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|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