E=\widetilde{C}\) or \(E=\widetilde{L}\), and to find the extremal polynomial. The solution of this problem is given by Theorem 6.

Theorem 6. For every \(n\geqslant 2\), \(\tau_n=8\). The extremal polynomial has the form \(\Phi(t)=\cos t+2\cos nt\).

Leningrad Institute of Soviet Trade
named after Fr. Engels

Received
29 X 1965

References

\({}^{1}\) F. Riesz, C. R., 158 (1914).
\({}^{2}\) D. L. Berman, DAN, 163, No. 3 (1965).
\({}^{3}\) D. L. Berman, DAN, 161, No. 5 (1965).
\({}^{4}\) D. L. Berman, DAN, 138, No. 4 (1961).
\({}^{5}\) D. L. Berman, Matem. sborn., 60 (102), No. 3 (1963).
\({}^{6}\) L. Fejér, Acta Lit. Sci. Univ. Hung., 2 (1925).