Mathematics

V. D. EROKHIN

ON CONFORMAL MAPPINGS OF RINGS AND ON THE BASIC BASIS OF THE SPACE OF FUNCTIONS ANALYTIC IN AN ELEMENTARY NEIGHBORHOOD OF AN ARBITRARY CONTINUUM

(Presented by Academician A. N. Kolmogorov, 27 I 1958)

1. Let \(K\) be a bounded continuum in the \(z\)-plane, not separating the plane, and let \(l_\rho\) be its level lines, i.e. the images of the circles \(|w|=\rho\) under the schlicht mapping \(w=\varphi(z)\) of the exterior of \(K\) onto the disk \(|w|>1\), \(\varphi(\infty)=\infty\). Then, as was first clarified by S. N. Bernstein \((^1)\), the interiors of the curves \(l_\rho\) are natural neighborhoods of the continuum from the point of view of uniform approximation, on it, of analytic functions by means of polynomials. In other words, from a certain rate of best polynomial approximations on \(K\) one can conclude that the function is analytic inside \(l_\rho\), and conversely. This fact is closely connected with another: in the space of functions analytic inside \(l_\rho\) there is a basis consisting of polynomials, the expansions with respect to which converge on the continuum with the rate of best approximations; such a basis is formed, as is known, by the Faber polynomials \((^{2,3})\).

Now let a simply connected domain \(G\) contain \(K\), but in general not be the interior of \(l_\rho\). Consider the class \(A_G^K\) of all functions \(f(z)\) analytic in \(G\), and introduce the norm
\[ \|f\|=\max_{z\in K}|f(z)|. \]
From the theorem of S. N. Bernstein cited above it follows that it is in general unreasonable to approximate an arbitrary function of the class \(A_G^K\) by polynomials. Generalizing G. Faber’s theorem, we shall show, however, that it is always possible to define a sequence \(e_0(z),\ldots,e_n(z),\ldots\) of functions analytic in \(G\) and linearly independent for polynomials
\[ \sum_{k=0}^{n} a_k e_k(z), \]
for which a proposition is valid (Theorem 3) analogous to S. N. Bernstein’s theorem. Moreover, the functions \(e_n(z)\) form a basis of the space \(A_G^K\), and this basis is “most efficient”: whatever basis is taken in the space \(A_G^K\), the expansions with respect to it of any function of the space cannot converge on the continuum faster (in the sense of a progression with a smaller denominator) than the expansions with respect to \(e_n(z)\). Finally, from the polynomials
\[ \sum_{k=0}^{n} a_k e_k(z) \]
with suitable \(n\), \(n\simeq \log \frac{1}{\varepsilon}\), one can choose an “asymptotically” minimal \(\varepsilon\)-net in the compact class of functions \(f(z)\in A_G^K\) such that \(|f(z)|\le M,\ z\in G\), for any \(M>0\).

The author considers it necessary to emphasize that the present note arose as a result of his work on the problem of the asymptotics of the \(\varepsilon\)-entropy of analytic functions posed by A. N. Kolmogorov.

2. On doubly connected domains

Let \(G\) be a certain doubly connected domain with boundary continua \(K\) and \(\Gamma\). We orient the domain \(G\), calling one of the boundary continua, for example \(K\), “inner,” the other “outer,” and in accordance with this write \(G=(K,\Gamma)\). By \(\Gamma(K)\) we denote the simply connected domain with boundary \(\Gamma\) and containing \(K\); the notation \(K(\Gamma)\) has an analogous meaning. It is known that every doubly connected domain \(G=(K,\Gamma)\) with nondegenerate inner continuum \(K\) can be conformally and univalently mapped onto a circular annulus \(1<|w|<R\) with preservation of orientation, where the mapping function \(w=\varphi(z)\) (up to a factor \(e^{i\theta}\)) and the number \(R\) \((1<R\leq+\infty)\), called the modulus of the domain \(G\), are determined by the domain itself uniquely.

Theorem 1. For any doubly connected domain \(G=(K,\Gamma)\) the function \(w=\varphi(z)\) can be represented in the form \(\varphi(z)=\varphi^{2}(\varphi^{1}(z))\), where the functions \(t=\varphi^{1}(z)\) and \(w=\varphi^{2}(t)\) are univalent, respectively, in the simply connected domains \(\Gamma(K)\) and \(K^{1}(\Gamma^{1})\). Here the function \(t=\varphi^{1}(z)\) is determined by the domain \(G\) uniquely up to an arbitrary fractional-linear transformation of the plane \((t)\).

It is easy to see that Theorem 1 can be expressed otherwise:

Theorem \(1'\). Let \(B\) be an arbitrary simply connected domain and \(K\) an arbitrary continuum, \(K\subset B\). The domain \(B\) can be conformally transformed into the interior of some level line of the transformation by the same transformation of the continuum. The indicated transformation is unique up to a repeated linear transformation.

For the proof of Theorem \(1'\) let us first map the domain \(B\) onto the disk \(|z_{0}|<r_{0}\), \(z_{0}=f_{0}(z)\). Let \(K_{0}=f_{0}(K)\). We may suppose that \(K_{0}\) does not separate the plane. Let \(\widetilde K_{0}\) be the inverse of \(K_{0}\) with respect to the circle \(|z_{0}|=r_{0}\). Map conformally the domain complementary to \(\widetilde K_{0}\) onto the disk \(|z_{1}|<r_{1}\), \(z_{1}=f_{1}(z_{0})\), \(f_{1}(0)=0\), \(f'_{1}(0)=1\). Let \(K_{1}=f_{1}(K_{0})\). Let \(\widetilde K_{1}\) be the inverse of \(K_{1}\) with respect to \(|z_{1}|=r_{1}\). Next the domain complementary to \(\widetilde K_{1}\) is mapped onto the disk \(|z_{2}|<r_{2}\), \(z_{2}=f_{2}(z_{1})\), \(f_{2}(0)=0\), \(f'_{2}(0)=1\). Here \(K_{2}=f_{2}(K_{1})\), and so on. Put
\[ \varphi^{1}_{n}(z)=f_{n}(f_{n-1}(\ldots(f_{0}(z)))). \]
Applying the general theorems on univalent functions, we prove that \(\varphi^{1}_{n}(z)\to\varphi^{1}(z)\). The proof that \(\varphi^{1}(B)\) is the interior \(t_{R}\) for \(\varphi^{1}(K)\) is obtained by means of the symmetry principle and the general theorems on convergence.

3.

In what follows, \(K\) denotes an arbitrary continuum in the plane \((z)\), degenerate neither into a point nor into the whole plane. By \(D_{q}\) \((q=0,1,\ldots)\) we denote the sequence of domains adjacent to \(K\).

We shall call an “elementary neighborhood of the continuum \(K\)” an arbitrary domain \(G\supset K\) of such a kind that inside each of the domains \(D_{q}\) there is located no more than one component of the boundary \(\Gamma\) of the domain \(G\). If \(G\) is an elementary neighborhood of \(K\), then \(\Gamma_{q}=\Gamma\cap D_{q}\). Those \(q\) for which \(\Gamma_{q}\) is empty are henceforth excluded from consideration. Put \(G_{q}=G\cap D_{q}\) and \(K_{q}=K\cap [G_{q}]\). Obviously, each of the domains \(G_{q}=(K_{q},\Gamma_{q})\) is doubly connected. Let the modulus of \(G_{q}\) be equal to \(R_{q}\).

Theorem 2. Whatever the continuum \(K\) and whatever its elementary neighborhood \(G\), there exists a double sequence of functions \(e_{00}(z)=1\), \(e_{qn}(z)\) \((n=1,2,\ldots)\) such that:

I. All functions \(e_{qn}(z)\) are analytic and single-valued in the domain \(G\); more precisely, for any \(q\) all \(e_{qn}(z)\) are analytic in the simply connected domain \(\Gamma_{q}(K_{q})\).

II. Every function \(f(z)\), regular in \(G\), has an expansion
\[ f(z)=a_{00}+\sum_{q}\sum_{n=1}^{\infty} a_{qn}e_{qn}(z), \]
converging uniformly and absolutely “inside” \(G\).

III. The expansion II is unique: \(a_{qn}=a_{qn}(f)\).

IV. The functions \(e_{qn}(z)\) and the coefficients \(a_{qn}\) possess the following properties (in special cases they can be strengthened):

\(1^\circ.\ \|e_{qn}\|< C(\delta)(1+\delta)^n.\)

\(2^\circ.\ \displaystyle \sup_{z\in G}|e_{qn}(z)|< C(\delta)(1+\delta)^n R_q^n.\)

\(3^\circ.\ |a_{qn}|< M_q C(\delta)(1+\delta)^n R_q^{-n},\quad M_q=\displaystyle \sup_{z\in G_q}|f(z)|.\)

\(4^\circ.\ |a_{qn}|< C_q\|f\|,\) if the Hausdorff length of \(K_q\) is \(<+\infty\).

Here \(\delta>0\) is arbitrarily small and the constants \(C(\delta)\) and \(C_q\) depend only on \(K\) and \(G\); \(q\) and \(n\) are arbitrary; the case \(R_q=+\infty\) is not formally excluded.

Lemma. Let \(G\) be a simply connected domain with boundary \(K\), containing the point \(z=0\) and not containing \(\infty\). Let the function \(w=\varphi(z)\) map \(G\) one-to-one onto the circle \(|w|>1\) in such a way that \(\varphi(0)=\infty\). By \(l_\rho\) denote the corresponding level lines. Consider the functions

\[ \varphi_n(z)=\frac{1}{2\pi i}\int_{l_\rho}\frac{|\varphi(\zeta)|}{\zeta-z}\,d\zeta =\frac{\beta_n^{(n)}}{z^n}+\frac{\beta_{n-1}^{(n)}}{z^{n-1}}+\cdots+\frac{\beta_1^{(n)}}{z}, \qquad z\in l_\rho(K). \]

Then every function \(f(z)\), analytic inside some \(l_\rho\) and such that \(f(\infty)=0\), has a unique expansion

\[ f(z)=\sum_{n=1}^{\infty} a_n\varphi_n(z), \]

where the coefficients are determined by the formulas

\[ a_n=\frac{1}{2\pi i}\int_{l_{\rho'}} f(\zeta)\, \frac{\varphi'(\zeta)}{[\varphi(\zeta)]^{n+1}}\,d\zeta, \qquad \rho'<\rho. \]

In the annulus \(1<|w|<\rho\) the function \(f(\varphi^{-1}(w))\) has a Laurent expansion. Consequently, in the domain \((K,l_\rho)\),

\[ f(\zeta)=\sum_{n=-\infty}^{+\infty} a_n[\varphi(\zeta)]^n, \qquad a_n=\frac{1}{2\pi i}\int_{|w|=\rho'} f(\varphi^{-1}(w))w^{-n-1}\,dw. \]

Multiply by \(\dfrac{1}{2\pi i}\dfrac{1}{\zeta-z}\), where \(z\in l_{\rho'}(K)\), and integrate with respect to \(\zeta\in l_{\rho'}\). Since \(f(\zeta)/(\zeta-z)\), as a function of \(\zeta\), has at \(\infty\) a zero of at least second order, its residue with respect to \(\infty\) is \(0\). Therefore we obtain

\[ f(z)=\sum a_n\varphi_n(z). \]

It remains to note that \(\varphi_n(z)\equiv 0\) for \(n\le 0\). Indeed, denoting \(\varphi^{-1}=\psi\), we have

\[ \varphi_n(z)=\frac{1}{2\pi i}\int_{|w|=\rho} \frac{\psi'(w)}{\psi(w)-z}\,w^n\,dw; \]

here

\[ \psi(w)=\frac{b_1}{w}+\frac{b_2}{w^2}+\cdots \quad (|w|>1). \]

The uniqueness of the expansion is proved as in Faber’s theorem \((^3)\).

General case. For definiteness assume: \(\infty\notin G,\ \infty\in D_0\). Let \(w=\varphi_q(z)\) be a function conformally mapping the domain \(G_q=(K_q,\Gamma_q)\) onto the annulus \(1<|w|<R_q\), with preservation of orientation. By Theorem 1 we have \(\varphi_q(z)=\varphi_q^2(\varphi_q^1(z))\). Performing, if necessary, additional fractional-linear transformations, we choose the functions \(\varphi_q^1\) and \(\varphi_q^2\) in the following “canonical” way. For \(q\ne 0\) the function \(t_q=\varphi_q^1(z)\) maps one-to-one the domain \(\Gamma_q(K_q)\) onto the domain \(\Gamma_q^1(K_q^1)\), containing the point \(t_q=\infty\), so that \(\varphi_q^1(\infty)=\infty\); moreover \(0\in K_q^1(\Gamma_q^1)\). The function \(w=\varphi_q^2(t_q)\) maps \(K_q^1(\Gamma_q^1)\) one-to-one onto the circle \(|w|>1\) in such a way that \(\varphi_q^2(0)=\infty\). The function \(t_0=\varphi_0^1(z)\) maps one-to-one the domain \(\Gamma_0(K_0)\) onto the domain \(\Gamma_0^1(K_0^1)\), which is the interior of the level line \(\Gamma_0^1=l_{R_0}^0\) of the bounded continuum \(K_0^1=\varphi_0^1(K_0)\).

Let \(z=\psi_q(w)\), \(z=\psi_q^1(t_q)\), \(t_q=\psi_q^2(w)\) be the functions inverse, respectively, to \(\varphi_q,\varphi_q^1,\varphi_q^2\). By \(L_{\rho_q}^q\) \((1<\rho_q<R_q)\) we denote the level lines of the domain \(G_q=(K_q,\Gamma_q)\); \(l_{\rho_q}^q=\psi_q^2(\{|w|=\rho_q\})\) \((\rho_q>1)\). All contours are oriented in accordance with the mappings.

Definition of the basis functions

\[ e_{qn}(z)=\varphi_{qn}^2\bigl(\varphi_q^1(z)\bigr) \qquad (q=0,1,\ldots;\ n=1,2,\ldots), \]

where

\[ \varphi_{0n}^2(t_0)=\frac{1}{2\pi i}\int_{l_{\rho_0}^0} \frac{[\varphi_0^2(\tau_0)]^n}{\tau_0-t_0}\,d\tau_0 = \alpha_n^{(n)}t_0^n+\alpha_{n-1}^{(n)}t_0^{n-1}+\cdots+\alpha_0^{(n)}, \]

\[ \varphi_{qn}^2(t_q)=\frac{1}{2\pi i}\int_{l_{\rho_q}^q} \frac{[\varphi_q^2(\tau_q)]^n}{\tau_q-t_q}\,d\tau_q = \frac{\beta_{q,n}^{(n)}}{t_q^n} + \frac{\beta_{q,n-1}^{(n)}}{t_q^{\,n-1}} +\cdots+ \frac{\beta_{q,1}^{(n)}}{t_q}, \]

or

\[ e_{qn}(z)=\frac{1}{2\pi i}\int_{L_{\rho_q}^q} \frac{[\varphi_q(\zeta)]^n}{\varphi_q^1(\zeta)-\varphi_q^1(z)} \frac{d}{d\zeta}\varphi_q^1(\zeta)\,d\zeta, \qquad z\in L_{\rho_q}^q(K_q). \]

Formulas for the coefficients

\[ a_{qn}=\frac{1}{2\pi i}\int_{|w|=\rho_q} f_q(\psi_q(w))\frac{1}{w^{n+1}}\,dw, \qquad \text{where}\quad f_q(z)=\frac{1}{2\pi i}\int_{L_{\rho_q}^q}\frac{f(\zeta)}{\zeta-z}\,d\zeta . \]

The functions \(f_q(\psi_q^1(t_q))\) are regular, respectively, in the domains \(T_q^1(K_q^1)\), and for \(q\ne0\), \(f_q(\psi_q^1(\infty))=\infty\). To the function \(f_0(\psi_0^1(t_0))\) we apply Faber’s theorem, and to the remaining ones, the lemma. Taking into account Cauchy’s formula \(f(z)=\sum_q f_q(z)\), we obtain the basic expansion. To prove uniqueness, multiply by \(\dfrac{1}{2\pi i}\dfrac{1}{z-\zeta}\) and integrate with respect to \(z\in L_{\rho_q}^q\). Properties \(1^\circ\)—\(4^\circ\) can be derived from the explicit formulas.

4. Theorem 3. Let a continuum \(K\) and a simply connected domain \(G\) containing it be given. Let \(e_0(z)\equiv1,\ e_1(z),\ldots,e_n(z),\ldots\) be the corresponding principal basis. In order that a function \(f(z)\), defined on \(K\), be analytic in the domain \(G\), it is necessary and sufficient that for every \(\delta>0\) and every natural \(n\) there exist a polynomial \(\sum_{k=0}^n a_k e_k(z)\) such that

\[ \left|f(z)-\sum_{k=0}^n a_k e_k(z)\right| \le C(\delta)\left(\frac{1+\delta}{R}\right)^n, \qquad z\in K \]

(the same can be said about polynomials \(\sum_{k=0}^n b_k[\varphi^1(z)]^k\)). Here \(C(\delta)\) depends only on \(\sup_{z\in G}|f(z)|\), if the function is bounded in \(G\).

Received
11 I 1958

References Cited

¹ S. N. Bernstein, Communications of the Kharkov Mathematical Society, ser. 2, 13 (1912). ² G. Faber, Math. Ann., 57, 389 (1903). ³ A. I. Markushevich, Theory of Analytic Functions, 1950.