SOME QUESTIONS OF APPROXIMATION ON SETS OF THE COMPLEX PLANE

E. Sh. Chatskaya

(Presented by Academician I. N. Vekua on 18 I 1965)

Let (E) be a compact set of the complex plane (Z); let (\mu(e)) be a finite positive measure defined on the Borel subsets (e) of the compact set (E); (L_p(\mu)), (p \geqslant 1), is the space of functions (\omega(z)) satisfying the condition

[
|\omega|=\left{\int_E |\omega(z)|^p\,d\mu\right}^{1/p}<\infty .
]

We shall call the measure (\mu) reduced with respect to the set (E) when the measure of every nonempty portion of (E) is positive.

Recall that a compact set (E) has zero analytic capacity (\Omega(E)=0) if in the complement (Z\setminus E) there are no bounded analytic functions different from constants.

In the works of S. Ya. Khavinson ((^{1})) and V. P. Khavin ((^{2})) the following was shown.

Every continuous function (\Phi(z)) on a set (E) of zero analytic capacity can be approximated by such a sequence of rational functions

[
\sum^{(k)} \frac{\lambda_j^{(k)}}{z-a_j^{(k)}}
]

with poles outside (E), that

[
\max_{z\in E}\left|\Phi(z)-\sum^{(k)}\frac{\lambda_j^{(k)}}{z-a_j^{(k)}}\right|\xrightarrow[k\to\infty]{}0
\tag{1}
]

and, at the same time,

[
\sum^{(k)} |\lambda_j^{(k)}|\xrightarrow[k\to\infty]{}0 .
\tag{2}
]

Conversely, if on a compact set (E), about which it is additionally assumed that its length (in the sense of Painlevé, see ((^{1}))) is finite, the constant one can be approximated by rational fractions in such a way that conditions (1) and (2) are fulfilled, then (\Omega(E)=0).

It follows from this that on a set (E), (\Omega(E)=0), for every measure (\mu) and every (\omega(z)\in L_p(\mu)), (p\geqslant 1), there exists a sequence of fractions

[
\sum^{(k)} \frac{\lambda_j^{(k)}}{z-a_j^{(k)}}
]

with poles outside (E), satisfying the relations

[
\int_E \left|\omega(z)-\sum^{(k)}\frac{\lambda_j^{(k)}}{z-a_j^{(k)}}\right|^p d\mu \xrightarrow[k\to\infty]{}0,
\tag{3}
]

[
\sum^{(k)} |\lambda_i^{(k)}|\xrightarrow[k\to\infty]{}0 .
\tag{4}
]

Let us pose the converse problem. Suppose that, for some measure (\mu) (which, of course, must be regarded as reduced), conditions (3) and (4) hold for all (\omega(z)\in L_p(\mu)), (p\geqslant 1). Does it follow from this that (\Omega(E)=0), i.e., are (3) and (4) not only necessary but also sufficient conditions for (\Omega(E)=0)? We give a negative answer to this question.

Theorem 1. For every nowhere dense compact set (E) there exists a reduced measure (\mu) such that, for all (p \geq 1), for any function (\omega(z)\in L_p(\mu)) one can construct a sequence of aggregates
(\sum^{(k)} \dfrac{\lambda_j^{(k)}}{z-a_j^{(k)}}), for which

[
\int_E \left|\omega(z)-\sum^{(k)} \frac{\lambda_j^{(k)}}{z-a_j^{(k)}}\right|^p d\mu \xrightarrow[k\to\infty]{}0,
\qquad
\sum^{(k)} |\lambda_j^{(k)}| \xrightarrow[k\to\infty]{}0.
]

We shall give the idea of the proof. Let (K={z_n}_1^\infty) be a countable everywhere dense subset of (E); let (\omega_n(z)) be the characteristic function of the set consisting of the single point (z_n). From the fact that
(\Omega({z_1,z_2,\ldots,z_k})=0) and the set (E) is nowhere dense, for each (\omega_n(z)), (n=1,2,\ldots), there is a fraction

[
\sum^{(k)} \frac{\lambda_j^{(n,k)}}{z-a_j^{(n,k)}}
]

with poles outside the set (E), satisfying the conditions:

[
\left|\omega_n(z)-\sum^{(k)} \frac{\lambda_j^{(n,k)}}{z-a_j^{(n,k)}}\right|
\leq \frac{1}{2^{k-1}}
\quad \text{for } z=z_1,\ldots,z_k,
]

[
\sum |\lambda_j^{(n,k)}|\leq \frac{1}{2^{k-1}}.
]

Denote

[
\max_{z\in E}\left|\omega_n(z)-\sum^{(k)} \frac{\lambda_j^{(n,k)}}{z-a_j^{(n,k)}}\right|
= M_k^{(n)},\qquad n=1,2,\ldots
]

Since (\omega_n(z_1)=\omega_n(z_2)=\cdots=\omega_n(z_k)=0), (n=k+1,k+2,\ldots), the function

[
\sum^{(k)} \frac{\lambda_j^{(n,k)}}{z-a_j^{(n,k)}}
]

may be taken to be the same for all (\omega_n(z)), beginning with (n=k+1). Therefore there exists a finite

[
\max_{n=1,2,\ldots}{M_k^{(n)}}
=
\max\left{
M_k^{(1)},\, M_k^{(2)},\,\ldots,\, M_k^{(k)},\,
\max_E\left|1-\sum^{(k)}\frac{\lambda_j^{(k+1,k)}}{z-a_j^{(k+1,k)}}\right|,
\right.
]

[
\left.
\max_E\left|\sum^{(k)}\frac{\lambda_j^{(k+1,k)}}{z-a_j^{(k+1,k)}}\right|
\right}
= M_k.
]

Put

[
\mu(z_1)=1,\qquad
\mu(z_{k+1})=\min\left[\frac{\mu(z_k)}{2},\,
\frac{1}{2^k(M_k)^k}\right],
\qquad k=1,2,\ldots
]

In this way we obtain a nonzero measure on the countable subset (K\subset E); next we extend it to the whole plane (Z) in the usual manner:
(\mu(e)=\mu(e\cap K)).

It can be verified that the measure (\mu) satisfies the conditions of our theorem. At the same time, if (E) has finite length, then the fulfillment of conditions (3) and (4) for all measures (\mu) is sufficient in order that (\Omega(E)=0). More precisely, the following theorem holds:

Theorem 2. Let (E) have finite length. If, whatever the measure (\mu) may be, there is a sequence

[
\sum^{(k)} \frac{\lambda_j^{(k)}}{z-a_j^{(k)}}
]

(to each measure (\mu) there corresponds its own sequence) satisfying the relations

[
\int_E \left|1-\sum^{(k)} \frac{\lambda_j^{(k)}}{z-a_j^{(k)}}\right|d\mu
\xrightarrow[k\to\infty]{}0,
]

[
\sum^{(k)}|\lambda_j^{(k)}| \xrightarrow[k\to\infty]{}0,
]

then (\Omega(E)=0).

The measure (\mu) constructed in Theorem 1 had as its essential support a countable set of points ({z_n}) (i.e., (\mu(E \setminus {z_n}_1^\infty)=0)). It turns out that any measure for which conditions (3) and (4) are satisfied must be concentrated on sufficiently rarefied subsets of (E). Denote by (A={\mu}) the class of all measures on the nowhere dense compact set (E) for which (3) and (4) are satisfied.

Theorem 3. Every measure (\mu \in A) is singular with respect to the Hausdorff (h)-measure defined by the function (h(r)), if

[
\int_0 \frac{h(r)}{r^2}\,dr < \infty .
]

(For Hausdorff measures see (3).)

I consider it my pleasant duty to express my gratitude to Prof. S. Ya. Khavinson, under whose supervision this work was carried out.

Engineering-Construction Institute
named after V. V. Kuibyshev

Received
12 I 1965

REFERENCES

1. S. Ya. Khavinson, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 60, 304 (1961).
2. V. P. Khavin, Sibirsk. matem. zhurn., 2, No. 4, 622 (1961).
3. R. Nevanlinna, Single-Valued Analytic Functions, Moscow, 1941.