MATHEMATICS

N. U. ARAKELYAN

ON UNIFORM AND ASYMPTOTIC APPROXIMATION BY ENTIRE FUNCTIONS ON UNBOUNDED CLOSED SETS

(Presented by Academician M. V. Keldysh on 8 II 1964)

One of the fundamental problems in the theory of uniform approximation by entire functions is the following: on which closed sets (E) does every function continuous on (E) (except, possibly, at the point (z=\infty)) and analytic at the interior points of (E) admit uniform approximation by entire functions? As Carleman first established ((^{1})), for (E) one may take any Jordan curve beginning and ending at infinity. This result was substantially strengthened by M. V. Keldysh and M. A. Lavrent’ev ((^{2})), who established a necessary and sufficient condition on a nowhere dense continuum (E) in order that every function continuous on (E) should admit uniform approximation by entire functions. This condition is the following.

Condition K. We shall say that a closed set (E) satisfies condition K, or is a (K)-set, if there exists a function (r_E(t)\ge 0) increasing to (+\infty) such that every point (z\in CE) can be joined with infinity by a Jordan line lying in (CE) and outside the circle (|\zeta|\le r_E(|z|)).

For sets (E) containing interior points, M. V. Keldysh established ((^{3,4})) that condition K for continua is sufficient for the possibility of approximation. It turns out that condition K in the general case is not only sufficient but also necessary for the solution of the problem formulated at the beginning of this note.

Theorem 1. In order that every function continuous on the closed set (E) (except, perhaps, at the point (z=\infty)) and analytic at the interior points of (E) admit uniform approximation on (E) by entire functions, it is necessary and sufficient that (E) satisfy condition K or coincide with the whole plane.

The proof of the necessity of condition K is based on the following lemma.

Lemma 1. Let (\gamma) be a Jordan arc lying in the annulus

[
S:\quad r\le |z|\le R
]

and joining the boundary circles of (S). For the function (\varphi(z)), (\varphi(0)=0), mapping the exterior of (\gamma) onto the unit disk, the estimate holds

[
1-|\varphi(z)|<\left(\frac{8r}{R-r}\right)^{1/2},\qquad |z|>R.
]

If the set (E) does not satisfy condition K, then one constructs a function (f(z)), analytic on (E), for which the inequality

[
|f(z)-G(z)|<1,\qquad z\in E, \tag{1}
]

is not satisfied for any entire function (G(z)). In the case when the complement of (E) contains a bounded component (G) lying in the circle (|z-z_0|<d), (z_0\in G), we choose (f(z)=\dfrac{2d}{z-z_0}). If, however, such a component (G) does not exist, then first we choose a certain sequence

Jordan arcs ({\gamma_k}1^\infty \in CE), (\min|z|\to\infty), and a sequence of points ({a_k}_1^\infty \in CE), (a_k\to\infty). Next we form the infinite product

[
\prod_{k=1}^{\infty}
\frac{f_k(a_k)-f_k(z)}{1-\overline{f_k(a_k)}\,f_k(z)}
\frac{\overline{f_k(a_k)}}{|f_k(a_k)|},
]

where (f_k(z)) is the function mapping the exterior of (\gamma_k) onto the unit disk, with (f_k(0)=0).

With the aid of Lemma 1 it is proved that this product converges uniformly outside the arcs ({\gamma_k}_1^\infty) to a certain analytic function (\mathfrak{B}(z)), all of whose zeros lie at the points ({a_k}_1^\infty). Finally, it is proved that for the function (f(z)=\dfrac{2}{\mathfrak{B}(z)}) there exists no entire function (G(z)) satisfying inequality (1).

The sufficiency of condition K for Theorem 1 is proved according to a scheme which is a synthesis of certain constructive schemes of M. V. Keldysh and S. N. Mergelyan, developed by them in paper ((^4)). In doing so we use the following lemmas.

Lemma 2. Let (\sigma) be a Jordan domain lying in the annulus

[
S:\ r\leq |z|\leq R,
]

and joining the boundary circles of (S). If (20r\leq R), there exists a rational function (R(z)) with a pole in (\sigma), satisfying the estimates

[
\left|\frac{1}{z}-R(z)\right|<c\,\frac{R^2}{|z|^3}
\quad \text{for } |z|\geq R,
]

[
|R(z)|<\frac{c}{r}
\quad \text{everywhere outside } \sigma,
]

where (c) is an absolute constant.

This lemma, similar to an important lemma of S. N. Mergelyan ((^4)) on approximation of the Cauchy kernel, is easily proved with the aid of Lemma 1.

Lemma 3. Let (E) be a closed set satisfying condition K with the aid of the function (r_E(t)). If (R_\zeta(z)) is a rational function with a single pole at the point (\zeta\in CE), then for any (\varepsilon>0) there exists an entire function (G_\zeta(z)) such that

[
|R_\zeta(z)-G_\zeta(z)|<\varepsilon
]

on the set (E) and in the disk (|z|\leq r_E(|\zeta|)).

Theorem 1 can be applied to the question of asymptotic approximation by entire functions with contact at infinity. A closed set (E) is said to be a Carleman set if every function continuous on (E) (possibly excluding the point (z=\infty)) admits an approximation of the form

[
|f(z)-G(z)|<\varepsilon(|z|),\quad z\in E,
\tag{2}
]

where (\varepsilon(t)) is any function with positive lower bound on every finite interval.

In the case where (E) is a continuum, the problem of describing Carleman sets was completely solved in paper ((^2)) by M. V. Keldysh and M. A. Lavrent'ev. The following theorem extends their result to arbitrary closed sets.

Theorem 2. In order that a closed set (E) be a Carleman set, it is necessary and sufficient that (E) be nowhere dense and satisfy condition K.

The necessity follows from Theorem 1. To prove sufficiency, take an arbitrary function (\varepsilon(t)) with positive lower bound

on any finite interval and denote

[
f_1(z)=\ln\left{\int_{|z|}^{|z|+1}\left[\inf_{t\le r}\varepsilon(t)\right]\,dr\right}.
\tag{3}
]

The function (f_1(z)) is continuous at every point (z); therefore, by Theorem 1, there exists an entire function (G_1(z)) such that

[
|f_1(z)-G_1(z)|<1,\qquad z\in E.
\tag{4}
]

The entire function (G_2(z)=e^{G_1(z)-1}) has no zeros in the whole plane and, according to (3) and (4), satisfies the inequality

[
|G_2(z)|<\varepsilon(|z|),\qquad z\in E.
\tag{5}
]

Now take an arbitrary function (f(z)) continuous on (E). The function (f(z)/G_2(z)) is also continuous on (E), so that, by Theorem 1, there exists an entire function (G_3(z)) such that

[
\left|\frac{f(z)}{G_2(z)}-G_3(z)\right|<1,\qquad z\in E.
\tag{6}
]

The required entire function satisfying (2) is obtained from (5) and (6) by setting (G(z)=G_2(z)G_3(z)).

In the case of asymptotic approximation of the form (2) on sets containing interior points, the deviation (\varepsilon(t)) cannot be an arbitrarily rapidly decreasing function. M. V. Keldysh established ((^{3,4})) the possible orders of decrease of (\varepsilon(t)) depending on the metric properties of the set (E). These orders were refined in ((^5)). Theorem 1 makes it possible to formulate the results in a more complete form. The following theorem is a strengthening of Theorem 1 (it is assumed that (\varepsilon(t)) is a nonincreasing function).

Theorem 3. In order that every function (f(z)), continuous on a closed set (E) (except, possibly, for the point (z=\infty)) and analytic at the interior points of (E), admit an approximation of the form (2) with the condition

[
\int_{1}^{\infty}\frac{\ln\varepsilon(t)}{t^{3/2}}\,dt>-\infty,
]

it is necessary and sufficient that (E) be a (K)-set or coincide with the whole plane. Moreover, if

[
\int_{1}^{\infty}\frac{\ln\varepsilon(t)}{t^{3/2}}\,dt=-\infty,
]

then there exist (K)-sets (E) and functions (f(z)), analytic on (E), such that (2) does not hold for any entire function (G(z)).

The necessity of condition (K) follows from Theorem 1; the sufficiency can be proved, as in ((^5)), by applying Theorem 1.

In conclusion, the author expresses sincere gratitude to M. M. Dzhrbashyan for discussing the results of the present work.

Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR

Received
6 II 1964

REFERENCES

(^1) T. Carleman, Ark. Math., Astr. och Fis., 20, No. 4 (1927).
(^2) M. V. Keldysh, M. A. Lavrentiev, DAN, 23, No. 8 (1939).
(^3) M. V. Keldysh, DAN, 47, No. 4 (1945).
(^4) S. N. Mergelyan, UMN, 7, issue 2 (1952).
(^5) N. U. Arakelyan, Matem. sborn., 53, 95, No. 4 (1961).