MATHEMATICS

I. Yu. RYZHAKOV

ON AN ANALOGUE OF A PROBLEM OF E. I. ZOLOTAREV

(Presented by Academician V. I. Smirnov on 16 VII 1964)

E. I. Zolotarev \((^1)\) posed and solved the following problem: among algebraic polynomials of the form \(x^n-\sigma x^{n-1}+\cdots,\ 0\leq \sigma<\infty\), find the one which deviates least from zero on the interval \([-1,1]\). Its extremal polynomials turned out, for \(0\leq \sigma\leq n\tg^2 \frac{\pi}{2n}\), to be the polynomials

\[ \frac{1}{2^{\,n-1}\alpha^n}T_n[(1+x)\alpha-1],\quad T_n(x)=\cos n\arc\cos x,\quad \sigma=n\frac{1-\alpha}{\alpha},\quad \alpha\in\left[\cos^2\frac{\pi}{2n},\,1\right], \]

and for \(\sigma>n\tg^2\frac{\pi}{2n}\), the Zolotarev polynomials \(Z_n(\sigma,x)\).

The present note is devoted to the analogous problem for trigonometric polynomials \(P_n(t)\)

\[ P_n(t)=\sum_{k=0}^{n}(a_k\cos kt+b_k\sin kt)=\sum_{k=-n}^{n}c_k e^{ikt}, \]

where \(a_k\) and \(b_k\) are real; namely: among \(P_n(t)\) whose coefficients \(a_n, b_n, a_{n-1}\), and \(b_{n-1}\) are fixed, find the one which deviates least from zero on \([-\pi,\pi]\). To exclude elementary cases, we assume \(n\geq 3\). It is clear that such a problem is equivalent to the problem: to construct those \(P_n(t)\) which have on \([-\pi,\pi)\) at least \(2n-2\) points of deviation and form at them a Chebyshev alternance. Since only \(A\cos(nt-\psi)\), \(A,\psi\) real, have \(2n\) points of deviation, the matter is the construction of the sets \(K_{n,2n-1}\) and \(K_{n,2n-2}\) of polynomials \(P_n(t)\) which have on \([-\pi,\pi)\), respectively, exactly \(2n-1\) or \(2n-2\) points of deviation and form at them a Chebyshev alternance.

Theorem 1. If \(P_n(t)\) has on \([-\pi,\pi)\) exactly \(2n-1\) points of deviation, then this is a polynomial of the form \(AT_n(1+\cos(t-\psi))\alpha-1]\), \(\alpha\in(\cos^2(\pi/2n),1)\), \(A,\psi\) real.

Suppose that on some interval \((a,b)\) there are given \(2n-1\) real functions \(\{u_k(\xi)\}_0^n\) and \(\{v_k(\xi)\}_1^n\), and let \(\lambda_k(\xi)=u_k(\xi)+iv_k(\xi)\), \(k=0,1,\ldots,n\), \(v_0(\xi)\equiv 0\), \(\lambda_{-k}(\xi)\equiv \overline{\lambda_k(\xi)}\).

Taking any fixed \(\xi\in(a,b)\), consider the set \(K_n(L^\xi)\) of polynomials \(P_n(t)\) whose coefficients are connected by the linear dependence \(L^\xi\):

\[ L^\xi(P_n)=\sum_{k=-n}^{n}c_k\lambda_k(\xi)=1. \]

By \(\pi_n(\xi,t)\) denote the polynomial least deviating from zero on \([-\pi,\pi]\) among the polynomials of the set \(K_n(L^\xi)\). According to the criterion expressed by V. S. Videnskii \((^2)\), on the unit circle there exist points \(\{\varepsilon_j(\xi)\}_1^m\), \(\varepsilon_j(\xi)=e^{i\tau_j(\xi)}\), \(-\pi\leq \tau_1(\xi)<\tau_2(\xi)<\cdots<\tau_m(\xi)<\pi\), \(1\leq m\leq 2n+1\), such that the system

\[ \sum_{j=1}^{m}\varepsilon_j^k(\xi)\delta_j=\lambda_k(\xi),\quad k=0,\pm1,\ldots,\pm n, \]

is consistent, and its solution \(\{\delta_j(\xi)\}_1^m\) satisfies the conditions

\[ \delta_j(\xi)\neq 0,\quad \operatorname{sign}\delta_j(\xi)=\operatorname{sign}\pi_n(\xi,\tau_j(\xi)),\quad j=1,2,\ldots,m. \]

The proof of the following theorem is based on this property of the polynomial \(\pi_n(\xi,t)\); in content and in the method of proof it is adjacent to the well-known theorem of S. N. Bernstein \(({}^{3}),\) p. 40).

Theorem 2. If: 1) \(\{u_k(\xi)\}_0^n\) and \(\{v_k(\xi)\}_1^n\) are functions analytic on \((a,b)\), and 2) the number \(m\) of points \(\{\varepsilon_j(\xi)\}_1^m\) is constant on \((a,b)\) and greater than \(n\), then the coefficients \(\{a_k(\xi)\}_0^n\) and \(\{b_k(\xi)\}_1^n\) of the polynomial \(\pi_n(\xi,t)\), as well as
\[ \pi_n(\xi)=\sup_{t\in[-\pi,\pi]}|\pi_n(\xi,t)| \]
are analytic functions on \((a,b)\).

Let now \(\lambda_k=0,\ k=0,1,\ldots,n-2,\ \lambda_{n-1}=-1,\ \lambda_n=w\), where \(w\) is a complex variable parameter. Taking any fixed \(w\), consider the set \(K_n(L^w)\) of polynomials \(P_n(t)\) whose coefficients are connected by the linear relation
\[ L^w:\ L^w(P_n)=\sum_{k=-n}^{n}c_k\lambda_k=1,\quad \lambda_{-k}=\overline{\lambda_k},\quad \text{i.e. } c_nw-c_{n-1}-c_{-n+1}+c_{-n}\overline{w}=1. \]

Denote by \(\pi_n(w,t)\) the polynomial least deviating from zero among the polynomials of the set \(K_n(L^w)\); and by \(B_{2n-2}\) the set of points of the \(w\)-plane which is obtained if from the closed domain bounded by the rectilinear segments connecting the point \(w=0\) with the points
\[ w=2-2(n-1)\tan^2\frac{\pi}{2n} \]
and
\[ w=2\cos\frac{\pi}{2n}e^{i\pi/2(n-1)} \]
and by the curve
\[ w=w(\psi)=e^{i(2n-1)\psi}+\left[1-2(n-1)\frac{\cos^2(n-1)\psi\cos^2\pi/2n}{\cos^2\pi/2n}\right]e^{i\psi}, \quad \psi\in\left[0,\frac{\pi}{2n(n-1)}\right] \tag{1} \]
the points of the curve (1) are excluded. The following simplest transformations of the trigonometric polynomial \(P_n(t)\) are denoted by \((*)\): 1) multiplication of \(P_n(t)\) by a real number; 2) replacement of the argument \(t\) by \(t-\psi\), where \(\psi\) is real; 3) replacement of \(t\) by \(2\pi-t\).

Theorem 3. 1) If \(w\in B_{2n-2}\), then \(\pi_n(w,t)\) is unique for every \(w\) and \(\pi_n(w,t)\in K_{n,2n-2}\); 2) whatever polynomial \(P_n^*(t)\in K_{n,2n-2}\) may be, there exists such a \(w^*\) in \(B_{2n-2}\) (moreover, only one for each \(P_n^*(t)\)) that \(\pi_n(w^*,t)\) can be transformed into \(P_n^*(t)\) using only \((*)\).

By Theorems 2 and 3, the coefficients of the polynomial \(\pi_n(w,t)\), as well as
\[ \pi_n(w)=\sup_{t\in[-\pi,\pi]}|\pi_n(w,t)|, \]
are functions analytic in each of the arguments \(u\) and \(v\), \(w=u+iv\), for \(w\in B_{2n-2}\). Therefore the deviation points of the polynomial \(\pi_n(w,t)\), \(w\in B_{2n-2}\), will be roots of
\(\partial\pi_n^*(w,t)/\partial w\) and
\(\partial\pi_n^*(w,t)/\partial v\), where
\(\pi_n^*(w,t)=\pi_n(w,t)/\pi_n(w)\).

Theorem 4. The polynomial
\[ \pi_n^*(w,t)=\sum_{k=0}^{n}\bigl(a_k^*(u,v)\cos kt+b_k^*(u,v)\sin kt\bigr) =\sum_{k=-n}^{n}c_k^*(u,v)e^{ikt}, \]
\[ c_{-k}^*(u,v)=\overline{c_k^*(u,v)},\quad w=u+iv,\quad w\in B_{2n-2}, \]
satisfies in \(B_{2n-2}\) the equations
\[ \frac{\partial \pi_n^*(w,t)}{\partial u} = \frac{1}{i}\, \frac{ \dfrac{\partial c_n^*}{\partial u}e^{it} +\dfrac{\partial c_{n-1}^*}{\partial u} -w\dfrac{\partial c_n^*}{\partial u} -\dfrac{\partial c_{-n}^*}{\partial u}e^{-it} }{ nc_n^*e^{it}+(n-1)c_{n-1}^*-nwc_n^*+nc_{-n}^*e^{-t} } \, \frac{\partial \pi_n^*(w,t)}{\partial t}, \]
\[ \frac{\partial \pi_n^*(w,t)}{\partial v} = \frac{1}{i}\, \frac{ \dfrac{\partial c_n^*}{\partial v}e^{it} +\dfrac{\partial c_{n-1}^*}{\partial v} -w\dfrac{\partial c_n^*}{\partial v} -\dfrac{\partial c_{-n}^*}{\partial v}e^{-it} }{ nc_n^*e^{it}+(n-1)c_{n-1}^*-nwc_n^*+nc_{-n}^*e^{-it} } \, \frac{\partial \pi_n^*(w,t)}{\partial t}. \tag{2} \]

Each of equations (2) splits into a system of first-order linear differential equations with respect to the coefficients \(\{a_k^*(u,v)\}_0^n\) and \(\{b_k^*(u,v)\}_1^n\) of the polynomial \(\pi_n^*(w,t)\). One can indicate initial—

necessary conditions for integrating such a system. Thus, if \(w\) lies on the segment of the real axis between the points \(w=0\) and \(w=2-2(n-1)\operatorname{tg}^{2}\frac{\pi}{2n}\), then \(\pi_n^*(w,t)\) is known, namely:
\[ \pi_n^*(w(\sigma),t)=\frac{1}{Z_n(\sigma)}Z_n(\sigma,\cos t), \]
where
\[ Z_n(\sigma)=\sup_{x\in[-1,1]} |Z_n(\sigma,x)|,\qquad \sigma\in\left[n\operatorname{tg}^{2}\frac{\pi}{2n},\infty\right), \]
\[ w(\sigma)=-2\sum_{j=1}^{n}x_j(\sigma), \]
and \(\{x_j(\sigma)\}_1^n\) are the deviation points of \(Z_n(\sigma,x)\) on \([-1,1]\).

When \(\sigma\) increases in \(\left[n\operatorname{tg}^{2}\frac{\pi}{2n},\infty\right)\), \(w(\sigma)\) moves along the indicated segment from
\[ w\left(n\operatorname{tg}^{2}\frac{\pi}{2n}\right) = 2-2(n-1)\operatorname{tg}^{2}\frac{\pi}{2n} \]
to \(w(\infty)=0\). Moreover, at the points \(w(\psi)\) of the curve (1),
\[ \pi_n^*(w(\psi),t)=T_n\bigl[(1+\cos(t-\psi))\alpha^*(\psi)-1\bigr], \]
\[ \psi\in\left[0,\frac{\pi}{2n(n-1)}\right], \qquad \alpha^*(\psi)=\frac{\cos^2\pi/2n}{\cos^2(n-1)\psi}. \]
When \(\psi\) increases in \(\left[0,\frac{\pi}{2n(n-1)}\right]\), \(w(\psi)\) moves along the curve (1) from
\[ w(0)=2-2(n-1)\operatorname{tg}^{2}\frac{\pi}{2n} \]
to
\[ w\left(\frac{\pi}{2n(n-1)}\right)=2\cos\frac{\pi}{2n}\,e^{i\pi/2(n-1)}. \]

As follows from Theorem 3, every polynomial of the set \(K_{n,2n-2}\) can be obtained by applying \((*)\) to some polynomial \(\pi_n^*(w,t)\), \(w\in B_{2n-2}\).

We note that, by a similar method, equations analogous to (2) for Zolotarev polynomials and certain other extremal algebraic polynomials were obtained by E. V. Voronovskaya \((^4)\).

Received
5 VII 1964

REFERENCES

\(^{1}\) E. I. Zolotarev, Application of elliptic functions to questions concerning functions least and most deviating from zero, Collected Works, Publishing House of the Academy of Sciences of the USSR, 1932.
\(^{2}\) V. S. Videnskii, DAN, 126, No. 2 (1951).
\(^{3}\) S. N. Bernstein, Extremal properties of polynomials, vol. 1, 1937.
\(^{4}\) E. V. Voronovskaya, The method of functionals and its applications, L., 1963.