UDC 517.51

MATHEMATICS

E. V. VORONOVSKAYA

STRUCTURAL PROPERTIES OF SOME DEFINING FUNCTIONALS

(Presented by Academician S. L. Sobolev on 24 VII 1967)

It was proved in \((^{1})\) that a segment-functional of the form

\[ (\mu_i)_0^n = 0_0,\ldots,0_{s-1},1_s,\theta_1,\ldots,\theta_l \qquad (s+l=n;\ s>n/2+1) \tag{\(*_s^+\)} \]

determines, for the variables \((\theta_i)\) in a bounded \(l\)-dimensional domain, all extremal polynomials \(\{Q_n(x)\}\) of class II \([0,s,0]\), and only them.

Since at each point \((\theta_i)_1^l\) of the infinite space the segment \((*_s^+)\) is also served by a unique extremal polynomial (only of another passport) \((^{1})\), it is of interest to determine the capacity of such service in view of the fact that segments of type \((*_s^+)\) are convenient for the analytic construction of extremal polynomials \((^{1})\), and also to generalize the form \((*_s^+)\).

Recall the general criterion of extremality \((^{1})\): \(Q_n(x)=\sum_{i=0}^{n} q_i x^i\) serves \((\mu_i)_0^n\) if and only if it is reduced, i.e.

\[ \max_{[0,1]} |Q_n(x)|=1, \]

and, for its distribution \(\overset{+}{\sigma_i}\) (i.e. for the points \(0\leq \sigma_1<\cdots<\sigma_s\leq 1\), where \(Q_n\overset{+}{\sigma}=+1;\ Q_n(\sigma)=-1\)), the system of linear equations

\[ \sum_{i=1}^{s} \delta_i \sigma_i^{k}=\mu_k \qquad (k=0,1,\ldots,n) \tag{V} \]

gives, for the loads \((\delta_i)\), solutions with \(\operatorname{sgn}\delta_i=Q_n(\sigma_i)\) or \(\delta_i=0\). If \(\delta_k\ne0\), we shall call \(\sigma_k\) an active node of \(Q_n(x)\). The totality of active nodes forms a subdivision, or selection, for \(Q_n(x)\) and is the true (unique) distribution of the segment-functional \((\mu_i)_0^n\). The whole system (V) is compatible.

Introduce the following abbreviated notation. If some segment \((a)\) is served by the polynomial \(Q_n(x)\), then the segment multiplied termwise by \(\alpha>0\), i.e. \(\alpha\cdot(a)\), is also served. If another segment \((b)\) has the same \(Q_n(x)\), then the segment

\[ (c)=\frac{\alpha\cdot(a)+\beta\cdot(b)}{\alpha+\beta} \]

also has it \((\alpha>0;\ \beta>0)\); moreover, if \((a)\) and \((b)\) have the form \((*_s^+)\), then so does \((c)\). If \((\delta_i')\) and \((\delta_i'')\) are the loads of \((a)\) and \((b)\), then the loads of \((c)\) are

\[ \frac{\alpha\cdot\delta_i' + \beta\cdot\delta_i''}{\alpha+\beta}. \]

Hence follows

Remark 1. The domain of service of \((*_s^+)\) by a certain selection \((\sigma_i)_1^{s_1}\) of the polynomial \(Q_n(x)\) forms a simply connected (convex) set.

In what follows we shall consider segments \((\nu_i)_0^n\) in which \(s\) \((1\leq s\leq n)\) parameters at any prescribed places are equal to zero. Among them there is always \(\nu_0=0\). The remaining parameters \((\theta_k)_{1}^{\,i+n-s}\) are variable, with the exception of \(\theta_{k_1}=1\) or \(-1\) \((k_1<k_2<\cdots<k_{n+1-s})\). We shall denote such segments by \(\nu_{(s)}^+\) and \(\nu_{(s)}^-\), respectively. Separating out all their constant parameters, we obtain the following “bases”:

\[ 0_0,0_{l_1},\ldots,1_{k_1},\ldots,0_{l_{s-1}}, \tag{\(1^+\)} \]

\[ 0_0,0_{l_1},\ldots,-1_{k_1},\ldots,0_{l_{s-1}}\quad (k_1\leq s). \tag{\(1^-\)} \]

Remark 2. The bases \((1\pm)\) require that the number \(s_1\) of nodes in the segments \((\nu_{(s)}^\pm)\) be \(s_1 \geq s+1\); when \(s_1=s+1\), the corresponding loads \((\delta_i)_1^{s+}\) give alternants of the form \(\ldots + - +\) for \(\nu_{(s)}^+\) and \((\ldots - + -)\) for \((\nu_{(s)}^-)\). This is obvious if, from system (V), one selects \(s+1\) equations with right-hand side corresponding, taking account of the numbering, to the numbers \((1\pm)\). The determinant of the selected system is \(>0\).

Theorem 1. Let \((\nu_{(s)}^+)\) be served at the points \((\theta_{k_i}^{(0)})_{i=1}^{n+1-s}\) by a sample \((\sigma_i)_1^{s_1}\) of the polynomial \(Q_n(x)\), with all active nodes and with loads \((\delta_i)_1^{s_1}\), where \(s+1\leq s_1<S\) (the total number of nodes of \(Q_n(x)\)). Then, when any \(\sigma^*\) from \(Q_n(x)\) is added to the sample, there will always be found a point \((\theta_{k_i}^{(1)})_{i=1}^{n+1-s}\) at which \(\nu_{(s)}^+\) is served by the entire sample supplemented by the node \(\sigma^*\).

Indeed, extend the basis \((1+)\) by zeros, \(s_1-s\) in number, at arbitrary free numbers; we obtain an incomplete segment \((^1)\) with \(s_1+1\) parameters (zeros and ones). Decompose it with respect to the nodes \((\sigma_i)_1^{s_1}\) with the adjoined \(\sigma^*\) (it remains unnumbered). We obtain loads \((\Delta_i)_1^{s_1}\) and \(\Delta^*\) for \((\sigma^*)\), of alternating signs in the order of increase of all \(\sigma\), according to Remark 2. Next we fill this segment structurally by these nodes. This means: construct the segment \((\varepsilon_i)_0^n\), in which the same extended basis is present, and at any free place with number \(p\) put
\[ \varepsilon_p=\sum_{i=1}^{s}\Delta_i\sigma_i^p+\Delta^*\sigma^{*p}. \]
Compare \((\Delta_i)_1^{s_1}\) and \((\delta_i)_1^{s_1}\). There are no zeros among them. Find \(a_i>0\) such that \(a_i|\Delta_i|=|\delta_i|\). Choose \(0<\alpha<\min(a_i)\); then \(\alpha|\Delta_i|<|\delta_i|\). If \(\operatorname{sgn}\Delta^*=Q_n(\sigma^*)\), construct the segment
\[ \frac{\nu_s^+ + \alpha(\varepsilon_i)_0^n}{1+\alpha} \]
with the same basis as \((\nu_s^+)\), and with loads
\[ \frac{\alpha\Delta_i+\delta_i}{1+\alpha} \quad\text{and}\quad \frac{\alpha\Delta^*}{1+\alpha}. \]
Since \(\operatorname{sgn}(\alpha\Delta_i+\delta_i)=\operatorname{sgn}\delta_i\), \(Q_n(x)\) is its extremal. If \(\operatorname{sgn}\Delta^*=-Q_n(\sigma^*)\), require additionally in the choice of \(\alpha\) that \(\alpha<1\), and construct the segment
\[ \frac{\nu_{(s)}^+ - \alpha(\varepsilon_i)_0^n}{1-\alpha} \]
also with the same basis and with loads
\[ \frac{\delta_i-\alpha\Delta_i}{1-\alpha} \]
and
\[ -\frac{\alpha\Delta^*}{1-\alpha}, \]
i.e. \(Q_n(x)\) is its extremal.

Corollary. There will always be found points \((\theta_{k_i})_{i=1}^{n+1-s}\) at which \((\nu_{(s)}^+)\) is served by the original sample of nodes of \(Q_n(x)\) with the addition of any number of nodes of \(Q_n(x)\) not included in it.

Remark 3. A sample \((\sigma_i)_1^{s_1}\) containing exactly \(q=s\) alternants contains, in any \(s+1\) of its points with a continuous alternant, only either a \((+)\)-alternant or a \((-)\)-alternant. If, however, in the sample \(q>s\), then it contains both \(s+1\) points with a \((+)\)-alternant and \(s+1\) points with a \((-)\)-alternant.

Remark 4. Every sample of nodes of \(Q_n(x)\) consisting of \(s+1\) points with a \((+)\)-alternant serves only either \((\nu_{(s)}^+)\) or \((\nu_{(s)}^-)\), and moreover at only one point. Let \((\sigma_i)_1^{s+1}\) be this sample. The basis \((\nu_{(s)}^+)\), when decomposed with respect to these nodes, gives loads \((\delta_i)\), \(\operatorname{sgn}\delta_{s+1}=(-1)^{s+k}\), and the structural completion of the basis with respect to these nodes is unique.

Theorem 2. A subsample \((\sigma_i)_1^{s_1}\) of the polynomial \(Q_n(x)\), where \(s+1\leq s_1\leq S\), has the following properties: 1) if the number of alternants \(q=s\), then the sample serves one and only one of the two segments of the form \((\nu_{(s)}^+)\) or \((\nu_{(s)}^-)\); 2) if \(q<s\), the sample serves neither the one nor the other.

The proof is by induction.

Let \(s_1=s+1\). According to Remark 4, under the condition of a continuous alternant the sample serves one and only one of the segments of the form

\((v_{(s)}^{\pm})\) (and only at one point). Suppose the theorem has been proved for any sample with number of nodes \(\leq s_1\); we shall prove its validity for \((\sigma_i)_1^{s_1+1}\) with number of alternations \(q \leq s_1\).

1. Let \(q=s_1\), and, for definiteness, suppose that this is a \((+)\)-alternation (see Remark 3). Then one of the two intervals \((v_{(s)}^{\pm})\), at some point \((\theta_{k_i}^1)_{i=1}^{n+1-s}\), in accordance with Theorem 1, is served by the sample \((\sigma_i)_1^{\pm,s_1+1}\) with all active nodes and with loads \((\delta_i')\). Let this be an interval of the form \((v_{(s)}^+)\). Suppose now that \((\sigma_i)_1^{\pm,s_1+1}\) also serves \((v_{(s)}^-)\) at some point \((\theta_{k_i}^{\prime})_{i=1}^{n+1-s}\) with all active nodes and loads \((\delta_i'')\). Then one must have \(\operatorname{sgn}\delta_i'=\operatorname{sgn}\delta_i''\). In the equalities \(a_i\delta_i'=\delta_i''\) \((a_i>0)\), take \(\alpha=\min(a_i)=a_p\); then \(\alpha\delta_p'=\delta_p''\) and \(\alpha|\delta_i'|<|\delta_i''|\) for \(i\ne p\). Form the interval

\[ (\mu_i)_0^n=\frac{(v_{(s)}^-)-\alpha_p(v_{(s)}^+)}{1+\alpha}, \]

it has the nodes \((\sigma_i)\) for \(i\ne p\), with loads

\[ \frac{\delta_i''-\alpha_p\delta_i'}{1+\alpha} \]

in number \(\leq s_1\), while the number of alternations \(q'\leq s\). By the induction hypotheses, \(q'<s\) is impossible. If, however, \(q'=s\), then a \((+)\)-alternation has been preserved, and then \((\mu_i)_0^n\) of type \((v_{(s)}^-)\) is served by the same sample as is \((v_{(s)}^+)\); this too is impossible by the induction hypothesis.

1. Let in \((\sigma_i)_1^{\pm,s_1+1}\) we have \(q<s\). Suppose that at the point \((\theta_{k_i}^{(0)})_{i=1}^{n+1-s}\) the sample serves \((v_{(s)}^+)\). From the general system (V), which gives the structural decomposition of \((v_{(s)}^+)\) by the nodes \((\sigma_i)_1^{s_1+1}\), choose \(s_1+1\) equations with \(s_1+1\) unknowns \((\Delta_i)\). In this system take all \(s+1\) equations with right-hand side equal to the \(s+1\) basis parameters. We shall vary one of the parameters \(\theta_l^{(0)}\) of this system by setting \(\theta_l^{(0)}=\theta\). Then all \(\Delta_i\) become linear functions of \(\theta\). Let us find, closest to \(\theta_l^{(0)}=\theta\), a value of \(\theta\) at which at least one of the loads \(\Delta_i\) becomes zero (the others retain their initial signs). Let this occur at \(\theta=\theta^*\). Then the interval \((v_{(s)}^+)\), in which \(\theta_l^{(0)}\) is replaced by \(\theta^*\), is served by \((\sigma_i)^{\pm}\) with one (at least) node removed, which does not increase the number of alternations; this is impossible by the induction hypothesis. Theorem 2 is proved.

For simplicity of the subsequent exposition, we shall consider only special intervals of the form \(0_0,\ldots,0_{s-1},\pm 1_s,\theta_1,\ldots,\theta_{n-s}=(\mu_{(s)}^{\pm})\). All results will also be valid in the general case. We shall compare \((\mu_{(s)}^{\pm})\), \((\mu_{(s+1)}^{\pm})\), and \((\mu_{(s-1)}^{\pm})\). If in \((\mu_{(s)}^{\pm})\) the \((\theta_i)\) are variables and range over a domain of the form \(\theta_i=\lambda\theta_i' + \theta_i''\), where \(\lambda\to\infty\) and \(\theta_i'\ne0\), we shall call such a domain a ray (the simplest cone).

Theorem 3. If \(Q_n(x)\) has a subdistribution \((\sigma_i)_1^{\pm,s_1}\) containing a number of alternations \(q\geq s+1\), then this sample serves an interval of the form \((\mu_{(s)}^{\pm})\) in an infinite domain of the \((\theta_i)\) of ray type.

Take from the sample two groups of \(s+1\) points (they may have common nodes), one forming a \((+)\)-alternation, the other a \((-)\)-alternation. Find the decompositions of the two bases \(0_0,\ldots,0_{s-1},\pm1\) respectively by these groups of points and, respectively, continue them in a structural way. We have:

\[ 0_0,\ldots,0_{s-1},1,\theta_1^{(1)},\ldots,\theta_{n-s}; \tag{2} \]

\[ 0_0,\ldots,0_{s-1},-1,\theta_1^{(2)},\ldots,\theta_{n-s}^{(2)}; \tag{3} \]

\(Q_n(x)\) is extremal for (2) and for (3). On the basis of Theorem 2 the original subdistribution also serves some interval

\[ 0_0,\ldots,0_s,1_{s+1},\theta_2^{(0)},\ldots,\theta_{n-s}^{(0)}. \tag{4} \]

Compose \(\lambda(4)+(2)\) (or \(\lambda\cdot(4)+(3)\)), where \(\lambda>0\) is arbitrary. We obtain
\[ 0_0,\ldots,0_{s-1},1,\lambda+\theta^{(1)}_1,\ldots,\lambda\theta^{(0)}_{n-s}+\theta^{(1)}_{n-s}, \]
and the required ray has been found, if all \(\theta_i^0\ne 0\); but this can always be achieved by a small change of the parameters in the system of equations (V).

Corollary. If \(Q_n(x)\) serves \((\mu^+_{s})\) in a finite domain \((\theta_i)\), then it does not serve at all a segment of the form \((\mu_{(s+1)})\) (i.e. one in which the number of alternations is \(q=s\)).

Theorem 4. If a sample \((\sigma_i)^{s_1}_1\) serves the segment \((\mu^+_{(s)})\) in an infinite domain, then this domain contains a ray.

Let us note that for \(s_1=s+1\) the sample can serve \((\mu^+_{(s)})\) only at one point. Let \(s_1=s+2\), and let the sample serve \((\mu^+_{(s)})\) in an infinite (one-dimensional!) domain. At any point of this domain the loads \((\Delta_i)^{s+2}_1\) are determined from the first \(s+2\) equations (V), and therefore are linear functions of one parameter \(\theta_1\), i.e. \(\Delta_i=\alpha_i\theta_i+\beta_i\) \((\alpha_i\ne 0)\); the remaining
\[ \theta_k=\sum_{i=1}^{s+2}\Delta_i\sigma_i^{k+s} =\theta_1\sum_{i=1}^{s+2}\alpha_i\sigma_i^{s+k} +\sum_{i=1}^{s+2}\beta_i\sigma_i^{s+k}. \]

Consequently, \(\theta_1\) assumes arbitrarily large values (according to what was noted above, one may assume \(\theta_k\ne 0\)). Then finally we obtain a functional of the form
\[ 0_0,\ldots,0_{s-1},1_s,\theta_1,\gamma_2\theta_1+\beta_2,\ldots,\gamma_{n-s}\theta_1+\beta_{n-s},\quad \gamma_i\ne 0, \tag{5} \]
i.e. a ray. These conclusions remain valid also for \(s_1>s+2\), since the domain \((\theta_i)\) served by a poorer subdistribution \(Q_n(x)\) is a boundary domain for the domain served by an enriched subdistribution (see Theorem 1).

Corollary 1. The sample of Theorem 4 also serves \((\mu_{(s+1)})\).

Indeed, divide (5) by \(|\theta_1|\), and in the limit as \(\theta_1\to\infty\) we obtain
\[ 0_0,\ldots,0_{s-1},0_s,\gamma_2,\ldots,\gamma_{n-s}. \]
The mentioned sample, serving \((\mu_{(s+1)})\), must contain not fewer than \(s+1\) alternations.

Corollary 2. A subdistribution \(Q_n(x)\) containing the number of alternations \(q_{\max}=s\) serves \((\mu_{(s)})\) in a finite domain, since otherwise there would, in the contrary case, be a ray in this domain, and then the subdistribution would serve \((\mu_{(s+1)})\), which is impossible.

Remark. The theorems proved extend in a completely analogous way to the case when the initial segments \((\mu_i)^n_0\) contain \(S\) zeros arranged arbitrarily, under the condition \(\mu_0=0\).

Finally, let us note an application of the proved theorems to the problem of V. A. Markov \((^2)\) on finding a polynomial \(Y_n(x)\) least deviating from 0 on \([0,1]\) among those whose coefficients \((y_i)\) are connected by the relation
\[ \sum_{i=0}^{n}\mu_i y_i=A\;(\ne 0). \tag{6} \]

Since this problem is identical to the problem of finding the extremal polynomial \(Q_n(x)\) of the segment-functional \((\mu_i)^n_0\) \((^1)\), we immediately obtain:

1. Whatever the relation (6), containing \(l\) coefficients (but not containing \(y_0\)), \(Y_n(x)\) has in its active distribution not fewer than \(n+1-l\) alternations.

2. If, in addition, (6) contains \(l<n/2\) coefficients, then the solution of Markov’s problem is unique.

Leningrad Electrotechnical
Institute of Communications

Received
24 V 1967

CITED LITERATURE

1. E. V. Voronovskaya, The Method of Functionals and Its Applications, Leningrad, 1963.
2. V. A. Markov, On Functions Least Deviating from Zero, 1892.