MATHEMATICS

S. Ya. GUSMAN

UNIFORM APPROXIMATION OF CONTINUOUS FUNCTIONS ON RIEMANN SURFACES

(Presented by Academician M. A. Lavrent’ev, 21 X 1959)

S. N. Mergelyan’s theorem ([1], p. 44) on the approximation of continuous functions by polynomials on closed point sets of the complex plane, which is a generalization of known theorems of M. A. Lavrent’ev and M. V. Keldysh, carries over to closed Riemann surfaces.

Theorem 1. In order that a function \(f(P)\), defined on a closed set \(E\) belonging to a closed Riemann surface \(R\), be expandable in a series of functions rational on \(R\) with a single pole at the point \(Q\), converging uniformly to \(f(P)\) on \(E\), it is necessary and sufficient that the complement of \(E\) consist of one domain containing the point \(Q\), and that \(f(P)\) be continuous on \(E\) and analytic at every interior point of the set \(E\).

The necessity of the condition of Theorem 1 is obvious; the sufficiency follows from the following theorem.

Theorem 2. If the complement of a closed set \(E\) relative to a closed Riemann surface \(R\) consists of \(n\) domains \(G_1, \ldots, G_n\), and \(Q_1, \ldots, Q_n\) are arbitrary points from these domains, then any single-valued function \(f(P)\), continuous on \(E\) and analytic at the interior points of \(E\), is expandable in a series of functions rational on \(R\), having poles only at the points \(Q_1, \ldots, Q_n\), converging uniformly to \(f(P)\) on \(E\).

For the proof of Theorem 2 one uses the lemma given in the paper [2], which is a transfer of Runge’s lemma to closed Riemann surfaces:

Lemma 1. Let \(D\) be a domain on a closed Riemann surface \(R\), bounded by a finite number of closed Jordan curves, and let the complement of \(\overline{D}\) relative to \(R\) consist of \(n\) domains \(G_1, \ldots, G_n\), and let \(Q_k\) be an arbitrary point of \(G_k\) \((k = 1, 2, \ldots, n)\). If the function \(f(P)\) is analytic on \(\overline{D}\), then for any \(\varepsilon > 0\) there exists a function \(\varphi(P)\), rational on \(R\), with poles only at the points \(Q_1, \ldots, Q_n\), satisfying the inequality

\[ \max_{P \in D} |f(P) - \varphi(P)| < \varepsilon . \]

Let \(P_1, \ldots, P_m\) and \(Q\) be arbitrary points of a closed Riemann surface \(R\), \(f(P)\) an arbitrary function, and \(dg(P)\) an arbitrary differential on \(R\), having poles at the points \(P_1, \ldots, P_m\); then there exist a function \(R(P)\), rational on \(R\), and an abelian differential \(dA(P)\) with poles only at the points \(P_1, \ldots, P_m\) and \(Q\), such that \(f_1(P) = f(P) - R(P)\) and \(dg_1(P) = dg(P) - dA(P)\) are regular at the points \(P_1, \ldots, P_m\). It can also be shown that for any closed set \(E\) on

on \(R\) there exists an Abelian differential of the first kind \(d\omega_1(P)\), having no zeros at the boundary points \(E\), and an Abelian differential \(d\omega_2(P)\), having no poles at the boundary points \(E\) and vanishing on \(R\) at the single point \(Q\), which can be chosen arbitrarily.

With the aid of \(d\omega_1(P)\) and \(R(P)\), Lemma 1 yields an analogous assertion for differentials:

Lemma 2. Let \(D\) be a domain on the closed Riemann surface \(R\), bounded by a finite number of closed Jordan arcs \(\partial y\), and let the complement of \(\overline D\) with respect to \(R\) consist of \(n\) domains \(G_1,\ldots,G_n\), and let \(Q_k\) be an arbitrary point of \(G_k\). If the differential \(dg(P)\) is analytic on \(\overline D\), then, for any fixed finite covering of \(R\) by cells and for any \(\varepsilon>0\), there exists an Abelian differential \(dh(P)\) with poles only at the points \(Q_k\), satisfying the inequality1

\[ \max_{P\in D}\left|\frac{dg(P)}{dz}-\frac{dh(P)}{dz}\right|<\varepsilon . \]

Next, almost as in the paper \((^1)\), one constructs the averaging of a discontinuous function \(f(P)\), defined on the closed set \(E\), and from it, with the aid of \(d\omega_1(P)\) and \(R(P)\), one obtains an averaging \(dH_\delta(P)\) for a differential \(dg(P)\) discontinuous on \(E\) and analytic at the interior points of \(E\).

Let now \(D\) be a domain on the closed Riemann surface \(R\), bounded by a finite number of smooth Jordan arcs; \(\Gamma\) the boundary of \(D\); \(dF(P,P_1)\) the elementary differential on \(R\)—a differential in \(P\) and a function in \(P_1\)—with characteristic point \(Q\) outside \(D\) (see \((^3)\)); \(dF(P,P_1)/dz\) the Cauchy kernel on \(R\). By means of Green’s formula one proves the important formula

\[ \varphi(P_1)=\frac{1}{2\pi i}\int_{\Gamma}\varphi(P)\,dF(P,P_1) -\frac{1}{\pi}\iint_D \frac{dF(P,P_1)}{dz}\, \frac{\partial\varphi(P)}{\partial \bar z}\,dx\,dy, \tag{1} \]

which is a generalization of the Cauchy formula for nonanalytic, but differentiable, functions, and the analogous formula for covariants

\[ \frac{d\psi(P)}{dz} =\frac{1}{2\pi i}\int_{\Gamma}\frac{dF(P,P_1)}{dz}\,d\psi(P_1) -\frac{1}{\pi}\iint_D \frac{dF(P,P_1)}{dz}\, \frac{\partial}{\partial \bar\zeta}\left(\frac{d\psi(P_1)}{d\zeta}\right)\,d\xi\,d\eta . \tag{2} \]

Removing from \(R\) an arbitrary closed domain having no common points with \(E\), we obtain an open surface \(R'\) containing \(E\). Mapping the universal covering surface \(\hat R'\) of the surface \(R'\) onto a disk, we obtain for \(R'\) a fundamental polygon \(\Pi\) and the corresponding Fuchsian group of the second kind. With the aid of automorphic differentials on \(\Pi\) one constructs a differential \(dF_{P'_1}(P)\), regular on \(E\) and approximating \(dF(P,P_1)\) in a neighborhood of the point \(P_1=P'_1\) the better, the closer \(P'_1\in E\) is to the boundary of this set.

After this we can prove the following theorem:

Theorem 3. If the complement of the closed set \(E\) with respect to the closed Riemann surface \(R\) consists of \(n\) domains \(G_1,\ldots,G_n\) and \(Q_1,\ldots,Q_n\) are arbitrary points from these domains, while \(dg(P)\) is a differential regular at every interior point of \(E\), except for a finite number of points \(P_1,\ldots,P_m\), where it has poles, and

continuous on \(E\) except at the same points, then, for any fixed finite covering of \(R\) by cells and for any \(\varepsilon>0\), there exists an Abelian differential \(dh(P)\) on \(R\), with poles only at the points \(P_i\) and \(Q_k\), satisfying the inequality

\[ \max_{P\in E}\left|\frac{dg(P)}{dz}-\frac{dh(P)}{dz}\right|<\varepsilon. \]

For \(m=0\) the theorem is proved as follows: we average \(dg(P)\); to the average \(dH_\delta(P)\) and to a domain \(B\) with smooth boundary contours, chosen so that the measure\(^*\) \(m(B-E)\) is sufficiently small, we apply formula (2); under the double-integral sign we replace \(dF(P,P_1)/dz\) by \(dF_{P_1'}(P)/dz\); by Lemma 2 we replace the differentials, regular on \(\overline B\), thus obtained by Abelian differentials on \(Q\) with poles at the points \(Q_1,\ldots,Q_n\). We obtain an Abelian differential \(dh(P)\) satisfying the inequality

\[ \max_{P\in E}\left|\frac{dg(P)}{dz}-\frac{dh(P)}{dz}\right|<N_1\omega(\delta)+\frac{N_2}{\delta}m(B-E). \]

For every \(\varepsilon>0\) there exists a \(\delta\) such that \(\omega(\delta)<\varepsilon/2N_1\), and a domain \(B\) such that \(m(B-E)<\varepsilon\delta/2N_2\).

The general case of Theorem 3 is obtained simply from the case \(m=0\) with the aid of \(dA(P)\).

From Theorem 3, with the aid of \(d\omega_2(P)\), Theorem 2 is obtained.

Theorem 2, with the aid of the function \(R(P)\), is generalized to the case of functions having poles at interior points of \(E\). This theorem is a generalization of Sakakihara’s theorem from the paper \((^2)\), where, instead of an arbitrary closed set \(E\), Jordan domains are considered.

With the aid of formula (1) the following theorem is proved.

Theorem 4. If the planar measure of the continuum \(E\) on \(R\) is zero, then any function \(f(P)\) continuous on \(E\) can be represented in the form of a uniformly convergent on \(E\) series \(\sum_{n=1}^{\infty} R_n(P)\), where \(R_n(P)\) are rational functions on \(R\).

The work was carried out under the supervision of Prof. L. I. Volkovyskii, to whom the author expresses sincere gratitude.

Received
12 X 1959

REFERENCES

\(^1\) S. N. Mergelyan, Uspekhi Mat. Nauk, 7, no. 2 (48), 31 (1952).
\(^2\) K. Sakakihara, J. Inst. Polytechn. Osaka Cit. Univ., 5, No. 1, Ser. A, 63 (1954); RZh Mat., No. 6646 (1958).
\(^3\) H. Tietze, J. f. reine u. angew. Math., 190, H. 1, 64 (1953).

\(^*\) The hyperbolic measure associated with the mapping \(R'\) onto the unit disk is considered.

1. Here and below \(z=x+iy\) denotes the local parameter belonging to the point \(P\). When we have to deal with two variable points, we shall denote the second by \(P_1\), and the corresponding local parameter by \(\zeta=\xi+i\eta\). ↩