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Reports of the Academy of Sciences of the USSR
- Volume 195, No. 5
MATHEMATICS
A. Ya. ISMAILOV, M. A. TAGIEVA
ON THE REPRESENTATION OF GENERALIZED ANALYTIC FUNCTIONS BY SERIES OF PSEUDOPOLYNOMIALS
(Presented by Academician I. N. Vekua on 1 VI 1970)
Generalized analytic functions in the sense of I. N. Vekua are solutions of an equation of the form
\[ \partial_{\bar z} W + AW + B\overline{W} = F, \tag{1} \]
where \(A, B, F \in L_p,\ p > 2\).
Solutions of these equations preserve a number of basic topological properties of analytic functions of one complex variable (the uniqueness theorem, the argument principle, etc.). Such analytic facts as Taylor and Laurent expansions, the Cauchy integral formula, etc., extend to these solutions. (For a detailed exposition see \((^1)\).)
In this article we consider questions of interpolation of generalized analytic functions by generalized polynomials. In his monograph I. N. Vekua first drew attention to the possibility of interpolating generalized analytic functions and indicated a method for constructing solutions of equation (1) that assume prescribed values at fixed points.
As is known, in the study of the convergence of interpolation processes the integral form of the remainder term is of great help.
In this article we give analogues of the Lagrange–Hermite and Abel–Goncharov formulas for a particular kind of generalized analytic functions, i.e., for functions that are solutions of equations of the form
\[ \partial_{\bar z} W + AW + B\overline{W} = 0, \tag{2} \]
where \(A, B \in L_{p,2},\ p > 2\) \((^1)\), and with their aid we prove several convergence theorems.
The theory of solutions of equation (2), which is a special case of equation (1) and is known in the literature under the name of the theory of pseudoanalytic functions, was the subject of investigations by L. Bers \((^2)\), who proceeded from other ideas.
In establishing the Lagrange–Hermite and Abel–Goncharov formulas we use the integration process for pseudoanalytic functions introduced by L. Bers and the Cauchy formula \((^2)\).
We introduce the following notation \((^3)\): a set \(B\) of type \(\mathfrak M\) is a closed set containing at least two points, whose complement in the extended plane is a simply connected domain containing the infinitely distant point; \(\overline{CB} = B'\); \(L\) is the boundary of \(B\); \(V = \varphi(z)\) is a function conformally mapping the domain \(B'\) onto \(|V| > 1\), \(\varphi(\infty)=\infty\), \(\varphi'(\infty)>0\), \([\varphi'(\infty)]^{-1}=c\); \(z=\psi(V)\) is the inverse function to \(\varphi(z)\); \(L_\rho,\ \rho>1\), are level lines of \(B'\); \(B_\rho\) is the domain bounded by \(L_\rho\); \(\{z_k^{(n)}\}_{k=0}^{\infty},\ z_k^{(n)} \in \overline{B}\), are interpolation nodes;
\[ \omega_n(z)=\prod_{k=1}^{n+1}\bigl(z-z_k^{(n)}\bigr);\quad M_n(\rho)=\sup_{z\in L_\rho}|\omega_n(z)|,\quad \rho>1, \]
\[ M_n=\sup_{z\in L}|\omega_n(z)|. \]
Consider the interpolation problem: it is required to construct a pseudopolynomial \(P_n(z)\) of degree \(n\), which at the prescribed \(n+1\) points \(z_1,z_2,\ldots,z_{n+1}\) would take prescribed values, i.e.,
\[ P_n(z_k)=W(z_k). \]
In papers \((^1,\ ^2)\) it is proved that such a polynomial exists and is unique. For the interpolation polynomial \(P_n(z)\) the following holds.
Theorem 1. Let \(B\) be a domain bounded by a closed piecewise-smooth Jordan curve \(L\). Let the function \(W(z)\) be pseudoanalytic in \(B\) and continuous in \(B+L\), and let the interpolation nodes \(z_k\) \((k=1,\ldots,n+1)\) (not necessarily distinct) lie in \(B\).
Then the interpolation polynomial \(P_n(z)\), satisfying the conditions
\(P_n(z_k)=W(z_k)\) \((k=1,\ldots,n+1)\), can be represented by the formula
\[ P_n(z)=\int_L Z^{(-1)} \left[ \frac{\omega(\zeta)-\omega(z)}{\omega(\zeta)}\, \frac{iW(\zeta)}{2\pi}\,d\zeta,\zeta;z \right], \qquad \omega(z)=\prod_{k=1}^{n+1}(z-z_k). \tag{3} \]
Moreover, for the remainder term of the interpolation the relation
\[ R_n(z)=\int_L Z^{(-1)} \left[ \frac{\omega(z)}{\omega(\zeta)}\, \frac{iW(\zeta)}{2\pi}\,d\zeta,\zeta;z \right]. \tag{4} \]
Formula (2) makes it possible to obtain the following estimate of the remainder term.
Theorem 2. If \(B+L\) is a set of type \(\mathfrak M\) and \(W(z)\) is a function regular pseudoanalytic in \(\overline{B}_{\rho_0}\), \(\rho_0>1\), then for the remainder term of interpolation the estimate
\[ |W(z)-P_n(z)|\leq K M(\rho_0)\, \frac{\rho_0+r}{\rho_0-r}\, \frac{\displaystyle \sup_{z\in L_r}|\omega_n(z)|} {\displaystyle \inf_{z\in L_{\rho_0}}|\omega_n(z)|}, \qquad |z|\in \overline{B}_r,\quad 1\leq r<\rho_0, \]
holds, where \(K\) is a constant depending on the generating pair \((F,G)\), and
\[ M(\rho_0)=\frac{1}{2\pi}\int_0^{2\pi} \left|W\left[\psi(\rho e^{i\theta})\right]\right|\,d\theta . \]
With the aid of this estimate one proves
Theorem 3. In order that the sequence of interpolation pseudopolynomials \(\{P_n(z)\}\), for any function regular pseudoanalytic on a set \(\overline{B}\) of type \(\mathfrak M\), converge uniformly to \(W(z)\), it is necessary and sufficient that, as \(n\to\infty\),
\[ \left|\sqrt[n+1]{\omega_n(z)/c\varphi(z)}\right| \equiv |\theta_n(z)| \rightrightarrows 1 \tag{5} \]
in \(B'\), or else
\[ \lim_{n\to\infty}\frac{\sqrt[n+1]{M_n}}{c}=1. \tag{6} \]
If the function \(W(z)\) is regular pseudoanalytic in \(B_{\rho_0}\), \(\rho_0>1\), and one of the conditions (3) or (4) is fulfilled, then for any \(\rho_1\), \(1<\rho_1<\rho_0\),
\[ |W(z)-P_n(z)|\leq M(\rho,k)/\rho_1^{\,n+1}, \qquad z\in \overline{B}, \]
where \(M(\rho_1,K)\) is a constant independent of \(n\), but dependent on the generating pair \((F,G)\) and \(\rho_1\).
Now consider the Abel–Goncharov interpolation problem \((^4)\): suppose we are given a pseudoanalytic function \(W(z)\), defined in some domain \(D\), and \(\{\lambda_n\}\) is an arbitrary sequence of complex numbers lying in this domain. Find a pseudopolynomial \(Q_n(z)\) of degree \(n\) having the property that
\[ Q^{(k)}(\lambda_k)=W^{(k)}(\lambda_k). \tag{7} \]
Here the derivatives are understood in the sense of L. Bers. By Bers’ theorem \((^2)\) such a polynomial exists and is unique. Let us clarify the circumstances under which \(Q_n(z)\) converges to \(W(z)\). We first state the following theorem, which gives an analogue of the Abel–Goncharov interpolation formula for pseudoanalytic functions.
Theorem 4. Let \(W(z)\) be a pseudoanalytic function defined in a domain \(D\), let \(\{\lambda_n\}\) be an arbitrary sequence of complex numbers lying in this domain, and let \(\{(F_k,G_k)\}_0^\infty\), \((F_0,G_0)=(F,G)\), be a sequence of generating pairs.
Then the interpolation pseudopolynomial satisfying condition (5) is determined as follows:
\[ Q_n(z)=\sum_{k=0}^{n}\left[\Phi(\lambda_k)P_k^*(z)+\Psi(\lambda_k)P_k^{**}(z)\right], \]
where
\[ P_k^*(z)=\int_{\lambda_0}^{z}\cdots\int_{\lambda_{k-1}}^{z_k} F_k(z_k)\,d_{(F_1,G_1)}z_1\ldots d_{(F_k,G_k)}z_k, \]
\[ P_k^{**}(z)=\int_{\lambda_0}^{z}\cdots\int_{\lambda_{k-1}}^{z_k} G_k(z_k)\,d_{(F_1,G_1)}z_1\ldots d_{(F_k,G_k)}z_k, \]
and the remainder term \(R_n(z)=W(z)-P_n(z)\) is determined by the formula
\[ R_n(z)=\int_{\lambda_0}^{z}\cdots\int_{\lambda_n}^{z_n} W^{(n+1)}(z_{n+1})\,d_{(F_1,G_1)}z_1\ldots d_{(F_{n+1},G_{n+1})}z_{n+1}. \]
Theorem 5. Let \(\lim_{n\to\infty}\lambda_n=\lambda\), and let \(W(z)\) be a pseudoanalytic function in some disk \(|z|<R\) with respect to a sequence of generating pairs \(\{(F_n,G_n)\}_0^\infty\), \((F_0,G_0)=(F,G)\).
Then, if
\[ 1)\ \sum_{n=0}^{\infty}|\lambda_{n+1}-\lambda_n|<\infty;\qquad 2)\ \lim_{n\to\infty}\sqrt[n]{\Pi(2M_k/m_k)^2}\le M;\qquad 3)\ \lim_{n\to\infty}\sqrt[n]{c_n}\le c, \]
then the series
\[ \sum_{n=0}^{\infty}\Phi(\lambda_n)P_n^*(z)+\Psi(\lambda_n)P_n^{**}(z) \]
converges uniformly to the pseudoanalytic function \(W(z)\) inside the disk \(|z|<R/CM\), where
\[ M_k=\max_{z\in D}\{|F_k|,\ |G_k|\},\qquad m_k=\min_{z\in D}|F_k\overline{G_k}-\overline{F_k}G_k|, \]
and \(c_n\) is a positive constant depending on the generating pair \((F,G)\).
Azerbaijan State University
named after S. M. Kirov
Baku
Received
27 IV 1970
CITED LITERATURE
\(^1\) I. N. Vekua, Generalized Analytic Functions, Moscow, 1959.
\(^2\) L. Bers, Theory of pseudo-analytic functions, Lecture notes, N.-Y., 1953.
\(^3\) V. I. Smirnov, N. A. Lebedev, Constructive Theory of Functions, Moscow, 1964.
\(^4\) M. A. Evgrafov, The Abel–Goncharov Interpolation Problem, Moscow, 1954.