E. P. Dolzhenko

ON APPROXIMATION ON CLOSED DOMAINS

AND ON ZERO SETS*

(Presented by Academician A. N. Kolmogorov on 27 XI 1961)

S. N. Mergelyan established \((^1)\) that if \(G\) is a finitely connected domain, moreover such that: 1) the complement \(C\bar G\) of the closure \(\bar G\) of this domain consists of a finite number of domains; 2) \(G\) coincides with the set of interior points of its closure \(\bar G\), then the following holds:

Theorem A. Every function analytic inside \(G\) and continuous on \(\bar G\) can be uniformly approximated on \(\bar G\), with any prescribed accuracy, by rational fractions.

Let us note that if the domains \(g_n\), complementary to \(\bar G\), are at a positive distance from one another, then this assertion follows from an earlier result of M. V. Keldysh \((^2)\).

Will this theorem remain true if no restrictions are imposed on the domain \(G\) (apart from the natural condition that \(G\) coincide with the set of interior points of its closure), or if even the condition of finite connectedness of \(\bar G\) is dropped, retaining this condition only for \(G\)? Below this question is answered in the negative. For a certain class \(B^{(1/3)}\) of infinitely connected domains \(G\), a complete structural characterization is given of the functions \(f(z)\) \((z\in \bar G)\) that admit uniform approximation on \(\bar G\) by rational fractions.

1. An example of a simply connected domain \(G\) for which Theorem A is not true. Take a function \(\varphi(z)\), distinct from a constant, defined and continuous in the whole extended complex plane \(Z\), analytic outside some bounded, nowhere dense continuum \(E\) in \(Z\) with connected complement, and such that \(\max |\varphi(z)| = 1\). As \(\varphi(z)\) one may take, for example, the function
\[ l\iint_E \frac{d\xi\,d\eta}{\xi+i\eta-z} \]
(see \((^3)\), p. 330), where \(E\) is a bounded nowhere dense continuum in \(Z\) with connected complement, having positive planar Lebesgue measure, and \(l\) is a constant.

Let the point \(a\in E\) be such that \(|\varphi(a)|=1\); let the circle \(\sigma_0\) be chosen so that \(E\subset \bar\sigma_0\) and on the boundary \(c_0\) of this circle there is a single point \(z_0\in E\), with \(z_0\ne a\). Take a system of open circles \(\{\sigma_n\}_{n=1}^{\infty}\) with the following properties: 1) \(\sigma_n\subset \sigma_0\setminus E\) \((n=1,2,\ldots)\); 2) on the boundary \(c_n\) of the circle \(\sigma_n\) there lies one and only one point \(z_n\in E\), with \(z_n\ne a\); 3) any two circles \(\sigma_n,\sigma_m\) \((m\ne n)\) are at a positive distance from each other and from \(c_0\); 4) in every neighborhood of each point \(z\in E\) there is an infinite number of circles \(\sigma_n\); 5) if \(r_n\) \((n\ge 0)\) denotes the distance from \(a\) to \(c_n\), and \(|c_n|\) the length of the curve \(c_n\), then
\[ \sum_{n=1}^{\infty} |c_n|/r_n < 1/4. \]
Obviously, the set
\[ G=\sigma_0\setminus\left(\bigcup_{n\ge 1}\bar\sigma_n\right) \]
is a simply connected domain. The set \(E\) forms part of the boundary of this domain, so that \(\varphi(z)\) is analytic in \(G\) and continuous on \(\bar G\). Choose a natural—

* The result of the paper was reported at the IV All-Union Mathematical Congress in Leningrad in July 1961.

so large that for \(z\in c=\{z:\ z\in c_0,\ |z-z_0|> \tfrac13 r_0\}\) one has
\(|\varphi(z)|^p< r_0/10|c_0|\) (this can be done, since \(|\varphi(z)|<1\) if \(z\in c_0\), \(z\ne a\)). We shall show that for \(f(z)=[\varphi(z)]^p\) and any rational function \(R(z)\),

\[ \max_{\overline G}|f(z)-R(z)|>r_0/|c_0|. \tag{1} \]

Assuming the contrary, for some rational function \(R(z)\) we have: a) for \(z\in \overline G\),
\(|R(z)|\le 1+r_0/|c_0|<4/3\); b) for \(z\in c\),
\[ |R(z)|<r_0/10|c_0|+r_0/|c_0|=11r_0/10|c_0|; \]
c) \(|R(a)|\ge 1-r_0/|c_0|>5/6\). If \(\sigma_{n_1},\sigma_{n_2},\ldots,\sigma_{n_k}\) are those of the circles \(\sigma_n\) \((n\ge 1)\) inside which poles of the fraction \(R(z)\) have fallen, then

\[ |R(z)|= \left|\frac{1}{2\pi i}\left[\int_{c_0}-\sum_{j=1}^{k}\int_{c_{n_j}}\right]\frac{R(z)\,dz}{a-z}\right| \le \frac{1}{2\pi}\left[\int_c+\int_{c_0\setminus c}+\sum_{n=1}^{\infty}\int_{c_n}\right]\frac{|R(z)|\,ds}{|a-z|} \]

\[ \le \frac{1}{2\pi}\left[ \frac{11r_0}{10|c_0|}\frac{|c_0|}{r_0} +\frac{4}{3}\frac{1}{r_0}\pi\frac{r_0}{30} +\sum_{n=1}^{\infty}\frac{4}{3}\frac{|c_n|}{r_n} \right] < \frac{11}{20\pi}+\frac{1}{45}+\frac{2}{3\pi}\frac{1}{4}<\frac13, \]

which contradicts inequality c): \(|R(a)|>5/6\). Thus, (1) is proved. Hence the following is true:

Theorem 1. There exists a simply connected domain \(G\) (coinciding with the set of interior points of its closure \(\overline G\)) for which theorem A does not hold.

2. Definitions and auxiliary propositions. Let \(G\) be a domain of the extended plane \(\overline Z\), coinciding with the set of interior points of its closure \(\overline G\); let \(\Gamma\) be the boundary of the domain \(G\); \(\Gamma=\overline G\setminus G\); let \(\{g_n\}\) be the collection of domains adjacent to \(\overline G\) \((\bigcup_n g_n=\overline Z\setminus \overline G)\); let \(\gamma_n\) be the boundary of the domain \(g_n\); \(\widetilde\Gamma=\bigcup_n\gamma_n\); and let \(K_p\) be a connected component of the set \(\widetilde\Gamma\) \((\bigcup_p K_p=\widetilde\Gamma,\ K_p\cap K_q\) empty for \(p\ne q;\ p,q=0,1,\ldots)\).

Let us introduce the set \(\Lambda(G)\). It consists of those and only those points of the plane \(\overline Z\) in every neighborhood of each of which there are points of infinitely many domains \(g_n\). The set \(\Lambda(G)\) will play an important role in what follows. Obviously, the closed domain \(\overline G\) is finitely connected if and only if \(\Lambda(G)\) is empty. Let us note here that \(\Lambda(G)\) may have no points in common with \(\widetilde\Gamma\). An example of such a domain is obtained from the domain \(G\) constructed in § 1 if we concentrically enlarge the circle \(\sigma_0\) and reduce the circles \(\sigma_n\) with \(n\ge 1\).

It is easy to prove

Theorem 2. In order that a set \(E\) coincide with the set \(\Lambda(G)\) for some domain \(G\subset\overline Z\), it is necessary and sufficient that: 1) \(E\) be closed; 2) \(E\) be entirely contained in the boundary of one of the domains adjacent to this set.

Definition 1. A bounded domain \(G\) belongs to the class \(B(\alpha)\) \((\alpha>0)\) if the following conditions are satisfied: 1) for every \(p\) the continuum \(K_p\) is the sum of a finite number of sets \(\gamma_n\) and is at a positive distance from the remaining part \(\Gamma\setminus K_p\) of the boundary of the domain \(G\); 2) if \(d_p\) denotes the diameter of the continuum \(K_p\) \((d_p>0!)\), then \(\sum_p d_p^\alpha<\infty\); 3) there exist positive numbers \(P=P(G)\) and \(c=c(G)\) such that, for \(p\ge P\), the \(cd_p^\alpha\)-neighborhood of the continuum \(K_p\) contains no points of the set \(\Gamma\setminus K_p\).

Obviously, any domain \(G\) with finitely connected closure \(\overline G\) belongs to all classes \(B(\alpha)\) \((\alpha>0)\); if \(\alpha<\beta\), then \(B(\alpha)\subset B(\beta)\). Let us also note that theorem 2 remains valid if one requires of the domain \(G\) that it belong to all classes \(B(\alpha)\) with \(\alpha>0\) (in this case, of course, one must require of \(E\) ...

boundedness). This is proved by a construction analogous to that carried out in §1. As usual, we shall call a bounded domain \(G\) a domain with finite perimeter if \(\gamma_n\) are rectifiable curves and

\[ \sum_n \operatorname{mes}_1 \gamma_n < \infty . \]

Definition 2. A functional property \(X\) is called regular if the following conditions are satisfied for it: 1) every function possessing property \(X\) on a set \(E \subset Z\) is continuous on \(E\); 2) if \(f(z)\) possesses property \(X\) on \(E\), then it possesses this property also on every set \(F \subset E\); 3) if \(f_1(z)\) and \(f_2(z)\) possess property \(X\) on \(E\), then their product \(f_1(z)\cdot f_2(z)\), and also the linear combination \(\alpha f_1(z)+\beta f_2(z)\) with real nonnegative numbers \(\alpha,\beta\), possess the same property \(X\) on \(E\); 4) if \(f(z)\) is continuous in the extended complex plane \(\overline Z\), possesses property \(X\) on a bounded closed set \(E \subset Z\), and is analytic outside the set \(E\), then \(f(z)\) possesses property \(X\) in the finite plane \(Z\); 5) every function \(f(z)\) uniformly differentiable* on a bounded set \(E\) possesses property \(X\) on \(E\).

Examples of regular properties are: 1) continuity, 2) uniform differentiability, 3) belonging to the class \(\operatorname{Lip}\alpha\). The last follows from the following lemma (which generalizes the theorem on p. 689 of [4]):

Lemma 1. If \(f(z)\) is analytic in an open set \(G\), not containing the point \(\infty\) on its boundary \(\Gamma\), is continuous on \(\overline G\) and on \(\Gamma=\overline G\setminus G\) satisfies the condition \(\operatorname{Lip}_M \alpha\) \((0<\alpha\leqslant 1)\), then also on \(\overline G\), \(f\in \operatorname{Lip}_M \alpha\) with the same constant \(M\).

For what follows we shall also need

Lemma 2. If \(f(z)\) is uniformly differentiable on a bounded set \(E\), analytic outside \(E\) and everywhere continuous in \(\overline Z\), then \(f(z)\equiv \mathrm{const}\).

Definition 3. The \(X\)-capacity of a bounded set \(E\subset Z\) is the quantity

\[ \gamma_X(E)= \sup_{f\in M(X,E)} \left\{|\lim_{z\to\infty} z f(z)|\right\} = \sup_{f\in M(X,E)} \left\{\left|\int_L f'(z)\,dz\right|\right\}. \]

Here \(L\) is a rectifiable Jordan contour enclosing \(E\), and \(M(X,E)\) is the class of functions \(f(z)\) with the properties: 1) \(f(z)\) is defined on \(\overline Z\) and possesses the regular property \(X\) on \(E\); 2) \(|f(z)|\leqslant 1\); 3) \(f(z)\) is analytic on \(\overline Z\setminus E\); 4) \(f(\infty)=0\) (cf. the definition of \(\mathfrak M(\overline{C}G,\infty)\) in [5, 7]).

It is easily shown that the class \(M(X,E)\) consists of the single function \(f(z)\equiv 0\) if and only if \(\gamma_X(E)=0\). Let us also note here that, in the case where \(X\) is the property \(\operatorname{Lip}\alpha\) \((0<\alpha<1)\), \(\gamma_X(E)=0\) if and only if the inner Hausdorff measure of order \(1+\alpha\) of this set \(E\) is zero: \(\operatorname{mes}^{1+\alpha} E=0\). In the case where \(X\) is the property of continuity, we shall call the \(X\)-capacity the \(AC\)-capacity.

3. Some theorems on approximation by rational fractions on domains of class \(B(\alpha)\). Theorem 3. Let \(G\) be a domain with finite perimeter such that every function \(f(z)\), analytic inside \(G\) and possessing a regular property \(X\) on \(G\), can be uniformly* approximated on \(\overline G\) by rational fractions. Then the \(X\)-capacity of the set \(\Lambda(G)\) is equal to zero.

* \(f(z)\) \((z\in E)\) is called uniformly differentiable on \(E\) if the ratio \([f(z)-f(\zeta)]/(z-\zeta)\), as \(z\to \zeta\) \((z,\zeta\in E)\), tends to its limit \(f'_E(\zeta)\) uniformly with respect to \(\zeta\in E\). Obviously, a function \(f(z)\) uniformly differentiable on \(E\) has at non-isolated points of this set a continuous derivative (on the set \(E\)).

** That is, analytic in some neighborhood of each point \(z\notin E\).

*** The words “with arbitrary prescribed accuracy” are omitted here and below.

Theorem 4. In order that every function \(f(z)\), analytic inside a domain \(G\in B(1/3)\) and possessing on \(\overline G\) a regular property \(X\), be uniformly approximable on \(\overline G\) by rational fractions, it is necessary and sufficient that the set \(\Lambda(G)\) have zero \(X\)-capacity.

Corollary 1. In order that, on a domain \(G\in B(1/3)\), every function analytic inside \(G\) and satisfying the condition \(\operatorname{Lip}\alpha\) \((0<\alpha<1)\) on \(\overline G\) be uniformly approximable by rational fractions, it is necessary and sufficient that
\[ \operatorname{mes}^{1+\alpha}\Lambda(G)=0. \]

Theorem \(4'\). In order that every function \(f(z)\), analytic inside a domain \(G\in B(1/2)\) and continuous on \(\overline G\), be uniformly approximable on \(\overline G\) by rational fractions, it is necessary and sufficient that the \(AC\)-capacity of the set \(\Lambda(G)\) be equal to zero.

Theorem 5. Let \(G\in B(1/3)\). A function \(f(z)\), analytic inside \(\overline G\) and continuous on \(\overline G\), is uniformly approximable on \(\overline G\) by rational fractions if and only if it is uniformly differentiable on \(\Lambda(G)\).

We note that, in this case, \(f(z)\) (uniformly approximable by rational fractions on \(\overline G\)) may fail to have a derivative at any point \(z\in\Lambda(G)\), if the derivative is taken with respect to \(\overline G\) \({}^{(6)}\). From Theorem 4 and a generalization of Theorem 5 we obtain the following “removal of singularities” theorem:

Theorem 6. Let \(G\in B(1/(m+2))\) (\(m\) natural) and \(\gamma_X(\Lambda(G))=0\). Then every function \(f(z)\), analytic inside \(G\) and possessing on \(\overline G\) a (regular) property \(X\), is \(m\) times uniformly differentiable on \(\Lambda(G)\).

Corollary 1. Let \(G\in B(1/(m+2))\) (\(m\ge 1\)) and let \(f(z)\) be continuous on \(\overline G\) and analytic inside \(G\). Then, if the \(AC\)-capacity of the set \(\Lambda(G)\) is equal to zero, \(f(z)\) is \(m\) times uniformly differentiable on \(\Lambda(G)\).

Corollary 2. If \(G\in B(1/(m+2))\) and \(\operatorname{mes}^{1+\alpha}\Lambda(G)=0\), then every function \(f(z)\) satisfying on \(\overline G\) the condition \(\operatorname{Lip}\alpha\) \((0<\alpha\le 1)\) and analytic inside \(G\) is \(m\) times uniformly differentiable on \(\Lambda(G)\).

In conclusion we note that theorems analogous to those stated above also hold for approximations in the metric \(L_p(G)\) for \(p\ge 2\). Thus, for example, there exist a simply connected domain \(G\) (as always, coinciding with the set of interior points of its closure \(\overline G\)) and a function \(f(z)\), analytic in \(G\) and continuous on \(\overline G\), such that
\[ \iint_G |f(z)-R(z)|^p\,d\sigma \ge k=\mathrm{const}>0 \]
for every rational \(R(z)\) and every \(p\ge 2\). It is interesting to compare this with a result of S. O. Sinanyan (reported at the Fourth All-Union Mathematical Congress), which consists in the following: let \(p\ge 1\) and \(\overline G\) be finitely connected; then every \(f\in L_p(G)\), analytic in \(G\), is approximable with arbitrary accuracy in the metric \(L_p(G)\) by rational fractions**.

Moscow State University
named after M. V. Lomonosov

Received
27 XI 1961

REFERENCES

1. S. N. Mergelyan, Uspekhi Mat. Nauk, 7, no. 2 (48), 31 (1952).
2. M. V. Keldysh, Matem. sbornik, 16 (58), 3, 249 (1945).
3. I. I. Privalov, Boundary Properties of Analytic Functions, Moscow–Leningrad, 1950.
4. J. L. Walsh, W. E. Sewell, Duke Math. J., 6, no. 3, 658 (1940).
5. L. V. Ahlfors, A. Beurling, Acta Math., 83, no. 1–2, 101 (1950).
6. E. P. Dolzhenko, Dokl. Akad. Nauk SSSR, 125, no. 5 (1959).
7. A. G. Vitushkin, Dokl. Akad. Nauk SSSR, 123, no. 5, 778 (1958).

* For \(p=2\) this result was obtained by Keldysh.