UDC 517.55

MATHEMATICS

E. M. CHIRKA

APPROXIMATION OF CONTINUOUS FUNCTIONS BY HOLOMORPHIC FUNCTIONS ON JORDAN ARCS IN \(C^n\)

(Presented by Academician M. A. Lavrent’ev on 14 VI 1965)

In this paper we study the possibility of uniform approximation of continuous functions on simple Jordan arcs in the space \(C^n\) of complex variables \(z=(z_1,\ldots,z_n)\) by functions holomorphic on these arcs. By \(C(X)\) we denote the algebra of all continuous complex-valued functions on a compact set \(X\), and by \(A(X)\), \(X\subset C^n\), the algebra of all uniform limits of functions holomorphic on \(X\).

1. Theorem 1. Let \(\gamma\) be a simple Jordan arc in \(C^2\) such that its projection \(z_1(\gamma)\) is a nowhere dense subset of \(C^1\). Then \(C(\gamma)=A(\gamma)\).

Proof. 1) We first show that for every polynomial \(p(z_1)\) and every positive integer \(m\), the function
\[ \sqrt[m]{p(z_1)} \]
belongs to \(A(\gamma)\).

For this it is enough to prove that
\[ \sqrt[m]{z_1-a}\in A(\gamma) \]
for any point \(a\in C^1\). Let \(a\in z_1(\gamma)\), and let \(a_\nu\), \(\nu=1,2,\) be such that \(a_\nu\in C^1\setminus z_1(\gamma)\) and \(a_\nu\to a\) as \(\nu\to\infty\). For each \(\nu\), in view of the connectedness of \(\gamma\), one can construct a function holomorphic on \(\gamma\),
\[ \sqrt[m]{z_1-a_\nu} \]
(see (1)). Let \(\gamma_a=\{z\in\gamma: z_1=a\}\), and let \(E_\delta\) be the subset of \(\gamma\) consisting of a finite number of connected components, containing the set \(\{z\in\gamma:\ |z_1-a|\ge \delta>0\}\) and not intersecting \(\gamma_a\). Then from the sequence of functions \(\sqrt[m]{z_1-a_\nu}\), \(\nu=1,2,\ldots\), one can choose a subsequence converging uniformly on \(E_\delta\). Taking \(\delta=1/n\) and letting \(n\to\infty\), by a diagonal process we extract a subsequence
\[ \sqrt[m]{z_1-a_{\nu_k}},\quad k=1,2,\ldots, \]
which converges uniformly on any compact subsets of \(\gamma\setminus\gamma_a\).

Let \(\varepsilon>0\) be given. Choose a neighborhood \(U\) of the set \(\gamma_a\) in \(C^n\) such that \(|z_1-a_\nu|<\varepsilon^m\) if \(z\in U\) and \(a_\nu\in z_1(U)\). Choose \(k_1\) so that \(a_{\nu_k}\in z_1(U)\), \(k>k_1\), and \(\delta>0\) so that \(\gamma\setminus U\subset E_\delta\). In view of the uniform convergence of the sequence \(\{\sqrt[m]{z_1-a_{\nu_k}}\}\) on \(E_\delta\), there is \(k_2\) such that
\[ \left|\sqrt[m]{z_1-a_{\nu_{k+1}}}-\sqrt[m]{z_1-a_{\nu_k}}\right|<\varepsilon \]
for \(k>k_2\). Then for \(k>\max(k_1,k_2)\) we obtain, uniformly on \(\gamma\),
\[ \left|\sqrt[m]{z_1-a_{\nu_{k+1}}}-\sqrt[m]{z_1-a_{\nu_k}}\right|<2\varepsilon, \]
whence it follows that
\[ \sqrt[m]{z_1-a}\in A(\gamma), \]
and therefore
\[ \sqrt[m]{p(z_1)}\in A(\gamma). \]

Consider the algebra generated by functions of the form
\[ \sqrt[m]{p(z_1)}, \]
where \(p\) is an arbitrary polynomial and \(m\) is any natural number. In a completely analogous way it is proved that roots of any degree of functions of this algebra belong to \(A(\gamma)\). From the roots obtained we construct a new algebra, etc. The union of all these algebras forms a subalgebra \(A_1\subset A(\gamma)\), which, together with each \(f\), also contains
\[ \sqrt[m]{f} \]
for any natural \(m\). By a diagonal process, as was done above, this property can be extended to the uniform closure \(\overline{A}_1\) of the algebra \(A_1\) on \(\gamma\).

2) Let \(V\) and \(E\) be connected compact subsets of \(\gamma\) such that \(|z_1-a|<\alpha\) for \(z_1\in z_1(V)\) and \(|z_1-a|>\alpha\) for \(z_1\in z_1(E)\). Put
\[ f_m=\alpha\sqrt[m]{\left(\frac{z_1-a}{\alpha}\right)^m+1}; \]
we have
\[ \lim_{m\to\infty}|f_m|=\max(|z_1-a|,\alpha) \]
and
\[ \lim_{m\to\infty} f_m(z)=\alpha \]
for \(z\in V\), in view of the connectedness of \(V\). Let
\[ f_{mN}=\sqrt[N]{\,f_m-\alpha\,}; \]
then
\[ \lim_{N\to\infty}\lim_{m\to\infty} f_{mN}(z)=0, \]
if \(z\in V\). Since the variation of the argument \(\Delta_E\operatorname{Arg}(f_m-\alpha)\) on the connected compact set \(E\) does not exceed \(\Delta_E\operatorname{Arg}(z_1-a)+2\pi\), choosing the necessary branches, we may assume that
\[ \lim_{N\to\infty}\lim_{m\to\infty} f_{mN}(z)=1, \]
if \(z\in E\). Therefore, for any \(\delta'>0\) one can choose \(m\) and \(N\) so that \(|f_{mN}|<1+\delta'\), \(|f_{mN}(z)-1|<\delta'\), \(z\in E\), and \(|f_{mN}(z)|<\delta'\), \(z\in V\).

Let now \(U_a\) be a neighborhood of \(\gamma_a\) in \(\gamma\), consisting of a finite number of connected components \(U_{a\nu}\), \(\nu=1,\ldots,n\), containing the set \(\{z\in\gamma: |z_1-a|\le\alpha,\ \alpha>0\}\) and contained in \(\{z\in\gamma: |z_1-a|<\alpha+\beta,\ \beta>0\}\). Let \(E_a\) be a subset of \(\gamma\), consisting of a finite number of connected compact sets \(E_{a\mu}\), such that
\[ \{z\in\gamma: |z_1-a|\ge\alpha+2\beta\}\subset E \]
and
\[ E\subset\{z\in\gamma: |z_1-a|>\alpha+\beta\}. \]
In view of the local connectedness of \(\gamma\), such sets can be constructed for any \(\alpha,\beta>0\). For each pair \(U_{a\nu}, E_{a\mu}\) and any \(\delta'>0\) we can now construct a function \(g_{\mu\nu}\) such that \(|g_{\mu\nu}|<1\), \(|g_{\mu\nu}(z)-1|<\delta'\), \(z\in E_{a\mu}\), and \(|g_{\mu\nu}(z)|<\delta'\), \(z\in\overline{U}_{a\nu}\).

Let
\[ g_\nu=\prod_\mu g_{\mu\nu} \]
and
\[ g=1-\sqrt[n]{(1-g_1)\cdots(1-g_n)}. \]
Then, if \(\delta''>0\), \(\delta'\) can be chosen so that \(|g|\le 1\), \(|g(z)-1|<\delta''\), if \(z\in E\), and \(|g(z)|\le\delta''\), if \(z\in\overline{U}_a\). Let \(1>\delta>0\), and let \(\varphi\) be a fractional-linear transformation of the unit disk onto itself such that \(\varphi(0)=-1+\delta\) and \(\varphi(1)=1\). Choose \(\delta''\) so that \(|\varphi(\lambda)-1|<\delta^4\), if \(|\lambda-1|<\delta''\), and \(|\varphi(\lambda)+1|<2\delta^4\), if \(|\lambda|<\delta''\), and put
\[ h_a=\sqrt[4]{\frac12(1-\varphi\circ g)}; \]
according to \(1)\), \(h_a\in A(\gamma)\). Moreover:
\[ |h_a|\le 1,\qquad |\arg h_a|\le \pi/8, \]
\[ |h_a(z)-1|<\delta,\quad \text{if } z\in\overline{U}_\alpha, \]
and
\[ |h_a(z)|<\delta,\quad \text{if } z\in E_a. \]

Let now \(f\in C(\gamma)\) and \(\varepsilon>0\). By the theorem of M. A. Lavrentiev \((^2)\), for every \(a\in z_1(\gamma)\) there is a polynomial
\[ p_a(z)\equiv p_a(z_1) \]
such that \(|f(z)-p_a(z)|<\varepsilon\), if \(z\in\gamma_a\). Choose \(\alpha\) and \(\beta\) so that \(|f-p_a|<\varepsilon\) on \(\gamma\setminus E_a\). From the covering of \(\gamma\) by the sets \(U_a\) select a finite subcover: \(U_1,\ldots,U_k\). Let \(p_1,\ldots,p_k\) be the corresponding polynomials and \(h_1,\ldots,h_k\) the functions constructed above for \(\delta\) and for the pairs \(U_i,E_i\); we assume that \(0<\delta<1/2\). Put
\[ e_i=h_i(h_1+\cdots+h_k)^{-1}. \]
Then
\[ e_1+\cdots+e_k\equiv 1 \]
on \(\gamma\), and
\[ |\arg e_i|<\pi/4,\qquad i=1,\ldots,k. \]
Moreover,
\[ |e_i(z)(f(z)-p_i(z))|<\varepsilon,\qquad z\in\gamma\setminus E_i, \]
and
\[ |e_i(z)(f(z)-p_i(z))|<2\max_\gamma |f-p_i|\cdot\delta,\qquad z\in E_i. \]
Let \(z\in\gamma\), and let \(i_1,\ldots,i_s\) be all indices such that \(z\notin E_{i_\nu}\), \(\nu=1,\ldots,s\). Then
\[ \left|f(z)-\sum_{i=1}^k e_i(z)p_i(z)\right| = \left|\sum_{i=1}^k e_i(z)(f(z)-p_i(z))\right| \le \]
\[ \le \left|\sum_{\nu=1}^s e_{i_\nu}(z)(f(z)-p_{i_\nu}(z))\right| + \left|\sum_{\nu=s+1}^k e_{i_\nu}(z)(f(z)-p_{i_\nu}(z))\right| < \]
\[ <\varepsilon\sum_{\nu=1}^k |e_\nu(z)| + \left(2k\max_i \max_\gamma |f-p_i|\right)\delta. \]
Since
\[ \sum_{\nu=1}^k |e_\nu(z)|\le \sqrt{2}, \]
with a proper choice of \(\delta\) we obtain uniformly on all of \(\gamma\)
\[ \left|f-\sum_{i=1}^k e_i p_i\right|<\varepsilon. \]
Since
\[ \left(\sum_{i=1}^k e_i p_i\right)\in A(\gamma), \]
the theorem is proved.

1. Theorem 2. Let \(\gamma\) be a simple Jordan arc in \(C^2\). Then, for every function \(j\) holomorphic on \(\gamma\), either the set \(j(\gamma)\) contains interior points, or \(C(\gamma)=A(\gamma)\).

In the proof of Theorem 1 we used Lavrentiev’s theorem on approximation by polynomials on plane compact sets. The proof of Theorem 2 rests on the following generalization of S. N. Mergelyan’s theorem \((^3)\):

Theorem 3. Let \(O\) be an open subset of \(C^n\) and \(S\) a one-dimensional analytic set in \(O\). Let a compact set \(X\subset S\) not divide the irreducible components of this set. Then: 1) the set \(X\) is polynomially convex in \(C^n\); 2) every function continuous on \(X\) and holomorphic at the points of \(X\) that are interior relative to \(S\) is uniformly approximable on \(X\) by polynomials in \(z_1,\ldots,z_n\).

Theorem 3 is proved using results of Rossi \((^4)\) and Bishop \((^5)\). It is interesting to compare Theorem 2 with the known examples of Wermer \((^6)\) and Rudin \((^7)\) of Jordan arcs in \(C^3\) and \(C^2\) on which not every continuous function is approximable by polynomials. However, in these examples, for every polynomial \(p(z)\), the plane set \(p(\gamma)\) contains no interior points.

1. Definition. Let \(A\) be a multiplicative group. We shall say that \(A\) is closed under extraction of (square) roots if, together with every \(f\in A\), \(A\) also contains an element \(\sqrt f\) such that \((\sqrt f)^2=f\). Let \(X\) be an arbitrary compact set, and let \(A_0\) be a subset of \(C(X)\). By the algebraic extension of the set \(A_0\) we mean the minimal subalgebra of \(C(X)\) that contains \(A_0\), is closed with respect to uniform convergence on \(X\), and is closed with respect to the operation of extracting square roots.

In the first part of the proof of Theorem 1, essentially the following was proved.

Theorem 4. Let \(X\) be a locally connected compact set, and let \(A\) be some family of functions continuous on \(X\), closed with respect to extraction of roots. Then \(\overline A\) (the uniform closure of \(A\) on \(X\)) is also closed with respect to this operation.

In the second part of the proof of Theorem 1 one can get by with roots of powers of the form \(2^k\), where \(k\) is a natural number. In doing so, only the following facts are essentially used: 1) the arc \(\gamma\) is locally connected; 2) there exists a family \(A_0\) of functions from \(A(\gamma)\) (namely, the functions \(z_1\)), whose algebraic extension lies in \(A=A(\gamma)\); 3) on every set \(\gamma_a\), where all functions from \(A_0\) are constant, \(C(\gamma_a)=A|\gamma_a\) (\(A|\gamma_a\) denotes the restriction of \(A\) to \(\gamma_a\)).

Thus the following holds.

Theorem 5. Let \(X\) be a locally connected compact set; let \(A\) and \(B\), \(A\subset B\), be algebras of continuous functions on \(X\). Suppose that in \(A\) there exists a subset \(A_0\) such that: 1) the algebraic extension of \(A_0\) lies in \(A\); 2) on every subset \(X_a\subset X\) on which all functions from \(A_0\) are constant, \(\overline{B|X_a}=\overline{A|X_a}\). Then \(\overline A=\overline B\).

If \(A_0\) separates the points of \(X\), then condition 2) is automatically satisfied, and therefore the following holds.

Theorem 5′. Let \(A\) be a separating algebra of continuous functions with identity on a locally connected compact set \(X\), closed with respect to extraction of square roots. Then \(\overline A=C(X)\).

In conclusion I express my deep gratitude to A. G. Vitushkin, A. A. Gonchar, and B. V. Shabat for discussion of this work.

Moscow State University
named after M. V. Lomonosov

Received
16 IV 1965

CITED LITERATURE

\(^1\) G. Stolzenberg, Acta Math., 109, 3—4 (1963).
\(^2\) M. A. Lavrentiev, Actual Sci. et Ind., No. 441, Paris, 1936.
\(^3\) S. N. Mergelyan, DAN, 78, No. 3 (1951).
\(^4\) H. Rossi, Trans. Am. Math. Soc., 100, 3 (1961).
\(^5\) E. Bishop, Trans. Am. Math. Soc., 102, 3 (1962).
\(^6\) J. Wermer, in: Some Questions of Approximation Theory, IIL, 1963, p. 120.
\(^7\) W. Rudin, ibid., p. 124.