MATHEMATICS

E. V. VORONOVSKAYA and M. Ya. ZINGER

ESTIMATION OF POLYNOMIALS IN THE COMPLEX PLANE

(Presented by Academician S. N. Bernstein, 29 XI 1961)

In the work \((^{1})\) of E. V. Voronovskaya, the extremal properties of the Chebyshev and Zolotarev polynomials on the disk of unit radius of the complex plane were studied, and the validity of analogous properties on a disk of arbitrary radius was also noted. In the present article the authors give a complete solution of the problem posed in the note \((^{1})\), namely: among the set of algebraic polynomials with real coefficients under the condition
\(\max_{[0,1]} |P_n(x)| = 1\) (reduced), to find that one which at the point \(z=\rho e^{i\varphi}\) has the greatest, in modulus, either real part or imaginary part.

It suffices to take \(0<\varphi<\pi\).

Theorem 1. If the interval-functionals on the set \(\{P_n(x)\}\) are denoted in the following way:

\[ F_{\cos}=1,\ \rho\cos\varphi,\ \rho^2\cos2\varphi,\ldots,\rho^n\cos n\varphi=(\mu_k)_0^n; \]

\[ F_{\sin}=0,\ \rho\sin\varphi,\ \rho^2\sin2\varphi,\ldots,\rho^n\sin n\varphi=(\nu_k)_0^n, \]

then the number of true nodes \((^{1})\) of \(F_{\cos}\) and \(F_{\sin}\) satisfies the condition \(s\ge n\) for any \(\rho>0\) and any \(0<\varphi<\pi\).

Indeed, \(\mu_k\) and \(\nu_k\) have the form
\(\Delta_1(e^{\varphi i})^k+\Delta_2(e^{-\varphi i})^k\), where
\(\Delta_1=1/2\) or \(1/2i\) and \(\Delta_2=1/2\) or \(-1/2i\), i.e. \((\mu_k)\) and \((\nu_k)\) have two fictitious nodes \((^{2})\). But then \(s+2>n+1\ (^{2})\), and the theorem is proved.

Corollary. The extremal polynomials for \(F_{\cos}\) and \(F_{\sin}\) are either \(\pm T_n(x)\), where
\[ T_n(x)=\cos n\arccos(2x-1), \]
or the passport polynomials \([n,n,p]\ (^{2,3})\), where \(p=0\) or \(1\); moreover, at each point \((\rho,\varphi)\) the extremal polynomial is unique.

Let us find necessary and sufficient conditions under which \(F_{\cos}\) is served by the polynomials \(\pm T_n(x)\) with deviation points \((\tau_k)_0^n\). Expanding \(F_{\cos}\) in the nodes \((\tau_k)_0^n\), where \(\tau_k=\sin^2(k\pi/2n)\), i.e. solving the system of equations

\[ \sum_{p=0}^{n}\delta_p\tau_p^k=\mu_k\quad (k=0,1,\ldots,n), \]

we obtain for the loads \((\delta_p)\) the formula

\[ \delta_p=(-1)^{\,n-p} \frac{ \frac12\left[ \prod_{k\ne p}(\rho e^{\varphi i}-\tau_k) + \prod_{k\ne p}(\rho e^{-\varphi i}-\tau_k) \right] }{ \prod_{k\ne p}|\tau_p-\tau_k| } \quad (p=0,1,\ldots,n). \]

Here the numerator is
\[ A_p=\operatorname{Re}\prod_{k\ne p}(\rho e^{\varphi i}-\tau_k). \]
The signs of \((\delta_p)\) will alternate if either \(A_p\ge 0\), or \(A_p\le 0\) for all \(p\).

Put

\[ \arg\prod_{k\ne p}(\rho e^{\varphi i}-\tau_k) = \sum_{k\ne p}\arg(\rho e^{\varphi i}-\tau_k) = \sum_{k\ne p}\varphi_k = \psi_p. \]

Thus, the following is valid:

Theorem 2. If \(E_T\) is the set of points in \((z)\) at which \(F_{\cos}\) is served by the polynomials \(+T_n(x)\) or \(-T_n(x)\), then a necessary and sufficient condition for \(\rho e^{\varphi i}\in E_T\) is that either all \(\psi_p\) lie in \([-\pi/2,+\pi/2]\) (right half-plane), or in \([\pi/2,3\pi/2]\) (left half-plane); moreover, in the first case \(+T_n(x)\) serves, and in the second \(-T_n(x)\).

Corollary. Fixing \(\rho\) and putting \(\varphi=\varphi_0\), we have
\(\varphi_0<\varphi_1<\cdots<\varphi_n\) and
\(\psi_n<\psi_{n-1}<\cdots<\psi_0\). The arguments of the boundaries of the Chebyshev arc (on the circle of radius \(\rho\)) are obtained when \(\psi_0\) or \(\psi_n\) touches the imaginary axis. Denote these arguments by \(a\) and \(b\), respectively; then for \(\varphi_0=a\) the node \(\tau_0=0\) drops out (i.e. \(\delta_0=0\)), and for \(\varphi_0=b\) the node \(\tau_n=1\) drops out (i.e. \(\delta_n=0\)). According to the theorem on continuous deformation \((^2)\), the family \(F_{\cos}\) is served by the polynomials of passport \([n,n,0]\) and only by them, i.e. by the Zolotarev polynomials \((^3)\).

For any \(\rho\ge 1\), the arguments of the boundaries of the Zolotarev arcs \((\alpha_k,\beta_k)\) and their number on the semicircle \((0,\pi)\) are determined from the conditions that \(\psi_0\) (or \(\psi_k\)) is equal to \(\pi/2, 3\pi/2,\ldots,(2n-1)\pi/2\); thus, on the semicircle there are altogether \(n\) Zolotarev arcs.

Remark 1. For \(F_{\sin}\) analogous results are obtained; thus, \(F_{\sin}\) is served by the polynomials \(\pm T_n(x)\) if and only if all \((\psi_p)_0^n\) lie either in the upper half-plane or in the lower; otherwise \(F_{\sin}\) is served by Zolotarev polynomials.

Let us note some elementary formulas for any \(z=\rho e^{\varphi i}\) in the upper half-plane. Put \(\overline{\varphi}_k=\varphi_k-\varphi_0=\psi_0-\psi_k\). Then

\[ \overline{\varphi}_k = \frac{\pi}{2}-\frac{\varphi_0}{2} -\operatorname{arctg}\left[ \frac{\rho-\tau_k}{\rho+\tau_k}\operatorname{ctg}\frac{\varphi_0}{2} \right], \tag{1} \]

and further

\[ \psi_0=n\varphi_0+\sum_{k=1}^{n}\overline{\varphi}_k, \tag{2} \]

\[ \psi_n=n\varphi_0+\sum_{1}^{n}\overline{\varphi}_k -\frac{\pi}{2}+\frac{\varphi_0}{2} +\operatorname{arctg}\left[ \frac{\rho-1}{\rho+1}\operatorname{ctg}\frac{\varphi_0}{2} \right]. \tag{3} \]

It is obvious that
\[ \max_{(\varphi_0)} \overline{\varphi}_k = \arc\sin\frac{\tau_k}{\rho} \quad (k=\mathrm{const},\ \varphi_k=\pi/2,\ \rho>1). \]
In \(\sum_{1}^{n}\overline{\varphi}_k\) the largest term is
\(\overline{\varphi}_n\le \arc\sin \frac{1}{\rho}\); thus (for \(n=\mathrm{const}\)) we have

\[ \sum_{1}^{n}\overline{\varphi}_k < n\arc\sin\frac{1}{\rho}, \qquad \lim_{\rho\to\infty}\sum_{1}^{n}\overline{\varphi}_k=0. \tag{4} \]

Theorem 3. On the semicircle \(0<\varphi_0<\pi\), as \(\rho\) increases, the arguments of the boundaries of the \(k\)-th Zolotarev arc \((\alpha_k<\beta_k)\) tend to \((2k-1)\pi/2n\).

Indeed, according to the corollary to Theorem 2, for \(\varphi_0=\alpha_k\) we have from (2)

\[ n\alpha_k+\sum_{p=1}^{a}\overline{\varphi}_p(\alpha_k) = \frac{2k-1}{2}\pi, \]

and for \(\varphi_0=\beta_k\) from (3)

\[ n\beta_k+\sum_{p=1}^{n}\overline{\varphi}_p(\beta_k) -\frac{\pi}{2} +\frac{\beta_k}{2} +\operatorname{arctg}\left[ \frac{\rho-1}{\rho+1}\operatorname{ctg}\frac{\beta_k}{2} \right] = \frac{(2k-1)\pi}{2}. \]

According to (4), we obtain

\[ \lim_{\rho\to\infty}\alpha_k = \lim_{\rho\to\infty}\beta_k = \frac{(2k-1)\pi}{2n}. \]

Theorem 4. In the plane \((z)\), the branches-boundaries of the Zolotarev regions \(\rho e^{\alpha_k(\rho)i}\) have a system of asymptotes issuing from the point \((n+1)/2n\) on the axis

\(Ox\), and for the branches \(\rho e^{\beta_k(\rho)i}\) the asymptotes are rays issuing from the point \((n-1)/2n\) (the angles of the asymptotes are determined in Theorem 3).

Indeed, assuming that \(l(\alpha_k)\) (respectively \(l(\beta_k)\)) is the length of the arc of radius \(\rho\) between the points \(\rho e^{(2k-1)\pi i/2n}\) and \(\rho e^{\alpha_k i}\) (or \(\rho e^{\beta_k i}\)), we have (on the basis of the formulas in Theorem 3):

\[ l(\alpha_k)=\rho\left(\frac{2k-1}{2n}\pi-\alpha_k\right) =\frac{\rho}{n}\sum_{p=1}^{n}\overline{\varphi}_p(\alpha_k), \]

\[ l(\beta_k)=\rho\left(\frac{2k-1}{2n}\pi-\beta_k\right) =\frac{\rho}{n}\sum_{p=1}^{n-1}\overline{\varphi}_p(\beta_k). \]

The oblique abscissa of the point \(\rho e^{\alpha_k i}\) is

\[ x_\alpha= \frac{\rho\sin\left(\frac{2k-1}{2n}\pi-\alpha_k\right)} {\sin\frac{2k-1}{2n}\pi} = \rho\, \frac{\sin\left[\frac1n\sum_{1}^{n}\overline{\varphi}_p(\alpha_k)\right]} {\sin\frac{2k-1}{2n}\pi}. \]

Here it is permissible first to pass to

\[ \lim_{\rho\to\infty} \left[ \rho\sin\frac1n\sum_{1}^{n}\overline{\varphi}_p(\varphi) \right] \]

for \(\varphi=\mathrm{const}\), and then to put \(\varphi=\frac{2k-1}{2n}\pi\). Since \(\sum_{p=1}^{n}\overline{\varphi}_p(\varphi)\to 0\), we have

\[ \lim \frac{\rho}{n}\sum_{1}^{n}\overline{\varphi}_p(\varphi) = \lim \rho\sum_{1}^{n} \left( \frac{\pi}{2}-\frac{\varphi}{2} -\operatorname{arc\,tg}\frac{\rho-\tau_k}{\rho+\tau_k}\operatorname{ctg}\frac{\varphi}{2} \right) = \]

\[ = \sum_{k=1}^{n}\tau_k \frac{2\operatorname{ctg}(\varphi/2)} {1+\operatorname{ctg}^{2}(\varphi/2)} = \sin\varphi\sum_{1}^{n}\tau_k . \]

Hence \(\lim x_\alpha=(n+1)/2n\). In exactly the same way we obtain \(\lim x_\beta=(n-1)/2n\), and the theorem is proved.

Corollary. The sum of the Zolotarev arcs on the semicircle of radius \(\rho\) tends to \(1/n\sin(\pi/2n)\) as \(\rho\to\infty\).

Indeed, the length of the \(k\)-th Zolotarev arc is

\[ l(\alpha_k)-l(\beta_k) = \frac{\rho}{n} \left[ \sum_{p=1}^{n}\overline{\varphi}_p(\alpha_k) - \sum_{p=1}^{n-1}\overline{\varphi}_p(\beta_k) \right] \to \frac1n\sin\frac{(2k-1)\pi}{2n}. \]

For the sum in the limit we have

\[ \frac1n\sum_{k=1}^{n}\sin\frac{2k-1}{2n}\pi = \frac{1}{n\sin(\pi/2n)}. \]

Thus, for large \(n\) this sum is close to \(2/\pi\).

Remark 2. For \(F_{\sin}\), the same method gives analogous results: 1) as \(\rho\to\infty\), the arguments of the boundaries of the Zolotarev arc \((\alpha_k,\beta_k)\) tend to \(k\pi/n\) \((k=1,2,\ldots,n-1)\); the \(n\)-th Zolotarev arc, for every \(\rho\), degenerates into the point \(\alpha_n=\beta_n=\pi\); 2) as \(\rho\to\infty\), the branches \(\rho e^{\alpha_k(\rho)i}\) and \(\rho e^{\beta_k(\rho)i}\) have asymptotes issuing respectively from the points \((n+1)/2n\) and \((n-1)/2n\) on the axis \(Ox\); 3) the sum of the lengths of the Zolotarev arcs tends to \(\operatorname{ctg}(\pi/2n)/n\) as \(\rho\to\infty\).

Remark 3. From the properties proved it follows that, for one and the same \(\rho\ge 1\), the Zolotarev arcs for \(F_{\cos}\) and for \(F_{\sin}\) cannot overlap.

In order to consider the boundaries of the Zolotarev regions for \(\rho<1\), let us note, restricting ourselves to the case \(F_{\cos}\), that in general the arguments of these boundaries \(\alpha(\rho)\)

and \(\beta(\rho)\) are respectively the roots of the equations

\[ F_{\cos}\left[\frac{R_{n+1}(x)}{x}\right]=0,\qquad F_{\cos}\left[\frac{R_{n+1}(x)}{x-1}\right]=0, \quad \text{where } \quad R_{n+1}(x)=\prod_{0}^{n}(x-\tau_k). \]

Putting

\[ \frac{R_{n+1}(x)}{x}=\sum_{0}^{n}S_{n-k}^{(1)}x^k,\qquad \frac{R_{n+1}(x)}{x-1}=\sum_{0}^{n}S_{n-k}^{(2)}x^k, \]

we have (for any \(\rho>0\))

\[ \rho^n\cos n\alpha+S_1^{(1)}\rho^{n-1}\cos(n-1)\alpha+\cdots+S_n^{(1)}=0, \]

\[ \rho^n\cos n\beta+S_1^{(2)}\rho^{n-1}\cos(n-1)\beta+\cdots+S_{n-1}^{(2)}\rho\cos\beta=0. \]

Putting: 1) \(\alpha=0\) and 2) \(\beta=0\), we have
\[ \prod_{1}^{n}(\rho-\tau_k)=0 \]
and
\[ \prod_{0}^{n-1}(\rho-\tau_k)=0. \]

Thus, the curves \(\rho e^{\alpha(\rho)i}\) and \(\rho e^{\beta(\rho)i}\) intersect \(Ox\) at the points \((\tau_p)_1^n\) and \((\tau_p)_0^{n-1}\), and the Chebyshev domains have on \([0,1]\) “sources” at the points \((\tau_k)_0^n\), while the Zolotarev domains contain the entire segment \([0,1]\), except for \((\tau_k)\). The sources of the branches for \(\alpha_k\) and \(\beta_k\) are \(\tau_{n-k+1}\) and \(\tau_{n-k}\).

For lack of space we shall confine ourselves to the indicated properties for \(\rho<1\). We are also unable to provide drawings illustrating the distribution of the mentioned domains in the \((z)\)-plane.

Finally, let us note the most immediate applications to classical extremal problems. Denote by \(E_{\cos}\) and \(E_{\sin}\), respectively, the Chebyshev sets of points in \((z)\) for \(F_{\cos}\) and \(F_{\sin}\), and by \(\mathscr E_{\cos}\) and \(\mathscr E_{\sin}\) the Zolotarev sets.

I. Among all algebraic polynomials \(\{P_n(x)\}\) reduced on \([0,1]\) and with real coefficients, find the one which at the point \(z_0\) attains the maximum modulus. If \(z_0\in(E_{\cos},E_{\sin})\), then this polynomial is, evidently, \(\pm T_n(x)\), and \(|T_n(z_0)|=N\), the norm of the functional \((z_0)^n\).

Problem I can be rephrased as follows:

I\(_A\). Among all \(\{P_n(x)\}\) taking at the point \(z_0\) the value \(|P_n(z_0)|=A\) in modulus, find the one which deviates least from zero on \([0,1]\). If the solution of problem I\(_A\) is \(Y_n(x)\) with deviation \(L\), and \(P_n(x)\) is the solution of problem I, then
\[ Y_n(x)/L=P_n(x),\qquad A=LN. \]

II. Among all \(\{P_n(x)\}\) reduced on \([0,1]\), find the one which at the point \(z_0\) gives: a) either \(\max|\operatorname{Re}P_n(z)|\), b) or \(\max|\operatorname{Im}P_n(z)|\). If \(z_0\in(E_{\cos},E_{\sin})\), then the desired polynomial is \(\pm T_n(x)\) for both variants a) and b). If \(z_0\in\mathscr E_{\cos}\), then the polynomial for variant a) belongs to the family \(Z_n(x,\vartheta)\), where \(\vartheta\) is the leading coefficient; for \(z_0=\rho e^{\varphi i}\) the value of \(\vartheta\) is found from the condition
\[ F_{\cos}[Z_n(x,\vartheta)]=\max(\vartheta), \]
i.e.
\[ \partial F_{\cos}(Z_n)/\partial\vartheta=0 \]
\((^{2,4})\), or, what is the same,

\[ F_{\cos}[R_n(z)]=0,\qquad \text{where } \quad R_n(x)=\prod_{1}^{n}(x-\sigma_k); \]
\(\sigma_k(\vartheta)\) are the nodes of the Zolotarev polynomial; in the case under consideration the extremal polynomial for variant b) is \(\pm T_n(x)\) (see Remark 3). For \(z_0\in\mathscr E_{\sin}\) analogous solutions are obtained.

Problem II can also be formulated differently: among all \(\{P_n(x)\}\) for which \(|\operatorname{Re}P_n(z_0)|=A\) or \(|\operatorname{Im}P_n(z_0)|=B\), find the one which deviates least from zero on \([0,1]\).

Received
12 XI 1961

CITED LITERATURE

\({}^{1}\) E. V. Voronovskaya, UMN, 12, no. 5, 254 (1957).
\({}^{2}\) E. V. Voronovskaya, Extremal Polynomials of Finite Functionals, dissertation abstract, LSU, 1955.
\({}^{3}\) E. V. Voronovskaya, DAN, 99, no. 2 (1954).
\({}^{4}\) E. V. Voronovskaya, Proceedings of the Third All-Union Mathematical Congress, 3, Publishing House of the Academy of Sciences of the USSR, 1958, p. 177.