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UDC 513.531
MATHEMATICS
V. D. KOROMYSLICHENKO
ON THE DIRECT AND INVERSE PROBLEM OF V. A. MARKOV IN THE COMPLEX DOMAIN
(Presented by Academician A. N. Kolmogorov on 4 V 1965)
1. Let continuous complex-valued functions \(f(z)\), \(\{\varphi_j(z)\}_0^n\) be given on a bicompact Hausdorff space \(G\) \((^1)\). We regard the functions \(\{\varphi_j(z)\}_0^n\) as linearly independent. Consider the following generalization of the work of V. A. Markov \((^2)\). Among all polynomials
\[ F(a;z)=\sum_{j=0}^{n} a_j\varphi_j(z),\quad z\in G, \]
whose coefficients satisfy the linearly independent relations
\[ \sum_{j=0}^{n} a_j\alpha_j^{(t)}=\alpha_t,\quad t=1,\ldots,p, \tag{1} \]
find a polynomial \(F(a^*;z)\) for which the condition
\[ \min_{a_j}\max_{z\in G}\left|\sum_{j=0}^{n}a_j\varphi_j(z)-f(z)\right| = \max_{z\in G}\left|\sum_{j=0}^{n}a_j^*\varphi_j(z)-f(z)\right| =\rho \tag{2} \]
is fulfilled.
Polynomials \(F(a;z)\) whose coefficients satisfy the relations (1) will be called admissible.
Works \((^{2-16})\) and others are devoted to the problem of Chebyshev approximation without constraints and with constraints.
In the present paper* the results of the author’s work \((^{11,12})\) are generalized to problem (2) with constraints (1). The following notation is used:
\[ \operatorname{Re}\varphi_j(z)=\varphi_j'(z),\quad \operatorname{Im}\varphi_j(z)=\varphi_j''(z),\quad \varphi_j(z)=\varphi_j'(z)+i\varphi_j''(z), \]
\[ \alpha_j^{(t)}=\alpha_j^{(t)'}+i\alpha_j^{(t)''},\quad \alpha_t=\alpha_t'+i\alpha_t'',\quad \delta(z_s)=F(a;z_s)-f(z_s), \]
\[ \operatorname{sgn}\delta(z_s)=e^{i\theta_s},\quad \theta_s=\arg\delta(z_s),\quad \varphi_j^*(z)=\varphi_j(z)\operatorname{sgn}\overline{\delta(z)}, \]
\[ \operatorname{sgn}\overline{\delta(z)}=e^{-i\theta_s},\quad \varphi_j^*(z)=\varphi_j^{*'}(z)+i\varphi_j^{*''}(z),\quad F^*(a;z)=F(a;z)\operatorname{sgn}\overline{\delta(z)}. \]
Consider the systems of equations:
\[ \sum_{s=1}^{r}K_s^{(t)}\varphi_j^{*'}(z_s) + \sum_{s=r+1}^{2n+2}K_s^{(t)}d_j'(s) = \alpha_j^{(t)'},\quad j=0,\ldots,n; \tag{3} \]
\[ \sum_{s=1}^{r}K_s^{(t)}\varphi_j^{*''}(z_s) + \sum_{s=r+1}^{2n+2}K_s^{(t)}d_j''(s) = \alpha_j^{(t)''},\quad j=0,\ldots,n;\ t=1,\ldots,p. \]
\[ \sum_{s=1}^{r}K_s^{(t)*}\varphi_j^{*}(z_s) + \sum_{s=r+1}^{2n+2}K_s^{(t)*}d_j(s) = -\alpha_j^{(t)''},\quad j=0,\ldots,n; \tag{4} \]
* The results of the work were reported at a meeting of the Academic Council of the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR on 23 I 1963.
\[ \sum_{s=1}^{r} K_s^{(t)*}\varphi_j^{*\,\prime}(z_s) + \sum_{s=r+1}^{2n+2} K_s^{(t)*}d_j^{\prime\prime}(s) = \alpha_j^{(t)\prime}, \qquad j=0,\ldots,n;\quad t=1,\ldots,p. \]
In the systems (3), (4) the number \(r\ge 1\), and the \(2n+2\) vectors
\[ \vec{\varphi}_s^{\,*} = [\varphi_0^{*\,\prime}(z_s),\varphi_0^{*\,\prime\prime}(z_s),\ldots, \varphi_n^{*\,\prime}(z_s),\varphi_n^{*\,\prime\prime}(z_s)], \qquad s=1,\ldots,r; \]
\[ \mathbf d_s = [d_0'(s),d_0''(s),\ldots,d_n'(s),d_n''(s)], \qquad s=r+1,\ldots,2n+2, \]
are linearly independent, where \(d_\nu'(s), d_\nu''(s)\) are real numbers.
Lemma 1. The vectors \(\mathbf K^{(t)}=[K_1^{(t)},\ldots,K_{2n+2}^{(t)}]\), \(\mathbf K^{(t)*}=[K_1^{(t)*},\ldots,K_{2n+2}^{(t)*}]\), \(t=1,\ldots,p\), are linearly independent.
From (3), (4) we obtain
\[ \sum_{s=1}^{r} K_s^{(t)}\varphi_j^*(z_s) + \sum_{s=r+1}^{2n+2} K_s^{(t)}d_j(s) = \alpha_j^{(t)}, \qquad j=0,\ldots,n;\quad t=1,\ldots,p; \tag{5} \]
\[ \sum_{s=1}^{r} K_s^{(t)*}\varphi_j^*(z_s) + \sum_{s=r+1}^{2n+2} K_s^{(t)*}d_j(s) = i\alpha_j^{(t)}, \qquad j=0,\ldots,n;\quad t=1,\ldots,p. \tag{6} \]
From (5), (6), analogously to what was done in \((^{11},{}^{12})\), we obtain
\[ \sum_{s=1}^{r} K_s^{(t)}\widetilde F^{*\,\prime}(\widetilde A;z_s) + \sum_{s=r+1}^{2n+2} K_s^{(t)}\widetilde\Phi^{\prime}(\widetilde A;s) \equiv 0, \qquad t=1,\ldots,p; \tag{7} \]
\[ \sum_{s=1}^{r} K_s^{(t)*}\widetilde F^{*\,\prime}(\widetilde A;z_s) + \sum_{s=r+1}^{2n+2} K_s^{(t)*}\widetilde\Phi^{\prime}(\widetilde A;s) \equiv 0, \qquad t=1,\ldots,p, \tag{8} \]
where \(\widetilde F^{*\,\prime}(\widetilde A;z_s)\), \(\widetilde\Phi^{\prime}(\widetilde A;s)\) are real forms with free parameters.
Theorem 1. In order that the vectors \(\mathbf K_s=[K_s^{(1)},\ldots,K_s^{(p)},K_s^{(1)*},\ldots,K_s^{(p)*}]\), \(s=1,\ldots,2p\), corresponding to the identities (7), (8), be linearly independent, it is necessary and sufficient that the forms \(\widetilde F^{*\,\prime}(\widetilde A;z_l)\), \(\widetilde\Phi^{\prime}(\widetilde A;m)\) \((l,m\ne s_1,\ldots,s_{2p})\) be linearly independent.
Let \(F(a^*;z)\) be a solution of problem (2). The identity is known \((^{6},{}^{7})\)
\[ \sum_{s=1}^{r} \lambda_s \widetilde F^*(\widetilde A;z_s)\equiv 0, \qquad \lambda_s>0, \qquad s=1,\ldots,r, \tag{9} \]
to which correspond the two identities
\[ \sum_{s=1}^{r} \lambda_s \widetilde F^{*\,\prime}(\widetilde A;z_s)\equiv 0, \qquad \lambda_s>0, \tag{10} \]
\[ \sum_{s=1}^{r} \lambda_s \widetilde F^{*\,\prime\prime}(\widetilde A;z_s)\equiv 0, \qquad \lambda_s>0, \tag{11} \]
with linear dependence in the narrow sense \((^7)\) among the real forms, where \(\{z_s\}_1^r\) are the Chebyshev deviation points* \((^7)\).
* For brevity, deviation points are the points of maximum deviation \(|F(a;z)-f(z)|\) on \(G\).
To the identities (10), (11) there corresponds the matrix
\[ \left\| \begin{array}{c} \varphi_j^{*'}(z_s)\\ \varphi_j^{*''}(z_s) \end{array} \right\|_{j=0,1,\ldots,n;\ s=1,2,\ldots,r}. \tag{12} \]
whose rank is not less than \(r-1\) by virtue of the linear dependence in the narrow sense of the forms \(\widehat F^{*'}(A;z_s)\) (or \(\widehat F^{*''}(A;z_s)\)).
Lemma 2. If the rank of the matrix (12) is equal to \(r-1\), i.e.
\[ \sum_{s=1}^{r} c_s \vec{\varphi}_s^{\,*}=0, \]
\(c_s\ne 0,\ s=1,\ldots,r\), then \(c_s=\lambda_s k,\ k=\mathrm{const}\).
Theorem 2. In order that an admissible polynomial \(F(a^*;z)\) be least deviating from the function \(f(z)\) on \(G\), it is necessary and sufficient that, for some subsystem of deviation points \(\{z_s\}_1^r\), one of the following two conditions be fulfilled:
1) in the case of linear independence of the vectors \(\vec{\varphi}_s^{\,*},\ s=1,\ldots,r\), corresponding to the chosen deviation points, for the numbers \(K_s^{(t)}, K_s^{(t)*}\) from (3), (4) the following conditions must be satisfied:
a) the rank \(m\) of the matrix
\[ \left\| \begin{array}{c} K_s^{(t)}\\ K_s^{(t)*} \end{array} \right\|_{t=1,2,\ldots,p;\ s=r+1,\ldots,2n+2}, \tag{13} \]
corresponding to the added vectors \(d_s\), is less than or equal to \(2p-1\);
b) the nonzero numbers
\[ \left| \begin{array}{cccc} K_{l_1}^{(t)} & \cdots & K_{l_{2p-1}}^{(t)} & K_s^{(t)}\\ K_{l_1}^{(t)*} & \cdots & K_{l_{2p-1}}^{(t)*} & K_s^{(t)*} \end{array} \right|_{t=1,\ldots,p;\ s=1,\ldots,2n+2} \tag{14} \]
must be of one sign, where in (14) the first \(2p-1\) columns are linearly independent and include \(m\) linearly independent columns of the matrix (13);
2) there exists a linear dependence in the narrow sense of the vectors \(\vec{\varphi}_s^{\,*}\)
\[ \sum_{s=1}^{r} c_s \vec{\varphi}_s^{\,*}=0, \tag{*} \]
where the numbers \(c_s,\ s=1,\ldots,r\), are of one sign.
From 1) of Theorem 2 there follows the identity
\[ \sum_{s=1}^{r} \lambda_s F^*(a;z_s)\equiv \sum_{j=0}^{n}\gamma_j a_j,\qquad \lambda_s>0,\qquad \gamma_j=\sum_{t=1}^{p}\mu_t\alpha_j^{(t)},\qquad j=0,\ldots,n; \tag{15} \]
\[ \lambda_s=\sum_{t=1}^{p}\left(\mu_t'K_s^{(t)}+\mu_t''K_s^{(t)*}\right),\qquad s=1,\ldots,r, \]
and the numbers \(\mu_t=\mu_t'+i\mu_t''\) are computed, in the notation of Theorem 2, from the system of equations
\[ \sum_{t=1}^{p}\left(\mu_t'K_\nu^{(t)}+\mu_t''K_\nu^{(t)*}\right)=0,\qquad \nu=1,\ldots,2p-1. \]
From (*) we obtain
\[ \sum_{s=1}^{r} c_s F^*(a;z_s)\equiv 0,\qquad c_s>0. \tag{16} \]
A number of other theorems have been obtained, formulated with and without added vectors \(d_s\), of the same type as in \((^{11,12})\).
For the value of the best approximation \(\rho\) in case 1) of Theorem 2, from (15) we obtain
\[ \rho=\left[\sum_{t=1}^{p}\mu_t\alpha_t-\sum_{s=1}^{r}\lambda_s\operatorname{sgn}\overline{\delta(z_s)}\,f(z_s)\right]\Big/\sum_{s=1}^{r}\lambda_s \tag{17} \]
and in case 2) of Theorem 2, from (16) we obtain:
\[ \rho=\left[-\sum_{s=1}^{r}c_s\operatorname{sgn}\overline{\delta(z_s)}\,f(z_s)\right]\Big/\sum_{s=1}^{r}c_s. \tag{18} \]
Corollary. If \(f(z)\equiv 0\) and in (1) \(\sum |a_t|>0\), then in Theorem 2 only case 1) occurs.
Remark 1. If \(f(z)\), \(\varphi_j(z)\), \(j=0,\ldots,n\), are only bounded and the set \(G\) is of arbitrary nature, then the study of the problem reduces to the preceding one if one considers all distinct vectors
\[
\vec{\varphi}^{(\nu)}(z_s)=
[\varphi_0^{(\nu)}(z_s),\ldots,\varphi_n^{(\nu)}(z_s),f^{(\nu)}(z_s)],
\quad z_s\in G,
\]
and takes their closure.
Remark 2. On the basis of the results obtained, a generalization to problem (2) has been obtained of the results of work \((^8)\) on the connection between Chebyshev approximations and the extremal moment problem.
2. Theorem 3. For any function \(f(z)\) and polynomial \(F(a;z)\) one can indicate \(p\) \((1\le p\le n+1)\) linearly independent relations under which the polynomial \(F(a;z)\) will be least-deviating from \(f(z)\) on \(G\).
Methods have been obtained for solving the inverse problem of V. A. Markov, analogous to those applied in the real domain \((^{12})\), etc.
Let us consider one of the methods for solving the inverse problem. For a given function \(f(z)\) and polynomial \(F(a^*;z)\), \(z\in G\), choose \(p\) linearly independent relations so that the points of maximal deviation \(\{z_s\}_1^r\), for which the vectors \(\varphi_s^*\), \(s=1,\ldots,r\), are linearly independent, form a Chebyshev subsystem of points of deviation, i.e., so that under the chosen relations the polynomial \(F(a^*;z)\) is a solution of problem (2).
To solve the problem, take numbers \(\lambda_s>0\), \(s=1,\ldots,r\), and put
\[
\sum_{s=1}^{r}\lambda_s\varphi_j^*(z_s)=\gamma_j,\quad j=0,\ldots,n.
\]
Take numbers \(\mu_t\) and \(p\) linearly independent vectors
\[
\vec{\alpha}^{(t)}=[\alpha_0^{(t)},\ldots,\alpha_n^{(t)}],
\quad t=1,\ldots,p,
\]
satisfying the condition
\[
\gamma_j=\sum_{t=1}^{p}\mu_t\alpha_j^{(t)},\quad j=0,\ldots,n.
\]
The sought relations will be
\[
\sum_{j=0}^{n}a_j\alpha_j^{(t)}=\alpha_t,\quad t=1,\ldots,p,
\]
where
\[
\alpha_t=\sum_{j=0}^{n}a_j^*\alpha_j^{(t)}.
\]
Institute of Mathematics
Academy of Sciences of the Ukrainian SSR
Received
20 IV 1965
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