UDC 517.537

MATHEMATICS

Academician of the Academy of Sciences of the Azerbaijan SSR I. I. IBRAGIMOV

ON THE COMPLETENESS OF A SYSTEM OF ANALYTIC FUNCTIONS

Let a system of functions \(\{\varphi_n(z)\}\), regular in a domain \(D\), be complete in this domain\(^*\). We note that from the definition of a complete system the following criterion for the completeness of a system of regular functions follows.

In order that a system of functions \(\{\psi_n(z)\}\) be complete in a domain \(D\), it is necessary and sufficient that for each function \(\varphi_m(z)\) there exist a sequence of linear combinations of the functions \(\psi_n(z)\) converging uniformly to the function \(\varphi_m(z)\) \((m=1,2,\ldots)\) in the domain \(D\). In particular, if \(D\) is a simply connected domain not containing the point at infinity, then as the system \(\{\varphi_n(z)\}\) in this criterion one may take the system \(\{z^n\}\) \((n=0,1,\ldots)\) (see \((^3)\) or \((^6)\)).

We record one consequence of this criterion.

If a system of functions \(\{\varphi_n(z)\}\), regular in a simply connected domain \(D\), is complete in this domain, then it is also complete in any simply connected subdomain of the domain \(D\); in particular, it is complete in each component of the intersection of the domain \(D\) with any circle.

We shall need one more definition. Let \(A(D)\) be the set of all functions regular in the domain \(D\). A system of functions \(\{\varphi_n(z)\}\), regular in a domain \(D_* \subset D\), is called complete in the class \(A(D)\) on the domain \(D_*\), if for any function \(f(z) \in A(D)\) there exists a sequence of linear combinations of the functions \(\{\varphi_n(z)\}\) converging uniformly to the function \(f(z)\) in the domain \(D_*\). From what was said above it follows that a system of functions \(\{\varphi_n(z)\}\), complete in the class \(A(D)\) on the domain \(D_* \subset D\), is complete in the domain \(D_*\), if the domains \(D\) and \(D_*\) are simply connected.

We shall prove two theorems on the completeness of a system of analytic functions of the form \(\{F(z,a_n)\}\), where \(\{a_n\}\) is a sequence of complex numbers.

Theorem 1. Let a system of functions \(\{\varphi_n(z)\}\), analytic in the finite disk \(|z|\le R\), be complete in this disk, and let the series

\[ F(z,u)=\sum_{n=0}^{\infty}\varphi_n(z)u^n \tag{1} \]

converge uniformly in \(z\) in the disk \(|z|<R\) and in \(u\) for \(|u|\le 1\). Then the system of functions \(\{F(z,a_n)\}\) is complete in the disk \(|z|<R\) for any set of points \(\{a_k\}\), where \(|a_k|\le 1\) \((k=0,1,\ldots)\), provided only that the series \(\sum_{k=0}^{\infty}(1-|a_k|)\) diverges.

Proof. Representing the function \(\partial^m F(z,u)/\partial u^m\) by the Cauchy integral with respect to \(u\) for fixed \(z\) \((|z|\le R)\) and then putting \(u=0\), we find:

\[ \left[\frac{\partial^m F(z,u)}{\partial u^m}\right]_{u=0} = m!\varphi_m(z) = \frac{m!}{2\pi i}\int_{|\xi|=1}\frac{F(z,\xi)}{\xi^{m+1}}\,d\xi . \tag{2} \]

\(^*\) Whatever regular function \(f(z)\) in \(D\) may be, there exists a sequence of linear combinations

\[ \sum_{n=1}^{p_k} c_{n,k}\varphi_n(z) \]

with constant coefficients, which converges uniformly in the domain \(D\) to the function \(f(z)\) (see, for example, \((^7)\), p. 274).

It is known that for \(|\xi|=1\) there exists a sequence of numbers \(\{A_k\}\) such that

\[ \frac{1}{\xi^{m+1}}=\sum_{k=0}^{n}\frac{A_k}{\xi-\alpha_k}+\varepsilon_n(\xi), \]

if the series \(\sum(1-|\alpha_k|)\) diverges, where \(|\varepsilon_n(\xi)|\to 0\) as \(n\to\infty\) \((|\xi|=1)\) (see \((^3)\), p. 48). Thanks to this, from equality (2) we find:

\[ \varphi_m(z)=\sum_{k=0}^{n} A_k F(z,\alpha_k)+\delta_n(z), \]

where the series

\[ \sum_{k=0}^{\infty}(1-|\alpha_k|) \]

diverges and \(|\delta_n(z)|\to 0\) as \(n\to\infty\) \((|z|\le R)\). Hence, by the completeness criterion for systems of analytic functions, it follows that the system \(\{F(z,\alpha_k)\}\) is complete in the disk \(|z|<R\) if and only if the series \(\sum(1-|\alpha_k|)\) diverges.

Corollary 1. Let

\[ f(z)=\sum_{n=0}^{\infty} c_n z^n,\qquad c_n\ne 0\quad (n\ne 0,1,\ldots), \]

be an analytic function in the finite disk \(|z|\le R\). Then the system \(\{f(\alpha_k z)\}\) is complete in the same disk \(|z|<R\) for any set of points \(\{\alpha_k\}\), where \(|\alpha_k|\le 1\) \((k=0,1,\ldots)\), if the series \(\sum(1-|\alpha_n|)\) diverges.*

Indeed, assuming that

\[ f(z)=\sum_{n=0}^{\infty} c_n z^n,\qquad c_n\ne 0\quad (n=0,1,\ldots), \]

is an analytic function in the disk \(|z|\le R\) and choosing \(\varphi_n(z)=c_n z^n\), we find

\[ F(z,u)=\sum_{n=0}^{\infty} c_n(zu)^n=f(zu)\qquad (|u|\le 1). \]

Theorem 2. Let the system of functions \(\{\varphi_n(z)\}\), analytic in a simply connected domain \(D\), be complete in this domain; let the function \(F(z,u)\) be defined by the series (1), converging uniformly in \(z\) in the domain \(D\) for any \(u\); let \(n(r)\) be the density function of the sequence \(\{\alpha_n\}\); and, moreover,

\[ M(F;D_*;r)=\max_{z\in D_*,\, |u|\le r}|F(z,u)|\qquad (D_*\subset D). \]

Then the system of functions \(\{F(z,\alpha_n)\}\) is complete in the class \(A(D)\) on the domain \(D_*\), if in the domain \(D_*\) the inequality

\[ \ln M(F;D_*;r/\theta)<c(\theta)n(r), \tag{3} \]

holds, where \(c(\theta)<\ln 1/\theta\) and \(\theta(0<\theta<1)\) is a fixed number.

Proof. In \((^3)\) it is shown that the function

\[ \Phi_n(t,u)=\prod_{k=1}^{n}\frac{t-\alpha_k}{u-\alpha_k}\, \frac{1}{|\alpha_n|^{2n}(u-t)} \prod_{k=1}^{n}\left[|\alpha_n|^2-\theta^2\overline{\alpha_k}(u-t)\right] \]

can be represented in the form

\[ \Phi_n(t,u)=\frac{1}{u-t}-\sum_{k=1}^{n}\frac{q_{n,k}(t)}{u^{-\alpha_k}}, \tag{4} \]

where

\[ q_{n,k}(t)= \prod_{\substack{j=1\\(j\ne k)}}^{n}\frac{t-\alpha_j}{\alpha_k-\alpha_j} \prod_{j=1}^{n}\left[1-\frac{\theta^2\overline{\alpha_k}(\alpha_j-t)}{|\alpha_n|^2}\right]. \]

\[ \text{* Proved by the author in }(^3). \]

Multiplying equality (4) by \(\dfrac{1}{2\pi i} F(z,u)\) and integrating along the circle \(\theta |u|=|\alpha_n|\) \((0<\theta<1)\), we obtain

\[ F(z,t)-\sum_{k=1}^{n} q_{n,k}(t)F(z,\alpha_k) = \frac{1}{2\pi i} \int_{\theta |u|=|\alpha_n|} \Phi_n(t,u)F(z,u)\,du . \]

We differentiate this equality \(m\) times with respect to \(t\), divide the result obtained by \(m!\), and then set \(t=0\). Then, putting
\(c_{k,m}=\dfrac{1}{m!}q_{n,k}^{(m)}(0)\), we obtain:

\[ \left|\varphi_m(z)-\sum_{k=1}^{n} c_{k,m}F(z,\alpha_k)\right| = |R_{n,m}(z)| \le \]

\[ \le \frac{|\alpha_n|}{\theta \rho^m} \max_{\theta |u|=|\alpha_n|,\ |\xi|=\rho} |\Phi(\xi,u)| \max_{\theta |u|=|\alpha_n|,\ z\in D} |F(z,u)|. \tag{5} \]

Using the estimate obtained in paper \((^3)\) for \(\max|\Phi(\xi,u)|\), it is not difficult to show that

\[ |R_{n,m}(z)| \le B\exp\left\{-n(r)\left[\ln\frac{1}{\theta} - \frac{\ln M(F;D_*;r/\theta)}{n(r)}\right]\right\}, \tag{6} \]

where \(B\) is a constant independent of \(r\). By the above-mentioned completeness criterion for systems of analytic functions, as inequalities (5) and (6) show, the system of functions \(\{F(z,\alpha_n)\}\) is complete in the domain \(D_*\) in the class \(A(D)\) when condition (3) is satisfied.

Corollary 1. Let the function \(\varphi_n(z)\) in the domain \(D\) satisfy the condition

\[ \lim_{n\to\infty} n^{1/\rho}|\varphi_n(z)|^{1/n} = (\sigma e\rho)^{1/\rho}|z| \qquad (z\in D) \tag{7} \]

uniformly in \(z\) in the domain \(D\), where \(\rho\) and \(\sigma\) are some positive numbers, the function \(F(z,u)\) is defined by equality (1), and the sequence of numbers \(\{\alpha_n\}\) is such that

\[ \overline{\lim_{n\to\infty}}\,\frac{n}{|\alpha_n|^\mu} = \nu<\infty . \]

Then*: 1) in the case \(\mu>\rho\), the system of functions \(\{F(z,\alpha_n)\}\) is complete in the class \(A(D)\) in every domain \(D_*\subset D\); 2) in the case \(\mu=\rho\), the system \(\{F(z,\alpha_n)\}\) is complete in the class \(A(D)\) on each component of the intersection of the domain \(D\) with the disk

\[ |z|<(\nu/\rho e\sigma)^{1/\rho}. \tag{8} \]

Proof. By condition (7), the series (1) converges uniformly for all \(z\in D\) and arbitrary \(u\), and, moreover, we have

\[ M(F;D_*;r)\le K(\varepsilon)\exp[(\sigma+\varepsilon)(|z|r)^\rho], \]

where \(z\in D_*\subset D\) and \(\varepsilon>0\).

Choosing \(r=(n/\nu)^{1/\mu}\) and \(\theta=e^{-1/\rho}\) and observing that \(n(r)=n\), from inequality (3) we obtain

\[ \sigma |z|^\rho (n/\nu)^{\rho/\mu}<n/\rho e \qquad (z\in D_*\subset D). \]

Hence it is seen that in the case \(\mu>\rho\) the system \(\{F(z,\alpha_n)\}\) is complete in the class \(A(D)\) in every domain \(D_*\subset D\), while in the case \(\mu=\rho\) the system \(\{F(z,\alpha_n)\}\) is complete in the class \(A(D)\) on each component of the intersection of the domain with the disk (8).

* The first part was proved by another method by A. A. Mirolyubov \((^4)\), while the second part is a refinement of his corresponding theorem.

Corollary 2. Let \(M(r)\) be the maximum modulus of the entire function

\[ f(z)=\sum_{k=0}^{\infty} c_k z^k \quad (c_k\ne 0,\ k=0,1,\ldots) \]

in the disk \(|z|\le r\); let \(n(r)\) be the density function of the sequence \(\{\alpha_n\}\). The system \(\{f(\alpha_n z)\}\) is complete in the disk \(|z|<R\), if the inequality

\[ \ln M(Rr/\theta)<c(\theta)n(r), \tag{9} \]

holds, where \(c(\theta)<\ln 1/\theta\) and \(\theta\) (\(0<\theta<1\)) is a fixed number.

Indeed, putting \(f(z)=\sum_{n=0}^{\infty} c_n z^n\) and \(\varphi_n(z)=c_n z^n\), from equality (1) we find

\[ F(z,u)=\sum_{n=0}^{\infty} c_n(zu)^n=f(zu). \]

In this case the domain \(D\) is the whole plane and the class \(A(D)\) consists of functions regular in a neighborhood of \(z=0\),

\[ M(F;\ |z|\le R;\ r)=\max_{|z|\le R,\ |u|<r}|f(zu)|=M(Rr), \]

and inequality (3) is written in the form (9).

This assertion was proved in the paper \((^3)\), and it was shown there that the corresponding theorems of A. O. Gelfond \((^1)\) and A. I. Markushevich \((^2)\) on the completeness of the system \(\{f(\alpha_n z)\}\) under various assumptions concerning the nature of \(f(z)\) and the sequence \(\{\alpha_n\}\) follow from it.

From Theorem 2 there also follows the following assertion (see \((^4)\), p. 282).

Corollary 3. If the entire function

\[ f(z)=\sum_{k=0}^{\infty} c_k z^k \]

satisfies the condition \(c_k\ne 0\) \((k=0,1,\ldots)\) and has type not exceeding \(\sigma\) for the refined order \(\rho(r)\), and \(\{\alpha_k\}\) is a sequence of complex numbers, then the system of functions \(\{f(\alpha_k z)\}\) is complete in the disk with center at zero and radius \(R\), determined by the equality (see \((^7)\), p. 283)

\[ R^\rho=\frac{1}{e\rho\sigma}\lim_{n\to\infty}\frac{n(|\alpha_n|)}{|\alpha_n|^{\rho(|\alpha_n|)}}. \]

Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR
Baku

Received
7 IV 1970

REFERENCES

\(^1\) A. O. Gelfond, Matem. sbornik, 4 (46), 1, 149 (1939).
\(^2\) A. I. Markushevich, Matem. sbornik, 17 (59), 2, 211 (1945).
\(^3\) I. I. Ibragimov, Izv. AN SSSR, ser. matem., 13, 45 (1949).
\(^4\) A. A. Mirolyubov, Matem. zametki, 3, No. 2, 125 (1968).
\(^5\) I. I. Ibragimov, M. V. Keldysh, Matem. sbornik, 20 (62), 2, 283 (1947).
\(^6\) I. I. Ibragimov, Izv. AN SSSR, ser. matem., No. 5–6, 553 (1939).
\(^7\) B. Ya. Levin, Distribution of zeros of an entire function, Moscow–Leningrad, 1956.