UDC 517.512.6

MATHEMATICS

M. Ya. ZINGER

A GENERALIZATION OF THE SHEFFER–DUFFIN PROBLEM TO FINITE FUNCTIONALS

(Presented by Academician V. I. Smirnov on March 9, 1966)

In the work \((^1)\), Sheffer and Duffin published the following result, which they regarded as a strengthening of a well-known theorem of V. A. Markov \((^2)\): let \(\{\lambda_i\}_{i=0}^n\) be the points of extremum of the polynomial \(L_n(x)=\cos n\arccos x\) on the interval \([-1,1]\); then, among polynomials of degree \(n\) satisfying the condition \(|P_n(\lambda_i)|\le 1\) \((i=0,1,\ldots,n)\), the maximum of the \(k\)-th derivative \((k=1,2,\ldots,n)\) on the interval \([-1,1]\) is attained on the polynomial \(L_n(x)\), i.e.
\[ |P_n^{(k)}(x)|\le L_n^{(k)}(1). \]

In the present article the general problem formulated below is solved and some of its special cases are considered. In what follows the consideration is carried out on the interval \([0,1]\).

I. Problem. Let on the set of algebraic polynomials with real coefficients of degree not exceeding \(n\) there be given a finite functional \(F=(\mu_i)_0^n\), where \(\mu_i=F(x^i)\), \((\mu_i)\) are real numbers, and let on \([0,1]\) be given \(n+1\) consecutive points \(0\le t_0<t_1<\cdots<t_n\le 1\), and to each point there be assigned a nonnegative number \(y_i\). In the class \(\mathcal P_t\) of polynomials \(P_n(x)\) of degree not exceeding \(n\) satisfying the condition \(|P_n(t_i)|\le y_i\) \((i=0,1,\ldots,n)\), find that polynomial on which \(\sup F[P_n]=N_F\) is attained.

Any polynomial solving the problem will be called extremal.

Any polynomial \(P_n(x)=\sum_{j=0}^n a_j^{(P)}x^j\in\mathcal P_t\) can be represented in the form
\[ P_n(x)=\sum_{i=0}^n \frac{\delta_i}{R'_{n+1}(t_i)}\,R_{n+1,i}(x), \tag{1} \]
where \(|\delta_i|\le y_i\);
\[ R_{n+1}(x)=\prod_{m=0}^n (x-t_m);\qquad R_{n+1,i}(x)=\frac{R_{n+1}(x)}{x-t_i};\qquad F[P_n]=\sum_{i=0}^n \delta_i K_i; \]
\[ K_i=\frac{F[R_{n+1,i}]}{R'_{n+1}(t_i)}. \]

Put
\[ M_n(x)=\sum_{i=0}^n \frac{y_i}{|R'_{n+1}(t_i)|}\,R_{n+1,i}(x) =\sum_{i=0}^n a_i^{(M)}x^i\in\mathcal P_t . \]

It is easy to see that for every polynomial \(P_n(x)\in\mathcal P_t\) the inequalities
\[ |a_j^{(P)}|\le |a_j^{(M)}|,\qquad j=0,1,\ldots,n. \tag{2} \]
hold.

From consideration of formula (1) it follows easily that

Theorem 1. For every functional \(F\), an extremal polynomial is any polynomial of the form (1), where \(\delta_i=y_i\operatorname{sign}K_i\) for \(K_i\ne0\); \(\delta_i\) arbitrary

from \([-y_i,y_i]\), if \(K_i=0\).

\[ N_F=\sum_{i=0}^{n} y_i|K_i|. \]

Remark. If all \(K_i\ne0\), then the extremal polynomial is unique; if some \(K_i=0\), then below in the extremal polynomial we shall put \(\delta_i=y_i\). In this way we ensure uniqueness of the extremal polynomial in all cases, and the number of extremal polynomials for all possible functionals does not exceed \(2^{n+1}\).

Let \(\mu_i=\mu_i(\xi)\) be a continuous function of the real argument \(\xi\), \(-\infty<\xi<+\infty\), \(i=0,1,\ldots,n\). Then \(F_\xi[P_n]\) is a continuous function of \(\xi\), and in this case, according to Theorem 1, the problem of finding the extremal polynomials \(\{G_n(x,\xi)\}\) is practically equivalent to the problem of finding the points of sign change of the functions \(\{F_\xi[R_{n+1,i}]\}_{i=0}^n\).

We proceed to the consideration of concrete interval-functionals.

II. \(F_\xi[P_n]=P_n^{(k)}(\xi)\), i.e.

\[ \mu_i=0,\quad i=0,1,\ldots,k-1;\qquad \mu_i=\frac{i!}{(i-k)!}\,\xi^{\,i-k},\quad i=k,k+1,\ldots,n. \tag{3} \]

Denote by \(\{x_i^{(l,k)}\}_{i=1}^{n-k}\) the roots of the polynomial \(R_{n+1,l}^{(k)}(x)\) \((l=0,1,\ldots,n)\).

Theorem 2. For \(\xi\in(-\infty,x_1^{(n,k)}),\ (x_{n-k}^{(0,k)},+\infty),\ (x_i^{(0,k)},x_{i+1}^{(n,k)})\) \((i=1,2,\ldots,n-k-1)\), one of the polynomials \(\pm M_n(x)\) is extremal; for \(\xi\in(x_i^{(l,k)},x_i^{(l-1,k)})\) \((i=1,2,\ldots,n-k;\ l=1,2,\ldots,n)\) the polynomial

\[ \Lambda_{l,n}(x)=M_n(x)+2\sum_{m=l}^{n}\frac{(-1)^{n-m+1}y_m}{R'_{n+1}(t_m)}\,R_{n+1,m}(x); \]

is extremal up to sign; for \(\xi\in\{x_i^{(l,k)}\}_{i=1}^{n-k}\) \((l=0,1,\ldots,n)\) one of the polynomials \(\pm M_n(x)\) or a polynomial of the form \(\Lambda_{l,n}(x)\) is extremal (chosen in accordance with the remark).

The proof of the theorem follows from the interlacing of the roots of the polynomials \(\{R_{n+1,l}^{(k)}(x)\}_{l=0}^n\).

Corollary 1. For \(k=1,2,\ldots,n-1\), the number of extremal polynomials (up to sign) is not greater than \(n+1\); for \(k=n\), on the whole real axis the extremal polynomial is \(M_n'(x)\).

Corollary 2. If all \(y_i>0\), then each of the extremal polynomials \(\Lambda_{l,n}(x)\) has \(n\) points (among \(\{t_i\}_{i=0}^n\)) at which the values \(|\Lambda_{l,n}(t_i)|=y_i\) are attained with successively opposite signs.

Corollary 3. The sum of the lengths of the intervals located on \([0,1]\), at each point of which one of the polynomials \(\pm M_n(x)\) is extremal, is equal to \(1-(1-k/n)(t_n-t_0)\); the sum of the lengths of the intervals at whose points the polynomial \(\Lambda_{l,n}(x)\) \((l=1,2,\ldots,n)\) is extremal up to sign is equal to \((1-k/n)(t_l-t_{l-1})\).

Corollary 4. For any \(\xi>0\)

\[ \sup_{P_n\in\mathscr P_t^{\,y}} F_\xi[P_n]=N_F(\xi)<|M_n^{(k)}(-\xi)|=N_F(-\xi). \]

III. If the points \(\{t_i\}_{i=0}^n\) are taken to be the points of extremum of the polynomial

\[ T_n(x)=\cos n\arccos(2x-1) \]

(denote them by \(\{\tau_i\}_{i=0}^n\)), and \(\{y_i\}_{i=0}^n=1\), then Theorems 1, 2 and their corollaries contain assertions directly supplementing the result of Sheffer and Duffin (1). In this case the extremal polynomials have the form

\[ P_{l,n}(x)=T_n(x)+\frac{x(x-1)T'_n(x)}{n^2 2^{2n-2}}\sum_{m=l}^{n}\frac{(-1)^{n-m+1}}{R'_{n+1}(\tau_m)(x-\tau_m)}, \tag{4} \]

\[ l=0,1,\ldots,n\qquad (P_{0,n}=-T_n(x)). \]

Besides the properties possessed by all extremal polynomials of the functional (3) (see Corollaries 2 and 3 of Theorem 2), the polynomials \(\{P_{l,n}(x)\}\) have certain specific properties.

Theorem 3. \(P'_{l,n}(\tau_i)P'_{l,n}(\tau_{i+1})<0\) \((i=1,2,\ldots,n-2;\ l=1,2,\ldots,n)\).

Proof. It suffices to consider \(l>n/2\), since

\[ P_{l,n}(x)=(-1)^{n-1}P_{n-l+1,n}(1-x)\qquad (l=1,2,\ldots,n). \]

We shall show that, for \(n/2<l\le n\),
\[ \operatorname{sign} P'_{l,n}(\tau_m)=(-1)^{n-m}\qquad (m=1,2,\ldots,n-1). \]
Taking into account that
\[ R'_{n+1}(\tau_m)=(-1)^{n-m}\frac{n}{2^{2n-1}}\qquad (m=1,2,\ldots,n-1); \]
\[ R'_{n+1}(0)=(-1)^n\frac{2n}{2^{2n-1}};\qquad R'_{n+1}(1)=\frac{2n}{2^{2n-1}}, \]
we have (see (4))
\[ P_{l,n}(x)=T_n(x)-\frac{2^{2n-1}R_{n+1}(x)}{n} \left[2\sum_{m=l}^{n}\frac{1}{x-\tau_m}-\frac{1}{x-1}\right]. \]

Let \(\tau_i<\tau_l\) \((i=1,2,\ldots,l-1)\). Then
\[ P'_{l,n}(\tau_i)=(-1)^{n-i-1} \left[2\sum_{m=l}^{n}\frac{1}{\tau_i-\tau_m} -\frac{1}{\tau_i-1}\right] \]
and
\[ \operatorname{sign} P'_{l,n}(\tau_i)=(-1)^{n-i}\qquad (i=1,2,\ldots,l-1). \]
Let \(\tau_i\ge \tau_l\) \((i=l,l+1,\ldots,n-1)\). Note that
\[ \left(\frac{R_{n+1}(x)}{x-\tau_i}\right)'_{x=\tau_i} = \frac{1-2\tau_i}{4\tau_i(1-\tau_i)}\,R'_{n+1}(\tau_i); \]
\[ P'_{l,n}(\tau_i)=(-1)^{n-i-1} \left[ 2\sum_{\substack{m=l\\ m\ne i}}^{n}\frac{1}{\tau_i-\tau_m} -\frac{1}{\tau_i-1} +\frac{1-2\tau_i}{2\tau_i(1-\tau_i)} \right], \]
and hence \(\operatorname{sign} P'_{l,n}(\tau_l)=(-1)^{n-l}\). Put \(\tau_i>\tau_l\) \((i=l+1,\ldots,n-1)\). Introduce the notation
\[ \Pi_i(x)=\prod_{\substack{m=l\\ m\ne i}}^{n}(x-\tau_m); \qquad \Lambda_l(x)=\prod_{m=0}^{l-1}(x-\tau_m). \]

Then
\[ \Pi_i(x)\Lambda_l(x)(x-\tau_i)=R_{n+1}(x); \qquad \sum_{\substack{m=l\\ m\ne i}}^{n}\frac{1}{x-\tau_m} = \frac{\Pi'_i(x)}{\Pi_i(x)}; \]
\[ \Pi_i(\tau_i)=\frac{R'_{n+1}(\tau_i)}{\Lambda_l(\tau_i)}; \qquad \Pi'_i(\tau_i)=\frac{R'_{n+1}(\tau_i)}{\Lambda_l(\tau_i)} \left[ \frac{1-2\tau_i}{4\tau_i(1-\tau_i)} - \frac{\Lambda'_l(\tau_i)}{\Lambda_l(\tau_i)} \right]; \]
\[ P'_{l,n}(\tau_i)=(-1)^{n-i} \left[ \frac{2\Lambda'_l(\tau_i)}{\Lambda_l(\tau_i)} -\frac{1}{\tau_i} \right]. \]

Noting that
\[ \frac{2\Lambda'_l(x)}{\Lambda_l(x)}-\frac{1}{x}>0 \quad\text{for } x\ge \tau_{l-1}, \]
we obtain
\[ \operatorname{sign} P'_{l,n}(\tau_i)=(-1)^{n-i} \qquad (i=l+1,l+2,\ldots,n-1), \]
and the theorem is proved.

Corollary 1. The polynomials \(P'_{l,n}(x)\) \((l=1,2,\ldots,n)\) have one root in each of the segments \([\tau_i,\tau_{i+1}]\) \((i=1,2,\ldots,n-2)\).

Corollary 2. For \(n/2<l<n\)
\[ \sum_{m=1}^{n-1}\left|P'_{l,n}(\tau_m)\right| = \frac12\left(\left|P'_{l,n}(0)\right|+\left|P'_{l,n}(1)\right|\right). \]

Corollary 3.
\[ \sum_{m=1}^{n-1}\left|P'_{n,n}(\tau_m)\right| = \frac12\left(\left|P'_{n,n}(0)\right|-P'_{n,n}(1)\right) = \frac23(n^2-1). \]

Remark. With the aid of Theorems 1–3 the proof of the Schaeffer–Duffin theorem is simplified and can be carried out by purely real-variable means (by analogy with the proof of S. N. Bernstein in [3]).

IV. \(F_{\rho,\varphi}[P_n]=\operatorname{Re} P_n^{(k)}(\rho e^{i\varphi})\), i.e.
\[ \mu_i=0,\quad i=0,1,\ldots,k-1;\qquad \mu_i=\frac{i!}{(i-k)!}\rho^{\,i-k}\cos(i-k)\varphi, \]
\[ i=k,k+1,\ldots,n, \tag{5} \]
where \(\rho>0,\ \varphi\in[0,2\pi]\). It suffices to consider \(\varphi\in[0,\pi]\).

From inequalities (2) there follows the following assertion for the functional (5): whatever the numbers \(\{t_i\}\) on \([0,1]\) and \(\{y_i\}\) on the circle of radius \(\rho\) may be, we have
\[ \max_{0\le\varphi\le 2\pi} N_F(\rho,\varphi) = \max_{0\le\varphi\le 2\pi}\sup_{P_n\in\mathscr{P}_t} \operatorname{Re} P_n^{(k)}(\rho e^{i\varphi}) \]
is attained at one of the polynomials \(\pm M_n(x)\).

It can also be shown that on \([0,\pi]\), for \(\rho\ge \rho_k\) (\(\rho_k\) is the greatest root of the polynomial \(R_{n+1}^{(k)}(x)\); \(\rho_k\le 1\)), there are \(n-k\) intervals \((\beta_m^{(k)}(\rho),\alpha_m^{(k)}(\rho))\), at whose points the extremal polynomials are polynomials different from \(\pm M_n(x)\), and moreover
\[ \beta_m^{(k)}(\rho),\alpha_m^{(k)}(\rho) \longrightarrow \frac{(2m-1)\pi}{2(n-k)} \quad (m=1,2,\ldots,n-k). \]

These intervals are noteworthy in that the order of growth of \(N_F^{(k)}(\rho,\varphi)\) (with respect to \(\rho\)) at each point of such an interval is equal to \(n-k-1\), i.e. is one less than the order of growth of
\[ \max_{0\le\varphi\le\pi} N_F^{(k)}(\rho,\varphi) = \left|M_n^{(k)}(-\rho)\right|. \]

V. \(F_\xi[P_n]=\displaystyle\int_0^\xi P_n(x)\,dx\), i.e.
\[ \mu_i=\frac{\xi^{i+1}}{i+1},\quad i=0,1,\ldots,n. \]

Consequently,
\[ \left|\sup_{\mathscr{L}_n\in\mathscr{P}_t}\int_0^\xi \mathscr{L}_n(x)\,dx\right| \le N_F(\xi) = \sum_{i=0}^n y_i \left| \frac{1}{R'_{n+1}(t_i)} \int_0^\xi R_{n+1,i}(x)\,dx \right|. \]

Denote by \(r\) the greatest of the roots of the polynomials
\[ \left\{\int_0^\xi R_{n+1,i}(x)\,dx\right\}_{i=0}^n \]
for \(-\infty<\xi<+\infty\). Obviously, \(r\ge0\). If \(r=0\), then at every point \(\xi\ne0\) of the real axis one of the polynomials \(\pm M_n(x)\) is extremal. If \(r>0\), then the polynomial \(M_n(x)\) (or \(-M_n(x)\)) is extremal at all points of the intervals \((-\infty,0)\), \((r,+\infty)\). With respect to the interval \((0,r]\) we note that in each particular case—when a point \(\xi_0\in(0,r]\) and a set of points \(\{t_i^{(0)}\}_{i=0}^n\) are prescribed—the determination of the extremal polynomial presents no difficulty.

Leningrad Electrotechnical Institute of Communications
named after M. A. Bonch-Bruevich

Received
3 III 1966

REFERENCES

1. A. C. Schaeffer, R. J. Duffin, Trans. Am. Math. Soc., 50, No. 3, 517 (1941).
2. V. A. Markov, On functions least deviating from zero in a given interval, St. Petersburg, 1892.
3. S. N. Bernstein, On V. A. Markov’s theorem, Collected Works, vol. 2, Publishing House of the Academy of Sciences of the USSR, 1954.