UDC 517.946

MATHEMATICS

E. V. VORONOVSKAYA

STRUCTURE OF EXTREMAL POLYNOMIALS WITH A COMMON SUBDISTRIBUTION

(Presented by Academician S. L. Sobolev on 10 II 1970)

The article preserves the terminology adopted in our monograph \((^1)\).

§ 1. A polynomial of a prescribed subdistribution. If \(P_n(x)\) is reduced on \([0,1]\), i.e. \(\max_{[0,1]} |P_n| = 1\), with full distribution
\[ (\overset{\pm}{\sigma_i})_1^s,\quad [P_n(\overset{+}{\sigma})=+1;\; P_n(\overset{-}{\sigma})=-1], \]
then its resolvent
\[ R_s(x)=\prod_1^s (x-\sigma_i) \]
is also called the resolvent of the distribution. A simple node of \(P_n(x)\) will mean a point \(\sigma \in [0,1]\) at which \(P_n(\sigma)=\pm 1\), and, if \(0<\sigma<1\), then \(P_n'(\sigma)=0,\; P_n''(\sigma)\ne 0\); while if \(\sigma=0\) or \(1\), then \(P_n'(\sigma)\ne 0\). In all cases the square (conditional) of the resolvent is called: for interior nodes
\[ R_s^2(x)=\prod_1^s (x-\sigma_i)^2; \]
for one endpoint node
\[ R_s^2(x)=x\prod_2^s (x-\sigma_i)^2 \]
or
\[ R_s^2(x)=(1-x)\prod_1^{s-1}(x-\sigma_i)^2, \]
for two endpoint nodes:
\[ R_s^2(x)=x(1-x)\prod_2^{s-1}(x-\sigma_i)^2. \]
In all cases on \([0,1]\) we have
\[ R_s^2(x)\ge 0. \]

Theorem 1. If \(P_n(x)\) is a polynomial of class I or II \((^1)\) (reduced!) with simple nodes and with full distribution \((\overset{\pm}{\sigma_i})_1^s\) \((s\ge 2\) and there is alternation), then there always exist reduced polynomials with the same full distribution of the form
\[ P_n(x)-\alpha R_s^2(x) \tag{1} \]
for sufficiently small \(|\alpha|\).

Indeed, (1) gives
\[ P_n(\overset{\pm}{\sigma})-\alpha R_s^2(\overset{\pm}{\sigma})=\pm 1; \]
it remains to satisfy the reducedness condition. Let \(\alpha>0\); then the condition
\[ 0\le \alpha R_s^2(x)\le P_n(x)+1, \]
which is always possible. For \(\alpha<0\) we have
\[ P_n(x)-1\le \alpha R_s^2(x)\le 0, \]
which is also possible.

Corollary. If \(P_n(x)\) is of class II \((n<\) degree of \(R_s^2(x))\), then (1) gives the nearest polynomials in degree with distribution \((\overset{\pm}{\sigma_i})_1^s\). If \(P_n(x)\) is of class I, there is always an infinite set of polynomials of degree \(n\) with the same \((\overset{\pm}{\sigma_i})_1^s\).

Denote by \(M_{n,s}\) the set of all reduced polynomials of exactly degree \(n\) and lower, containing \((\overset{\pm}{\sigma_i})_1^s\) as their subdistribution; then between any two elements of \(M_{n,s}\) the relation \((^1)\) holds
\[ Q_n^{(2)}(x)=Q_n^{(1)}(x)-\alpha\varphi_k(x)R_s^2(x), \tag{2} \]

where $\alpha \gtreqless 0$ and $\varphi_k(x)=x^k+\cdots$; $k+2s\le n$. In view of (2), $M_{n,s}$ is expressed through any $L_n(x)\in M_{n,s}$:

\[ M_{n,s}=\{L_n(x)-\alpha\varphi_k(x)R_s^2(x)\} \tag{3} \]

for all admissible $\varphi_k(x)$. We shall distinguish two cases. A. $M_{n,s}$ contains a unique $L_n(x)$. B. $M_{n,s}$ is a set, always infinite (for example, weighted means). An example of case A is given by the subdistribution
\[ T_n(x)=\cos n\arccos(2x-1) \]
with two neighboring nodes ([1], p. 65); for case B see [1], p. 73.

On the basis of Theorem 1: if $L_n(x)$ is of class I, then we always have case B. Case A is possible only when $L_n(x)$ is of class II (with a subdistribution of class I); but here case B is also possible.

Remark 1. Suppose $M_{n,s}$ has been constructed by formula (3). If $M_{p,s}$ exists for $p>n$ (the polynomials of exact degree $p$ must enter into $M_{p,s}$), then it is always an infinite set, and
\[ Q_p^{(2)}(x)=Q_p^{(1)}(x)=\alpha\varphi_k(x)R_s^2(x) \]
$(k=0,1,\ldots,p-2s)$, and if $k+2s<p$, then also

\[ Q_p^{(1)}(x)-\alpha\psi_l(x)\varphi_k(x)R_s^2(x)\in M_{p,s} \tag{4} \]

provided that $l+k+2s\le p$ and $0\le \psi_l\le 1$ on $(0,1)$.

Remark 2. If $M_{n,s}$ is a set, then $M_{n+1,s}$ exists and contains polynomials with different signs of the leading coefficients $(q_{n+1})$. Indeed, in the form (3), where $k+2s\le n$, in passing to the form (4) we choose $\psi(x)=x^{n+1-2s-k}$ and $\psi(x)=(1-x)x^{n-2s-k}$.

Theorem 2. If, for the constructed $M_{n,s}$, the set $M_{p,s}$ $(p\ge n)$ exists, then it always contains $Q_{p(\max)}^{(x)}(x)$ and $Q_{p(\min)}^{(x)}(x)$—polynomials with $q_p=\max$ and $q_p=\min$, and each of them is unique in $M_{p,s}$.

Indeed, $(q_p)$ is a bounded set, and $Q_{p(\max)}$, $Q_{p(\min)}$ exist.

Suppose there are two $Q_{p(\max)}^{(1)}(x)$ and $Q_{p(\max)}^{(2)}(x)$; then
\[ Q_{p(\max)}^{(2)}(x)=Q_{p(\max)}^{(1)}(x)-\alpha\varphi_k(x)R_s^2(x), \]
where $k+2s<p$; in the case $\operatorname{sgn}(-\alpha)=\operatorname{sgn}q_{p(\max)}$ we take, according to the scheme (4), $\psi(x)=x^{p-k-2s}$; in the case $\operatorname{sgn}\alpha=\operatorname{sgn}q_{p(\max)}$, we take $\psi(x)=(1-x)x^{p-k-2s-1}$. In both cases we obtain a polynomial with $q_p>q_{p(\max)}$.

Remark 3. Neither $q_{p(\max)}$ nor $q_{p(\min)}$ is equal to zero for $p>n$ (see Remark 2). But also for $p=n$: suppose $q_{n(\min)}=0$; then $M_{n,s}$ is lowered, i.e., there is $M_{n-1,s}\subset M_{n,s}$; and then in $M_{n,s}$ it is necessary that $q_{n(\max)}$ and $q_{n(\min)}$ have different signs.

Corollary 1. A necessary and sufficient condition for the nonlowerability of $M_{n,s}$ is that in it $\{q_n\}$ is strictly sign-constant.

Corollary 2. If $M_{n,s}$ is a set, then $Q_{n(\max)}(x)$ and $Q_{n(\min)}(x)$ are always of class II, since, if $L_n(x)\in M_{n,s}$ and is of class I, then, according to the form (1), its leading coefficient can give neither a max nor a min.

§ 2. Indices of subdistributions

Suppose a complete distribution $L_n(x)$ contains $p$ nodes; from it a subdistribution of class I
\[ (\sigma_i^{\pm})_1^s \]
is chosen; let
\[ (\sigma_i^{\pm})_1^{s'} \]
be the remaining subdistribution $(p=s+s')$. Then

\[ M_{n,s}=\{L_n(x)-\alpha\varphi_k(x)R_s^2(x)\}; \tag{5} \]

here $k+2s\le n$.

Definition. The index of $L_n(x)$ relative to $(\sigma_i^{\pm})_1^s$ is the integer nonnegative number $k_{(s)}^*$ equal to the least possible degree of $\varphi_k(x)$ in the scheme (5). This number is, obviously, always unique.

Corollary 1. Since the reducedness of the polynomials in scheme (5) requires that

\[ L_n(x)-1 \leq \alpha \varphi_k(x)R_s^2(x) \leq L_n(x)+1, \]

the curve \(\alpha\varphi_k(x)R_s^2(x)\) passes through, with tangency (one-sided or two-sided), all the points \((\bar{\sigma}_i^\pm)_1^s\), while the points \((\sigma_i^{\prime\pm})_1^{s'}\) must be passed through by means of \(\varphi_k(x)\). Thus, \(k^*_{(s)}\) is the minimum number of roots on \([0,1]\) necessary for this. Consequently, we have

\[ \varphi_k(x)=\prod_1^{k_s^*}(x-\lambda_i). \]

Corollary 2. If in scheme (5) it turns out that \(k_s^*>n-2s\), then \(\alpha=0\) and \(M_{n,s}\equiv L_n(x)\) (in this case \(L_n(x)\) is always of class II), and conversely. Thus, the necessary and sufficient condition for the existence of the set \(M_{n,s}\) is

\[ 0\leq k_s^* \leq n-2s. \]

Corollary 3. If \(M_{n,s}\) is a set, then it always contains “fully impoverished” polynomials, i.e. such polynomials whose complete distribution is

\[ (\sigma_i^\pm)_1^s. \]

Indeed, let us form the subset

\[ \left\{\,L_n(x)-\alpha\prod_1^{k^*}(x-\lambda_i)R_s^2(x)\,\right\}\subset M_{n,s}, \tag{6} \]

where \(k^*_{(s)}\leq n-2s\) and the \((\lambda_i)\) are all distinct in the admissible intervals on \([0,1]\). After choosing the \((\lambda_i)\), for sufficiently small \(|\alpha|\) we obtain fully impoverished polynomials.

Corollary 4. If \(L_n(x)\) has index \(k^*_{(s)}=n-2s\), then the entire set \(M_{n,s}\) is expressed in the form

\[ M_{n,s}=\left\{\,L_n(x)-\alpha\prod_1^{\,n-2s}(x-\lambda_i)R_s^2(x)\,\right\} \tag{7} \]

with a unique representation for each \(P_n(x)\in M_{n,s}\).

Theorem 3. If \(L_n(x)\) has index \(k^*_{(s)}(\leq n-2s)\), then in every subset of the form

\[ \left\{\,L_n(x)-\alpha\prod_1^{k_s^*}(x-\lambda_i)R_s^2(x)\,\right\}=\widetilde{M}_{n,s} \tag{8} \]

under all values of \((\lambda_i)\) and \(\alpha\) admissible by the reducedness conditions, the number \(\alpha\) remains sign-constant.

Let \(\widetilde{M}_{n,s}\) contain \(P_n^{(1)}(x)\) and \(P_n^{(2)}(x)\) with the corresponding \((\lambda_i),\alpha_0\) and \((\lambda_i'),\alpha_0'\). Put \(\alpha_0>0,\ \alpha_0'<0\); for all \(0<\alpha<\alpha_0\) and \(\alpha_0'<\alpha'<0\) the reducedness is preserved. Then the polynomial

\[ L_n(x)-\frac12\bigl[\alpha\prod(x-\lambda_i)+\alpha'\prod(x-\lambda_i')\bigr]R_s^2(x) \]

belongs to \(M_{n,s}\), and, choosing \(\alpha'=-\alpha\), we obtain polynomials of the form

\[ L_n(x)-\beta\varphi_{k_s^*-1}(x)R_s^2(x)\in\widetilde{M}_{n,s}, \]

which contradicts the condition.

Corollary. If \(k_s^*=n-2s\), then \(\widetilde{M}_{n,s}\equiv M_{n,s}\) and, consequently, \(L_n(x)\) is one of the extreme polynomials; for \(\alpha>0\), \(L_n\equiv Q_{n(\max)}\), for \(\alpha<0\), \(L_n\equiv Q_{n(\min)}\). Thus, the necessary and sufficient condition that \(L_n(x)\) be extreme is: its index \(k_{(s)}^*=n-2s\).

Sufficiency has been proved; necessity follows from the form

\[ M_{n,s}=\left\{\,Q_{n(\max)}(x)-\alpha\prod_1^{k_s^*}(x-\lambda_i)R_s^2(x)\,\right\}, \]

where, by virtue of the uniqueness of the polynomial with \(q_{n(\max)}\), it is necessary that \(k_s^*=n-2s\) and \(\alpha>0\) throughout \(M_{n,s}\). The case \(Q_{n(\min)}(x)\) is analogous.

Theorem 4. In \(M_{n,s}\), both extreme polynomials \(Q_{n(\max)}(x)\) and \(Q_{n(\min)}(x)\) have \((\overset{\pm}{\sigma_i})_1^s\) as their common greatest subdistribution.

Consider the unique representation

\[ Q_{n(\min)}(x)=Q_{n(\max)}(x)-\alpha^* \prod_1^{\,n-2s}(x-\lambda_i^*)R_s^2(x); \tag{9} \]

here \(\alpha^*=\max\). Suppose that, in addition to \((\overset{\pm}{\sigma_i})_1^s\), the extreme polynomials have at least one more common node, for definiteness \(\overset{+}{\sigma}\). Then formula (9) will take the form (after a change of numbering)

\[ Q_{n(\min)}(x)=Q_{n(\max)}(x)-\alpha^* \prod_1^{\,n-2s-2}(x-\lambda_i^*)R_s^2(x)(x-\sigma)^2 . \]

We shall prove that here \(\alpha^*\) can be increased by replacing \((x-\sigma)^2\) by \((x-\lambda')(x-\lambda'')\). Indeed, if, for example, we set \(\lambda'=\sigma-\varepsilon\) and \(\lambda''=\sigma+\varepsilon\), then \((x-\lambda')(x-\lambda'')=(x-\sigma)^2-\varepsilon^2\). Further, since the curve

\[ \alpha^* \prod_1^{\,n-2s-2}(x-\lambda_i^*)R_s^2(x)(x-\sigma)^2 \]

must lie within the bounds \(Q_{n(\max)}(x)\pm 1\), with some points of tangency, then upon replacing it by

\[ \alpha^* \prod_1^{\,n-2s-2}(x-\lambda_i^*)R_s^2(x)(x-\lambda')(x-\lambda''), \]

this curve remains within the same bounds for sufficiently small \(\varepsilon\), no longer having points of tangency with the bounds. Consequently, \(\alpha^*\) can be increased, which is impossible.

The results obtained answer the questions of uniqueness and non-reducibility of the solution in the problem of V. A. Markov \((^1)\). This problem is equivalent to finding an extremal polynomial of a not absolutely monotone segment-functional \((\mu_k)_0^n\). Replacing the segment by \(\mu_0,\mu_1,\ldots,\mu_{n-1},\theta\), we have, for any \(-\infty<\theta<+\infty\), a segment of class II except perhaps for only one point \(\theta=\theta^0\). Thus, only at \(\theta^0\) can the solution of the problem fail to be unique and can it be reducible. This point, as well as the true distribution \((\overset{\pm}{\sigma_i})_1^s\) of the corresponding segment, are found by an algebraic method \(((^1), p. 62)\).

The principal extremal polynomial \(Q_N(x)\) for such a distribution is constructed by the methods indicated in \(((^1), p. 21)\). By the very meaning of the problem, \(N\le n\). If \(N<n\), the solution is reducible, and the question of uniqueness is determined by the index of \(Q_N(x)\) relative to \((\overset{\pm}{\sigma_i})_1^s\); for \(k_s^*>n-2s\), the solution (of degree not exceeding \(n\)) is unique; for \(k_s^*\le n-2s\), there exists \(M_{n,s}\), the set of solutions of the problem. If \(N=n\), then for \(k_s^*>n-2s\) we have both uniqueness and non-reducibility; for \(k_s^*\le n-2s\), there exists a non-reducible set of solutions \(M_{n,s}\), and the point \(\theta^0\) has been called singular by us \(((^1), p. 72)\).

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

Received
29 I 1970

CITED LITERATURE

\(^1\) E. V. Voronovskaya, The Method of Functionals and Its Applications, L., 1963.