UDC 517.946

MATHEMATICS

E. V. VORONOVSKAYA

ON ZOLOTAREV STABILITY OF FUNCTIONALS

(Presented by Academician S. N. Bernstein on 29 IV 1966)

Let \(Z_n(x,\theta)\) denote the polynomials determined by the segment-functional
\(0_0,0_1,\ldots,0_{n-2},1,\theta\) for
\((n-1)/2\leq \theta\leq (n+1)/2\) (the critical interval). They all belong to the passport \([n,n,0]\), and the nature of their deformation with respect to \(\theta\) has been studied in detail in \((^1)\). Recall that, under a monotone increase of \(\theta\), all interior nodes (extremum points) shift to the right. The polynomials \(\pm Z_n(x,\theta)\) exhaust all polynomials of the indicated passport.

Definition. The segment-functional \(\mu_0,\ldots,\mu_{n-1},\theta\) possesses Zolotarev stability if, in the critical interval \(\mu_n'\leq \theta\leq \mu_n''\), the segment belongs to the passport \([n,n,0]\), and moreover is serviced either by all \(+Z_n(x,\theta)\), or by all \(-Z_n(x,\theta)\) (and only by them).

Theorem 1. Whatever the prescribed basis \(\mu_0,\mu_1,\ldots,\mu_{n-2}\), there always exist two numbers \(A_0'\leq A_0''\) such that a segment of the form \(\mu_0,\ldots,\mu_{n-2},A,\bar\theta\) is stable for \(A\geq A_0''\) and for \(A\leq A_0'\). In the first case it is serviced by \(Z_n(x,\theta)\), and in the second by \(-Z_n(x,\theta)\).

Expand \(\mu_0,\mu_1,\ldots,\mu_{n-2},A\) with respect to the nodes \([\sigma_i(\theta)]_1^n\) of any of the polynomials \(\pm Z_n(x,\theta)\) \((^1)\); we have

\[ \delta_j=R_{n-1}^{(j)}(\bar\mu,\theta)\,/\,R_n'(\sigma_j) \quad (j=1,2,\ldots,n), \tag{1} \]

where

\[ R_n(x)=\prod_1^n (x-\sigma_i);\qquad R_{n-1}^{(j)}(x)=R_n(x)/(x-\sigma_j). \]

The signs of the denominator in (1) alternate with \(j\); to obtain
\(\operatorname{Sgn}\delta_j=\pm Z_n(\sigma_j,\theta)\) it is necessary and sufficient that either
\(R_{n-1}^{(j)}(\bar\mu)\geq 0\) \((j=1,\ldots,n)\), or
\(R_{n-1}^{(j)}(\bar\mu)\leq 0\) \((j=1,\ldots,n)\) (zeros are also possible). If

\[ R_{n-1}^{(j)}(x,\theta)=x^n-s_1^{(j)}(\theta)x^{n-1}+s_2^{(j)}(\theta)x-\cdots, \]

we have

\[ R_{n-1}^{(j)}(\bar\mu,\theta) = A-s_1^{(j)}(\theta)\mu_{n-2} +s_2^{(j)}(\theta)\mu_{n-3} -\cdots +(-1)^{n-2}s_{n-1}^{(j)}(\theta)\mu_0 = A-M_j(\theta). \]

The family of continuous functions \([M_j(\theta)]_1^n\) on
\((n-1)/2\leq \theta\leq (n+1)/2\) is bounded. Let

\[ A_0''=\max_{(j,\theta)} M_j(\theta) \quad\text{and}\quad A_0'=\min_{(j,\theta)} M_j(\theta); \]

then for fixed \(A\geq A_0''\) or \(A\leq A_0'\) the \((\delta_i)\) have alternating signs. The parameter \(\bar\theta\) for each \(\theta\) is determined uniquely from the condition
\(R_n(\bar\mu)=0\), i.e.

\[ \bar\theta-s_1(\theta)A+s_2(\theta)\mu_{n-2}-\cdots+(-1)^n s_n(\theta)\mu_0=0 \]

for all \((n-1)/2\leq \theta\leq (n+1)/2\). This proves the theorem.

Corollary 1. For \(A_0'<A<A_0''\), the sign-alternation conditions for \((\delta_j)\) are not satisfied for the whole family \(+Z_n(x,\theta)\) (or \(-Z_n(x,\theta)\)); that is, the segment does not possess Zolotarev stability.

Thus, for every basis \(\mu_0,\mu_1,\ldots,\mu_{n-2}\) there exists a completely determined Zolotarev “critical” interval \((A_0',A_0'')\); then and only then is the segment \(\mu_0,\ldots,\mu_{n-2},\mu_{n-1},\bar\theta\) stable, if \(\mu_{n-1}\) is chosen outside or on the boundary of the interval \((A_0',A_0'')\).

Remark 1. \(A_0' = A_0''\) if and only if the basis \((\mu_k)_0^{n-2} \equiv 0\), since here the simultaneous possibility (for \(A=A_0'=A_0''\)) of two expansions with opposite signs is required.

Corollary 2. Introduce the following notation: \(A''(\theta)=\max_{(j)} M_j(\theta)\) and \(A'(\theta)=\min_{(j)} M_j(\theta)\). Then the necessary and sufficient condition that the segment \(\mu_0,\ldots,\mu_{n-2},\mu_{n-1},\bar\theta\) should under no \(\bar\theta\) belong to the passport \([n,n,0]\), i.e. should be completely “detached” from this passport, is the following: \(\mu_{n-1}-A''(\theta)<0\), \(\mu_{n-1}-A'(\theta)>0\) for all \(\theta\) in the interval \([(n-1)/2,(n+1)/2]\).

Remark 2. If the segment \(\mu_0,\ldots,\mu_{n-2},\mu_{n-1},\theta_0\) is served by some \(Z_n(x,\theta_0)\), then the segment \(\mu_0,\ldots,\mu_{n-2},\mu_{n-1}+A,\bar\theta+A\theta_0\) is served by the same polynomial for any \(A>0\).

Indeed, the segment \(0_0,\ldots,0_{n-2},1,\theta_0\) is served by the polynomial \(Z_n(x,\theta_0)\); consequently, the segment \(0_0,\ldots,0_{n-2},A,A\theta_0\) is also served, and then so is the termwise sum \(\mu_0,\ldots,\mu_{n-2},\mu_{n-1}+A,\bar\theta+A\theta_0\). Thus, serving by one polynomial is extended in two parameters.

Theorem 2. The Zolotarev interval \((A_0', A_0'')\) of any basis \(\mu_0,\mu_1,\ldots,\mu_{n-2}\) contains the critical (Chebyshev) interval of the parameter \(\mu_{n-1}\).

Thus, if \(\mu_{n-1}'\), \(\mu_{n-1}''\) are the endpoints of the critical interval for the variable parameter \(\mu_{n-1}\), then

\[ A_0' \leqslant \mu_{n-1}' < \mu_{n-1}'' \leqslant A_0''. \tag{2} \]

Indeed, the segment

\[ \mu_0,\ldots,\mu_{n-2},\mu_{n-1}''+h \tag{3} \]

for \(h>0\) is served by the polynomial \(T_{n-1}(x)\equiv Z_n(k,n/2)\) with all \(\delta_i\ne0\). By the theorem on continuous deformation (1), if \(\bar\theta^*(h)\) is the best continuation of (3) to the \(n\)-th place, then the segment \(\mu_0,\ldots,\mu_{n-2},\mu_{n-1}''+h,\bar\theta^*(h)\pm\varepsilon\) is served by some \(Z_n(x,n/2\pm\varepsilon)\). But if we take \(\mu_0,\ldots,\mu_{n-2},\mu_{n-1}''-h\) \((h>0)\), then \(T_{n-1}(x)\) is no longer suitable; consequently, some \(Z_n(x,n/2\pm\varepsilon)\) also will not be suitable. Hence \(\mu_{n-1}''-h=A<A_0''\) for every \(h>0\). Therefore, \(A_0''\geqslant\mu_{n-1}''\).

The left-hand part of inequality (2) is proved in the same way.

Corollary. A segment of the form \(\mu_0,\ldots,\mu_{n-2},\mu_n,\bar\theta\) has “partial Zolotarev stability in the interval \(\bar\theta^*\leqslant\bar\theta\leqslant\bar\theta\).”

Let us give some applications of the results obtained.

Example 1. The segment-functional

\[ 1,\rho,\ldots,\rho^{n-1},\rho^n+\bar\theta \tag{4} \]

for \(\rho>1\) is Zolotarev-stable, since here \(A-M_j(\theta)=R_{n-1}^{(j)}(\rho,\theta)>0\), i.e. the segment determines all \(Z_n(x,\theta)\). For \(0<\rho<1\) the segment \((\rho^i)_0^{n-1}\), when expanded with respect to arbitrary nodes \((\sigma_i)_1^n\) \((0\leqslant\sigma_i\leqslant1\) and \(\sigma_i\ne q)\), always gives one repetition of sign for the loads \((\delta_i)\). Consequently, the segment (4) is completely detached from the passport \([n,n,0]\). For \(\rho=1\) the segment (4) is served over the whole critical interval only by Chebyshev transforms \(T_n(ax)\), i.e. although (4) belongs only to the passport \([n,n,0]\), the requirement of completeness is not fulfilled—the serving is partial, but without involving extraneous passports.

Example 2. The segment-functional \((\mu_i)_0^n = 0_0,\ldots,0_{n-3},1_{n-2},n/2,\bar\theta\) gives an example of partial serving by Zolotarev polynomials, with the involvement of polynomials of another passport. Indeed, since here \(n/2=\mu_{n-1}''\), the basis \((\mu_i)_0^{n-1}\) is served by the polynomial \(T_{n-1}(x)\)

with one unloaded end node \(\sigma_1=0\). Let us find the boundaries of the Zolotarev critical interval from the conditions of Theorem 1, which here take the form
\[ A_0''=\max_{j,\theta}[\theta-\sigma_j(\theta)] \]
and
\[ A_0'=\min_{(j,\theta)}[\theta-\sigma_j(\theta)]. \]
For all \(\theta\) in the interval \([(n-1)/2,\ (n+1)/2]\) one has
\[ \theta-\sigma_n(\theta)\leq \theta-\sigma_{n-1}(\theta)\leq\cdots\leq \theta-\sigma_1(\theta). \]
Thus, the max is attained at \(\theta-\sigma_1(\theta)\), and the min at \(\theta-\sigma_n(\theta)\). Finally we have
\[ A_0''=(n+1)/2-\widetilde{\sigma}_1(>n/2); \]
\[ A_0'=(n-1)/2-\widetilde{\sigma}_{n-1}(<n/2-1), \]
where \(\widetilde{\sigma}_1\) and \(\widetilde{\sigma}_{n-1}\) are the corresponding nodes of
\[ T_n(x)=\cos n\arccos(2x-1). \]
Thus, the segment \((\mu_i)_0^n\) is served only by part of the polynomials \(Z_n(x,\theta)\), and only in part of the critical interval \(\bar{\theta}^{*}\leq \bar{\theta}\leq \theta''\). In the remaining part, service belongs to another passport.

Example 3. The segment \((\mu_i)_0^n=\alpha_0,\alpha_1,\ldots,\alpha_{n-1},\bar{\theta}\) with the amorphous basis \((\alpha_i)_0^{n-1}\) cannot, for any \(\bar{\theta}\), be served by primitive Zolotarev polynomials, i.e. by those among whose nodes \((\sigma_i)_1^n\) there are both \(\sigma_1=0\) and \(\sigma_n=1\). Indeed, if for some \(\bar{\theta}_0\) one has
\[ \alpha_k=\sum_1^{s_1}\delta_i'\sigma_i'^k-\sum_1^{s_2}\delta_i''\sigma_i''^k \quad (k=0,1,\ldots,n-1), \]
then the resulting equalities
\[ \alpha_k+\sum_1^{s_1}\delta_i''\sigma_i''^k = \sum_1^{s_1}\delta_i'\sigma_i'^k \quad (k=0,1,\ldots,n-1) \]
are impossible, since on the left we have an amorphous segment, and on the right a nodal one \((^2)\).

In this note the question of Zolotarev stability is studied from a somewhat different standpoint than in the article \((^3)\).

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

Received
6 III 1966

CITED LITERATURE

\(^1\) E. V. Voronovskaya, The Method of Functionals and Its Applications, L., 1963.
\(^2\) E. V. Voronovskaya, DAN, 166, No. 6 (1966).
\(^3\) E. V. Voronovskaya, DAN, 161, No. 2 (1965).