MATHEMATICS

M. A. EVGRAFOV and A. D. SOLOV’EV

ON A GENERAL CRITERION FOR A BASIS

(Presented by Academician A. N. Kolmogorov on 12 X 1956)

We shall say that a system of functions

\[ u_n(z)=z^n\varphi_n(z),\qquad \varphi_n(0)=1,\qquad n=0,1,2,\ldots, \tag{1} \]

regular in a domain \(G\), forms a basis in a domain \(G_1\subset G\), if every function \(f(z)\), regular in \(G_1\), is represented in this domain by a uniformly convergent series

\[ f(z)=\sum_{n=0}^{\infty} a_n u_n(z), \]

and this representation is unique.

Theorem 1. Let a system (1) be given, where the functions \(\varphi_n(z)\) are regular in the disk \(|z|<R\) and do not vanish inside this disk.

Write the system (1) in the form

\[ u_n(z)=z^n e^{-\lambda_n(z)}, \]

where the functions \(\lambda_n(z)\) are regular in the disk \(|z|<R\).

Introduce the notation

\[ \lambda_n(z)-\lambda_{n-1}(z)=\Delta_n(z)=\sum_{k=1}^{\infty}\Delta_{nk}z^k,\qquad \Delta_0(z)=\lambda_0(z); \]

\[ \Delta_n^0(r)=\sum_{k=1}^{\infty}|\Delta_{nk}|r^k,\qquad l_n(r)=\sum_{k=0}^{n}\Delta_k^0(r). \tag{2} \]

If the functions \(\lambda_n(z)\) satisfy the condition

\[ \lim_{n\to\infty}\frac{l_n(r)}{n}=0 \quad\text{for every } r<R, \tag{3} \]

then the system (1) forms a basis in the disk \(|z|<R\).

Proof. Let \(f(z)=\sum_{n=0}^{\infty}c_n z^n\) be an arbitrary function regular in the disk \(|z|<R\). Write the expansion

\[ f(z)=\sum_{n=0}^{\infty} a_n z^n e^{-\lambda_n(z)}. \]

From this formal identity, by comparing coefficients of like powers of \(z\), one can obtain finite recurrence relations from which the numbers \(a_n\) are determined successively (and moreover uniquely).

By virtue of condition (3),

\[ \lim_{n\to\infty}\sqrt[n]{\,|z^n\varphi_n(z)|\,}=|z|,\qquad |z|<R, \]

therefore, in order to prove the theorem it is enough to show that

\[ \lim_{n\to\infty}\sqrt[n]{|a_n|}\leq \frac1R . \]

Introduce the numbers \(a_{nk}\) as the coefficients of the series

\[ f(z)=\sum_{n=0}^{k-1} a_{nk}z^n e^{-\lambda_n(z)} +\sum_{n=k}^{\infty} a_{nk}z^n e^{-\lambda_k(z)} . \]

It is not difficult to verify that the property of the system (1) of being a basis in the disk \(|z|<R\) will not be violated if we change a finite number of functions \(\varphi_n(z)\) (while, of course, preserving the conditions: \(\varphi_n(z)\) are regular for \(|z|<R\) and \(\varphi_n(0)=1\)). Therefore our series converges in the same disk in which the series

\[ \sum_{n=0}^{\infty} \widetilde a_{nk}z^n=f(z)e^{\lambda_k(z)}, \]

does, i.e. for \(|z|<R\).

We note that \(a_{nk}=a_n\) for \(n<k+1\), since the coefficients \(a_n\) depend only on the first \(n\) functions \(\lambda_0(z),\lambda_1(z),\ldots,\lambda_{n-1}(z)\).

Write two identities:

\[ f(z)=\sum_{n=0}^{k-1} a_n z^n e^{-\lambda_n(z)} +\left(\sum_{n=k}^{\infty} a_{nk}z^n\right)e^{-\lambda_k(z)}, \]

\[ f(z)=\sum_{n=0}^{k-1} a_n z^n e^{-\lambda_n(z)} +\left(\sum_{n=k}^{\infty} a_{n,k-1}z^n\right)e^{-\lambda_{k-1}(z)} . \]

Comparing them, we obtain

\[ \sum_{n=k}^{\infty} a_{nk}z^n = \left(\sum_{n=k}^{\infty} a_{n,k-1}z^n\right)e^{\Delta_k(z)} . \tag{4} \]

We shall prove that every \(a_{nk}=a_{nk}(\Delta_{ij};c_m)\) is a polynomial in the \(\Delta_{ij}\) and \(c_m\) with positive coefficients. We carry out the proof by induction on the index \(k\).

\[ \sum_{n=0}^{\infty} a_{n0}z^n=f(z)e^{\Delta_0(z)}, \]

so that for the numbers \(a_{n0}\) this assertion is true. Suppose it is true for the numbers \(a_{n,k-1}\). Then from identity (4) it follows that it is true also for the numbers \(a_{nk}\).

Now assume that all coefficients \(c_m\) and \(\Delta_{ij}\) are nonnegative. Then all \(a_{nk}\geq 0\).

Denote

\[ f_k(z)=\sum_{n=k}^{\infty} a_{nk}z^n . \]

From (4) it follows that

\[ f_k(z)\ll f_{k-1}(z)e^{\Delta_k(z)}, \tag{5} \]

where the symbol \(\ll\) means that the series on the left is majorized by the series on the right, i.e. the coefficients of the left-hand series do not exceed the corresponding coefficients of the right-hand series.

From (5) it follows immediately that

\[ f_k(z)\ll e^{\lambda_k(z)}f(z), \]

whence we obtain

\[ a_{nk}\ll A_{nk}=\frac{1}{2\pi i}\int_{|z|=r<R} \frac{f(z)e^{\lambda_k(z)}}{z^{n+1}}\,dz \ll \frac{e^{\lambda_k(r)}}{r^n}f(r). \]

If now \(c_m\) and \(\Delta_{ij}\) are arbitrary complex numbers, then

\[ |a_{nk}(\Delta_{ij},c_m)|\ll a_{nk}(|\Delta_{ij}|;|c_m|) \ll \frac{e^{l_k(r)}}{r^n}M_0(r), \]

where

\[ M_0(r)=\sum_{m=0}^{\infty}|c_m|r^m. \]

In particular,

\[ |a_n|\ll M_0(r)\frac{e^{l_n(r)}}{r^n},\qquad r<R. \tag{6} \]

From this inequality it follows that

\[ \varlimsup_{n\to\infty}\sqrt[n]{|a_n|}\ll \frac1r, \]

whence, by virtue of the arbitrariness of \(r<R\),

\[ \varlimsup_{n\to\infty}\sqrt[n]{|a_n|}\ll \frac1R. \]

The theorem is proved.

Remark 1. Theorem 1 will also be valid in the case when, in each disk \(|z|\le r<R\), only a finite number of the functions \(\varphi_n(z)\) have zeros, since we can always change the functions having zeros for \(|z|\le r\) (putting, for example, \(\varphi_n(z)=1\)) without violating the property of system (1) of being a basis in the disk \(|z|<R\).

Remark 2. It can be shown that if \(|z|<R\) is the largest disk in which the conditions of Theorem 1 are fulfilled (weakened by Remark 1), then for the given functions \(\Delta_n^0(r)\) there exists a system (1) which in the disk \(|z|<R+\varepsilon\), for any \(\varepsilon>0\), no longer forms a basis.

Remark 3. Inequality (6) does not depend on condition (3) and has a universal character.

From Theorem 1 the following follows without difficulty:

Theorem 2. Suppose a system is given

\[ u_n(z)=u^n(z)\psi_n(z),\qquad \psi_n(0)=1,\qquad n=0,1,\ldots, \tag{7} \]

where \(u(z)=z+\ldots\) is a function regular and univalent in a simply connected domain \(G\), which is mapped by the function \(u(\zeta)\) onto a disk with center at the origin. The functions \(\psi_n(z)\) are regular in the domain \(G\), and on each closed set \(E\subset G\) only a finite number of these functions have zeros. Denote

\[ l_n(E)=\sum_{k=k_0}^{n}\max_{z\in E}\left|\ln\left[\frac{\psi_{k-1}(z)}{\psi_k(z)}\right]\right| \]

\[ (E\subset G\text{ is a closed set},\quad k_0=k_0(E)). \]

If

\[ \lim_{n\to\infty}\frac{l_n(E)}{n}=0 \quad \text{for every } E\subset G, \]

then the system (7) forms a basis in the domain \(G\).

Theorem 3. If in Theorem 1 condition (3) is replaced by the condition

\[ \overline{\lim_{n\to\infty}}\frac{l_n(r)}{n}=l(r), \]

then the system (1) will form a basis in the disk

\[ |z|\,e^{l_1(|z|)}<\rho, \]

where

\[ \rho=\sup_{r<R} r e^{-l(r)}, \qquad l_1(r)=\overline{\lim_{n\to\infty}}\frac{\max_{|z|=r}|\lambda_n(z)|}{n}\ll l(r). \]

Let us consider several examples of applications of these theorems.

1.

\[ u_n(z)=z^n\varphi(z)e^{\lambda_n z}. \tag{8} \]

Suppose that

\[ \sum_{k=0}^{n}|\lambda_k-\lambda_{k-1}|=o(n). \]

In this case the system forms a basis in the disk \(|z|<|\alpha_1|\), where \(\alpha_1\) is the zero of the function \(\varphi(z)\) nearest to the origin.

2.

\[ u_n(z)=u^n(z)e^{\lambda_n z}, \qquad u(z)=z+az^2+\cdots . \tag{9} \]

Again require that

\[ \sum_{k=0}^{n}|\lambda_k-\lambda_{k-1}|=o(n). \]

Then the system (9) forms a basis in the domain \(G\) in which the function \(\mu(z)\) is univalent and which is mapped by the function \(u(z)\) onto a disk with center at the origin.

3.

\[ u_n(z)=z^n\varphi^{\lambda_n}(z), \qquad \varphi(0)=1. \tag{10} \]

Let, again,

\[ \sum_{k=0}^{n}|\lambda_k-\lambda_{k-1}|=o(n). \]

Then the system will form a basis in the disk \(|z|<|\alpha_1|\) (\(\alpha_1\) is the zero of the function \(\varphi(z)\) nearest to the origin).

Received
12 X 1956

REFERENCES

1. A. O. Gel’fond, Calculus of Finite Differences, 1952.
2. M. A. Evgrafov, Izv. AN SSSR, Ser. Mat., 18, 449 (1954).
3. A. I. Markushevich, Matem. sborn., 17 (59), no. 2, 211 (1945).