V. K. DZYADYK

THEOREMS ON THE TRANSFORMATION AND APPROXIMATION OF ANALYTIC FUNCTIONS

(Presented by Academician S. L. Sobolev on 4 February 1963)

In this article we present without proofs the principal results of an investigation of the so-called analytic transformation \(f_g(z)\) of two functions \(f(z)\) and \(g(t)\), of which the first is analytic and the second periodic. It is established that the transformations \(f_{J_n}(z)\), where \(J_n(t)\) is the Jackson trigonometric kernel, are algebraic polynomials. With the aid of these polynomials a constructive characterization has been obtained for functions of the classes \(W^r H^\alpha\) (\(r\) an integer \(\geqslant 0\), \(0<\alpha<1\)) in domains with piecewise smooth boundary, as well as some other results*.

Let \(K\) be a bounded nondegenerate continuum with simply connected complement, and let \(w=\Phi(z)\) be a function effecting a conformal mapping of the exterior of \(K\) onto the exterior of some circle \(|w|\leqslant \rho\), where \(\rho\) is chosen so that
\[ \lim_{z\to\infty}\frac{\Phi(z)}{z}=1. \]
For simplicity we shall assume that \(\rho=1\). Denote by \(z=\Psi(w)\) the function inverse to \(\Phi(z)\).

In [1] it was established that, in the case when \(K\) is the closure of some domain \(G\), an integral of the form

\[ \frac{1}{4\pi^2 i}\int_{-\pi}^{\pi} K_n(\sigma)\,d\sigma \int_C \frac{f(\zeta_{[\sigma]})}{\zeta-z}\,d\zeta, \tag{1} \]

where \(\zeta_{[\sigma]}=\Psi[\Phi(\zeta)e^{-i\sigma}]\), \(K_n(\sigma)\) is a trigonometric polynomial of order \(\leqslant n\), and the function \(f(\zeta)\) is such that the function it generates,
\(\hat f(t)=f[\Psi(e^{it})]\), is summable on \([-\pi,\pi]\), is, for all \(z\in G\), an algebraic polynomial \(P_n(f;z)\) of order \(\leqslant n\); moreover, if

\[ K_n(\sigma)=\sum_{-n}^{n}\lambda_k e^{ik\sigma}, \tag{2} \]

\[ \hat f(t)\sim \sum_{-\infty}^{\infty} c_k e^{ikt}, \tag{3} \]

then

\[ P_n(f;z)=\frac{1}{4\pi^2 i}\int_{-\pi}^{\pi} K_n(\sigma)\,d\sigma \int_C \frac{f(\zeta_{[\sigma]})}{\zeta-z}\,d\zeta =\sum_0^n \lambda_k c_k \Phi_k(z), \tag{4} \]

where \(\Phi_k(z)\) is the \(k\)-th Faber polynomial for the domain \(G\).

The following questions arise:

a) under what conditions on the boundary \(C\) and the function \(f(\zeta)\), \(\zeta\in C\), does there exist

* These results were reported by us on 10 October 1962 at the Second All-Union Conference on Constructive Function Theory in Baku.

the so-called \(\sigma\)-transform \(F(z;\sigma)\) of the function \(f(\zeta)\):

\[ F(z;\sigma)^{\operatorname{def}}=\frac{1}{2\pi i}\int_C \frac{f(\zeta_{[\sigma]})}{\zeta-z}\,d\zeta, \tag{5} \]

and also the even \(\sigma\)-transform \(F(c;z;\sigma)\):

\[ F(c;z;\sigma)^{\operatorname{def}}=\frac{1}{2\pi i}\int_C \frac{f(\zeta_{[\sigma]})+f(\zeta_{[-\sigma]})}{2(\zeta-z)}\,d\zeta; \tag{6} \]

b) under what conditions on the periodic function \(g(t)\) and on the \(\sigma\)-transform \(F(z;\sigma)\) there exists a transform \(f_g(z)\) of the functions \(f(\xi)\), \(\xi\in C\), and \(g(t)\):

\[ f_g(z)^{\operatorname{def}}=\frac{1}{2\pi}\int_{-\pi}^{\pi} g(t)F(z;t)\,dt = \frac{1}{4\pi i}\int_{-\pi}^{\pi} g(t)\,dt\int_C \frac{f(\zeta_{[t]})}{\zeta-z}\,d\zeta; \tag{7} \]

c) whether a generalization of \(\sigma\)-transforms and of transforms of two functions is possible to the case when the function \(f(\xi)\) is given on the boundary \(C\) of a bounded continuum \(K\) with simply connected complement.

The answers obtained are:

a) if the boundary \(C\) of the domain \(G\) is rectifiable and \(\hat f(t)=f[\Psi(e^{it})]\) is summable and expands in the Fourier series (3), then the \(\sigma\)-transform \(F(z;\sigma)\) exists for almost all \(\sigma\in[-\pi,\pi]\), and in this case

\[ F(z;\sigma)\sim \sum_0^\infty c_k\Phi_k(z)e^{-ik\sigma}; \tag{8} \]

b) if one of the three functions \(\hat f(t)\), \(g(t)\), and \(\Psi'(e^{it})\) has bounded variation, and the other two are summable on \([-\pi,\pi]\), then the transform \(f_g(z)\) exists, is an analytic function in the domain \(G\), and in this case

\[ f_g(z)=\sum_{k=0}^\infty c_k d_k \Phi_k(z), \tag{9} \]

where \(d_k\) are the Fourier coefficients of the function \(g(t)\)

\[ g(t)\sim \sum_{-\infty}^{\infty} d_k e^{ikt}; \tag{10} \]

c) for a bounded continuum \(K\) with simply connected complement, it is expedient to define the \(\sigma\)-transform by means of the limit (provided it exists)

\[ F(z;\sigma)=\lim_{R\to 1+0}\frac{1}{2\pi i}\int_{C_R} \frac{f\left\{\Psi\left[\frac{1}{R}\Phi(\zeta)e^{-i\sigma}\right]\right\}}{\zeta-z}\,d\zeta, \]

where \(C_R\) is the level line of the continuum \(K\): \(|\Phi(\zeta)|=R\).

In the particular case when the Jackson kernel \(J_n(\sigma)\) serves as the kernel \(g(\sigma)\), as the transforms \(f_{J_n}(\sigma)\) we obtain algebraic polynomials which, depending on the continuum \(K\) under consideration, turn into the known trigonometric Jackson polynomials (\(K\) is a circle and \(z=e^{it}\)), into algebraic polynomials by means of which A. F. Timan \((^2)\) obtained a strengthening of Jackson’s theorem in the periodic case (\(K\) is the interval \([-1,1]\)), into algebraic polynomials convenient for approximating analytic functions continuous in domains with rectifiable boundary (see \((^1)\) and Theorems 1–3 of the present paper), and so on.

Theorem 1. Let \(G\) be an arbitrary finite domain with simply connected complement, possessing the property that the function \(\Psi(w)\) has, at all points of the circle \(|w|=1\), a nonzero continuous derivative \(\Psi'(w)\). Suppose, moreover, that the length of the arc between arbitrary two poi-

on the boundary \(C\) of the domain \(G\) does not exceed the distance between these points multiplied by some constant number:

\[ \int_{z_1}^{z_2}|d\zeta|\leq A_1|z_2-z_1|,\qquad z_1,z_2,\zeta\in C;\qquad A_1=\mathrm{const}. \]

Then, for any function \(f(z)\) analytic in \(G\) and having continuous derivatives up to order \(r\) inclusive on \(\overline G\) (\(r\) an integer, \(r\geq 0\)), for every natural \(n\) there exists an algebraic polynomial \(P_n(z)\) of degree not exceeding \(n\) such that, for all \(z\in G\),

\[ |f(z)-P_n(z)|\leq A_2\left( \frac{1+\omega\left(\Psi';\frac1n\right)\ln n}{n} \right)^r \left\{ \left[1+\omega\left(\Psi';\frac1n\right)\ln n\right] \omega\left(f^{(r)};\frac1n\right) + \int_0^{1/n} \frac{\omega(f^{(r)};\sigma)\,\omega(\Psi';\sigma)}{\sigma}\,d\sigma \right\} \tag{11} \]

and, in particular, if for the functions \(\Psi'(\omega)\) and \(f^{(r)}(z)\), for \(|\omega|=1\), \(z\in C\), the conditions

\[ 1)\ \omega\left(\Psi';\frac1n\right)=O\left(\frac1{\ln n}\right); \qquad 2)\ \int_0^{1/n} \frac{\omega(f^{(r)};\sigma)\,\omega(\Psi';\sigma)}{\sigma}\,d\sigma = O\left[\omega\left(f^{(r)};\frac1n\right)\right], \]

are fulfilled, then for all \(z\in G\)

\[ |f(z)-P_n(z)|\leq A_2'\frac{\omega(f^{(r)};1/n)}{n^r}, \tag{11'} \]

where the constants \(A_2\) and \(A_2'\) depend neither on \(n\) nor on the function \(f(z)\).

The special feature of this theorem is that the estimates established in it depend both on the properties of the derivative \(f^{(r)}(z)\) and on the properties of the boundary \(C\) (or, equivalently, on the properties of \(\Psi'(\omega)\)). This theorem contains, as a special case, a result proved by S. Ya. Al’per \(([^3]\), Theorem 4) on the approximation of functions analytic in \(G\) and continuous on \(\overline G\).

Theorem 2. Let \(G\) be a finite domain with a simply connected complement, whose boundary \(C\) consists of a finite number of arcs \(C^{(j)}\) \((j=1,2,\ldots,k)\) with continuous curvature, forming at their junction points \(z_j\) angles \(\alpha_j\pi\), \(0<\alpha_j<1\), and having the property that, in a neighborhood of each of the points \(\omega_j=\Phi(z_j)\), the function \(z=\Psi(\tau)\) can be represented in the form

\[ \Psi(\tau)=\lambda(\tau)(\tau-\omega_j)^{2-\alpha_j}+\Psi(\omega_j), \]

where \(\lambda(\tau)\) is a function continuous along \(|\tau|=1\) in some neighborhood of the point \(\omega_j\), together with its first and second derivatives \(\lambda'(\tau)\), \(\lambda''(\tau)\), and \(\lambda(\omega_j)\ne 0\).

Then, if on \(\overline G\) a function \(f(z)\) is given, analytic in \(G\) and continuous on \(\overline G\), with modulus of continuity \(\omega(t)=\omega(f;t)\), then for \(f(z)\), for every natural \(n\), one can construct an algebraic polynomial \(P_n(z)=P_n(f;z)\) of degree not exceeding \(n\) such that at all points \(z\in C\) the inequality

\[ |f(z)-P_n(z)|\leq A_3\omega\left[\rho_{1+1/n}(z)\right] \]

will hold, where \(\rho_{1+1/n}(z)\) is the distance from the point \(z\) to the level line \(|\Phi(z)|=1+\frac1n\), and \(A_3\) is a constant independent of both \(z\) and \(n\).

Theorem 3. If, under the conditions of the preceding theorem, in the domain \(G\) a function \(f(z)\), being analytic in \(G\), has on the closed domain \(\overline G\) continuous derivatives up to order \(r\) inclusive (\(r\) an integer, \(r\geq 0\)) and, moreover, for some \(a\in(0,1]\) the derivative \(f^{(r)}(z)\) satisfies a Hölder condition of order \(a\), then for the function \(f(z)\), for every natural \(n\), one can

to construct an algebraic polynomial \(P_n(z)\) of degree not exceeding \(n\) such that at all points \(z \in C\) the inequality

\[ |f(z)-P_n(z)| \leq A_4 [\rho_{1+1/n}(z)]^{r+\alpha}, \]

holds, where the constant \(A_4\) depends neither on \(z \in C\) nor on \(n\).

From this theorem and Theorems 4.1 and 2.4 from \({}^{(4)}\) there follows the following theorem, giving a constructive characterization in domains with corners for analytic functions \(f(z)\), whose \(r\)-th derivative belongs to the class \(H^\alpha\), \(0<\alpha<1\).

Theorem 4. In order that a function \(f(z)\), analytic in a domain \(G\) with corners satisfying the conditions of Theorem 2, have in the closed domain a continuous derivative of order \(r\) (\(r\) arbitrary integer), \(f^{(r)}(z)\), belonging to the Hölder class \(H^\alpha\), \(0<\alpha<1\), it is necessary and sufficient that for this function, for each \(n=1,2,\ldots\), there exist an algebraic polynomial \(P_n(z)\) of degree not exceeding \(n\) such that, for all \(z\) belonging to the boundary \(C\) of the domain \(G\), the inequality

\[ |f(z)-P_n(z)| \leq A_5 [\rho_{1+1/n}(z)]^{r+\alpha} \]

holds, where the constant \(A_5\) depends neither on \(z\) nor on \(n\), and \(\rho_{1+1/n}(z)\) denotes the distance from the point \(z \in C\) to the level line

\[ |\Phi(z)| = 1+\frac{1}{n}. \]

In addition, we have considered questions relating to the approximation of analytic functions given in domains with corners \(>\pi\), on the simplest Riemann surfaces, and on a finite number of continua.

Theorems analogous to Theorem 2 also hold for the approximation of logarithmic potentials by harmonic polynomials.

Institute of Mathematics
Academy of Sciences of the Ukrainian SSR

Received
1 II 1963

CITED LITERATURE

\({}^{1}\) V. K. Dzyadyk, Izv. AN SSSR, ser. matem., 26, 797 (1962).
\({}^{2}\) A. F. Timan, DAN, 78, 17 (1951).
\({}^{3}\) S. Ya. Al’per, Izv. AN SSSR, ser. matem., 19, 423 (1955).
\({}^{4}\) V. K. Dzyadyk, Izv. AN SSSR, ser. matem., 23, 697 (1959).