MATHEMATICS

A. A. GONCHAR

ON QUASIANALYTIC CONTINUATION OF ANALYTIC FUNCTIONS

(Presented by Academician A. N. Kolmogorov, 19 VII 1961)

Let \(E\) be an arbitrary continuum in the plane \(P\) of the complex variable \(z\); let \(G\) be a finite or infinite domain not intersecting the continuum \(E\); suppose that a sequence of rational functions \(r_n(z)\) of the form

\[ r_n(z)=\frac{a_{n0}z^n+a_{n1}z^{n-1}+\cdots+a_{nn}} {(z-\alpha_{n1})(z-\alpha_{n2})\cdots(z-\alpha_{nn})}, \qquad \alpha_{nk}\in \overline{E},\quad \{\alpha_{nk}\}'\cap G=0, \tag{1} \]

converges uniformly on \(E\) to a function \(f_1(z)\) (continuous on \(E\) and analytic on the set of interior points of \(E\)) and inside the domain \(G\) to a function \(f_2(z)\) (analytic in \(G\)):

\[ \lim_{n\to\infty} r_n(z)= \begin{cases} f_1(z), & z\in E,\\ f_2(z), & z\in G. \end{cases} \]

We note that no restrictions are imposed on the location of the poles \(\alpha_{nk}\) of the rational functions \(r_n(z)\) in \(P\setminus(E\cup G)\). The question is posed as follows: under what conditions on the rate of convergence of the sequence \(r_n(z)\) on the continuum \(E\) are the values of the function \(f_2(z)\), \(z\in G\), uniquely determined by the values of the function \(f_1(z)\) on the continuum \(E\), and in this sense the function \(f_2(z)\) is a quasianalytic continuation of the function \(f_1(z)\); in particular, under what conditions does it follow from \(f_1(z)=0\), \(z\in E\), that also \(f_2(z)=0\), \(z\in G\).

Some results of this type were obtained, in particular, by Borel ((\(^{1}\)); see also (\(^{6}\))) for functions of the form

\[ \sum_{n=1}^{\infty}\frac{A_n}{z-z_n}; \]

by S. N. Mergelyan (\(^{2}\)) for the case of approximation by polynomials in tangent domains; in a number of cases the solution of the problem posed follows from the results on overconvergence of sequences of rational functions given in (\(^{6}\)).

Theorem 1. Let \(E\) be an arbitrary continuum; let \(L_R\), \(R>1\), be the image of the circle of radius \(R\) with center at \(0\) under the mapping of the exterior of the unit circle onto the component of the complement of \(E\) containing the infinitely distant point; let \(G\) be an arbitrary domain having as one of the components of its boundary the level line \(L_R\). Suppose that a sequence of rational functions \(r_n(z)\) of the form (1) converges on \(E\) to a function \(f_1(z)\) and in the domain \(G\) to an analytic function \(f_2(z)\), and moreover

\[ \overline{\lim}_{n\to\infty}\left[\max_{z\in E}|f_1(z)-r_n(z)|\right]^{1/n}=q<\frac{1}{R}. \]

Then, if \(f_1(z)=0\), \(z\in E\), then also \(f_2(z)=0\), \(z\in G\).

We note that Theorem 1 remains valid if, instead of a sequence of rational functions \(r_n(z)\), one considers a sequence of functions of the form \(p_n(z)/a_n(z)\), where \(p_n(z)\) is a polynomial of degree \(n\), and \(a_n(z)\), \(n=1,2,\ldots\), is an arbitrary sequence of functions analytic-

functions in a simply connected domain \(D\) containing \(E\) and \(G\) (in particular, an arbitrary sequence of entire functions).

In the theorem one may consider a sequence of rational functions of the form (1) for \(n=n_1,n_2,\ldots\); thereby the sign of the upper limit may be replaced by the sign of the lower limit.

Theorem 1 is sharp in a certain sense; indeed, the sequence of rational functions
\[ r_n(z)=\frac{z^n}{z^n-R^n},\qquad R>1, \]
converges to zero in the closed unit disk \(|z|\leqslant 1\) and to one in the domain \(|z|>R\), and
\[ \lim_{n\to\infty}\left[\max_{|z|\leqslant 1}|r_n(z)|\right]^{1/n}=\frac1R. \]

Corollary 1. Let \(E\) be an arbitrary continuum, \(D\) a domain containing this continuum in its interior; \(G=D\cap D_\infty\), where \(D_\infty\) is the component of the complement of the continuum \(E\) that contains the point at infinity. If a sequence of rational functions \(r_n(z)\) of the form (1) converges on \(E\) to a function \(f_1(z)\) and in the domain \(G\) to an analytic function \(f_2(z)\), and moreover
\[ \overline{\lim}_{n\to\infty}\left[\max_{z\in E}|f_1(z)-r_n(z)|\right]^{1/n}=q<1, \]
then the values of the function \(f_1(z)\), \(z\in E\), uniquely determine the values of the function \(f_2(z)\) in the domain \(G\) (\(f_2(z)\) is a quasianalytic continuation of \(f_1(z)\)); in particular, if \(f_1(z)=0\), \(z\in E\), then also \(f_2(z)=0\), \(z\in G\).

Remark 1. The function \(f_2(z)\) in Corollary 1 is not necessarily an analytic continuation of the function \(f_1(z)\), since the set of limit points of the sequence of poles of the rational functions \(r_n(z)\) may consist of the continuum \(E\) and a sequence of isolated points whose derived set coincides with \(E\) (the domain \(G\) is then infinitely connected). It can be shown that if the conditions of the corollary are fulfilled and the function \(f_1(z)\) is analytic on the continuum \(E\) (and therefore also in some domain containing the continuum \(E\)), then the function \(f_2(z)\), which is a quasianalytic continuation of the function \(f_1(z)\), coincides in some neighborhood of \(E\) with the analytic continuation of the function \(f_1(z)\).

Remark 2. The last remark can be applied to the following particular problem: let \(D\) be an arbitrary bounded Jordan domain; \(\Gamma\) the boundary of the domain \(D\); \(G\) a simply connected domain containing the closed domain \(\overline D\) in its interior; and let the function \(f(z)\) be analytic in the domain \(G\). Wolff \((^4)\) showed (see also \((^3)\)) that the function \(f(z)\) in the domain \(D\) can be represented in the form
\[ f(z)=\sum_{n=1}^{\infty}\frac{A_n}{z-z_n},\qquad \sum_{n=1}^{\infty}|A_n|<\infty,\qquad z\in D, \tag{2} \]
where \(\{z_n\}\) is a sequence of points lying in \(G\setminus \overline D\), and all limit points of this sequence belong to \(\Gamma\) (so that \(\{z_n\}'=\Gamma\)).

Thus, the series \(\sum \frac{A_n}{z-z_n}\) may converge in the domain \(D\) to a function \(f(z)\) analytically continuable to the larger domain \(G\supset \overline D\), while \(\sum \frac{A_n}{z-z_n}\) has an infinite set of poles. From the assertion given in Remark 1 it follows that, when the coefficients \(A_n\) tend to zero sufficiently rapidly (at the rate of a geometric progression), one may assert that \(\Gamma\) is a cut for the function
\[ f(z)=\sum \frac{A_n}{z-z_n},\qquad z\in D. \]
More precisely, the following is true:

Corollary 2. Let an analytic function \(f(z)\) in a simply connected domain \(D\) be represented in the form

\[ f(z)=\sum_{n=1}^{\infty}\frac{A_n}{z-z_n}, \qquad z\in D, \]

where \(\{z_n\}\) is a sequence of points lying outside \(\overline D\) and such that \(\{z_n\}'=\Gamma=\overline D\setminus D\).

If

\[ \overline{\lim_{n\to\infty}}\sqrt[n]{|A_n|}<1, \tag{3} \]

then the domain \(D\) is the full domain of analyticity of the function \(f(z)\).

In other words, in Wolff’s theorem on the representability of a function \(f(z)\), analytic in the closed domain \(\overline D\), by means of functions of the form (2), the coefficients \(A_n\) tend to zero sufficiently slowly (in any case,

\[ \overline{\lim_{n\to\infty}}\sqrt[n]{|A_n|}=1 \]

).

It can be shown that condition (3) in Corollary 2 cannot be substantially weakened (cf. \(\left({}^5\right)\)).

Theorem 2. Let \(G_1\) and \(G_2\) be arbitrary domains whose boundaries have at least one point of contact; let a sequence of rational functions \(r_n(z)\) of the form (1) converge to an analytic function \(f_i(z)\) for \(z\in G_i,\ i=1,2\), and suppose that

\[ \overline{\lim_{n\to\infty}}\left[\max_{z\in F}|f_1(z)-r_n(z)|\right]^{1/n}\leq q<1 \tag{4} \]

for every closed set \(F\) belonging to the domain \(G_1\). Then the values of the function \(f_1(z)\), \(z\in G_1\), uniquely determine the values of the function \(f_2(z)\), \(z\in G_2\) (\(f_2(z)\) is a quasianalytic continuation of \(f_1(z)\)); in particular, if \(f_1(z)=0\), \(z\in G_1\), then also \(f_2(z)=0\), \(z\in G_2\).

Condition (4) cannot be weakened for any order of contact of the boundaries of the domains \(G_1\) and \(G_2\); indeed, consider the sequence of rational functions

\[ r_n(z)=\frac{z^n}{z^n-1}, \]

and let \(G_1:\ |z|<1,\ G_2:\ |z|>1\). We have

\[ \lim_{n\to\infty} r_n(z)= \begin{cases} 0, & z\in G_1,\\ 1, & z\in G_2. \end{cases} \]

At the same time, for any function \(\mu(n)\) tending to infinity arbitrarily slowly as \(n\to\infty\), we have

\[ \overline{\lim_{n\to\infty}}\left[\max_{z\in F}|r_n(z)|\right]^{\mu(n)/n}=0 \]

for every closed set \(F\subset G_1\). Note that in this case the domains \(G_1\) and \(G_2\) have the common boundary \(|z|=1\), i.e. the order of contact of the boundaries is, in a certain sense, maximal.

Received
21 VI 1961

References

\({}^1\) E. Borel, Leçon sur la théorie des fonctions, Paris, 1914.
\({}^2\) S. N. Mergelyan, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 37 (1951).
\({}^3\) S. L. Walsh, Interpolation and Approximation, N. Y., 1935.
\({}^4\) J. Wolff, C. R., 173, 1327 (1921).
\({}^5\) A. Denjoy, C. R., 174, 95 (1922).
\({}^6\) A. A. Gonchar, DAN, 141, No. 5 (1961).