Abstract
Full Text
Yu. F. Korobeinik
On the Properties of the Limit Function of a Sequence of Linear Aggregates
(Presented by Academician I. N. Vekua on October 22, 1963)
A. F. Leont’ev posed the following problem \((^1)\). Suppose that \(f(z)\) is an entire function and \(\lambda_k\) are complex numbers such that the system \(\{f(\lambda_k z)\}\) is not complete in the disk \(|z-z_0|<R\). Suppose further that the sequence of linear aggregates
\[ \mathscr{P}_n(z)=\sum_{k=1}^{q_n} c_{k,n} f(\lambda_k z), \qquad n=1,2,\ldots, \tag{1} \]
converges uniformly in the disk \(|z-z_0|\le R\). It is required to determine what specific properties the limit function of this sequence possesses.
In the works of Pólya, Valiron, and Leont’ev \((^{2-4})\), the properties of the limit function of the sequence (1) were studied in the simplest, but perhaps the most important, special case, when \(f(z)=e^z\). In \((^5)\) it is obtained as a consequence of a more general theorem that if \(f(z)\) has the form
\[ f(z)=\sum_{n=0}^{\infty} a_n z^n =1+\sum_{n=1}^{\infty} \frac{z^n}{\psi(1)\cdots\psi(n)}, \tag{2} \]
\(\psi(x)=\sum_{l=1}^{p} a_l x^l,\ p>1,\ \psi(k)\ne 0,\ k=1,2,\ldots,\) the numbers \(\lambda_n\) satisfy the condition \(\lim_{n\to\infty}\frac{n^p}{|\lambda_n|}=0\), and the sequence (1) converges uniformly in some domain, then the limit function is single-valued everywhere and its domain of existence is simply connected.
Using the theory of generalized derivatives in the sense of A. O. Gel’fond and A. F. Leont’ev \((^6)\), one can obtain a fairly broad generalization of the last result.
Suppose that the entire function \(f(z)\) has the form (2), where \(\psi(z)\) is an entire function of growth not exceeding first order of minimal type such that
\[ \psi(0)=0,\qquad \psi(n)\ne 0,\qquad n=1,2,\ldots, \tag{3} \]
\[ \lim_{n\to\infty} |\psi(1)\cdots\psi(n)|^{1/n}=\infty. \tag{4} \]
Condition (4) will be satisfied, for example, if \(|\psi(x)|\to\infty\) as \(x\) increases without bound along the positive real axis.
Let \(\{\lambda_n\}\) be a sequence of pairwise distinct and nonzero complex numbers such that \(\sum_{n=1}^{\infty}\frac{\alpha_n}{|\lambda_n|}<\infty\), where \(\alpha_n\) are natural numbers, \(\alpha_n\ge 1,\ n=1,2,\ldots\). Denote by \(\omega(z)\) the function \(\sum_{n=1}^{\infty}\psi(n)z^{n-1}\); po
by the Wigert–Löw theorem, \(\omega(z)\) is an entire function of \(\dfrac{1}{1-z}\). Further, let
\[ \nu(R)=\max_{|1-z|=1/R}|\omega(z)|, \]
and let \(\mu(R)\) be the function inverse to \(\nu(R)\).
Introduce the functions
\[ \mathcal L(x)=x^{p_1}\prod_{k=1}^{\infty}\left(1+\frac{x}{\lambda_k}\right)^{\alpha_k} =\sum_{k=p_1}^{\infty} c_k x^k, \]
where \(p_1\) is a natural number, \(p_1\ge 0\), and
\[ \widetilde{\mathcal L}(x)=x^{p_1}\prod_{k=1}^{\infty}\left(1+\frac{x}{|\lambda_k|}\right)^{\alpha_k} =x^{p_1}\prod_{k=1}^{\infty}\left(1+\frac{x}{\mu_k}\right) =\sum_{k=p_1}^{\infty}\sigma_k x^k . \]
Suppose that the numbers \(\lambda_n\) satisfy the condition
\[ \lim_{n\to\infty}\frac{n}{\mu\left(n/\sqrt[n]{\sigma_n}\right)}=\tau<\infty . \tag{5} \]
Consider the sequence
\[ \mathfrak P_n(z)=\sum_{k=1}^{l_n}\sum_{j=0}^{\alpha_k-1} z^j f^{(j)}(z\lambda_k)c_{k,j}^{(n)},\qquad n=1,2,\ldots, \tag{6} \]
where \(l_n\) are natural numbers.
Denote by \(d(\tau,z)\) the function defined as follows:
1) for a transcendental entire function \(\psi(z)\) of class \([1,0]\),
\[ d(\tau,z)= \begin{cases} (e^\tau-1)|z|, & \text{if } \tau<\ln 2,\\ e^\tau |z|, & \text{if } \tau\ge \ln 2; \end{cases} \]
2) for
\[ \psi(z)=\sum_{l=1}^{p}\alpha_l z^l,\qquad p>1, \]
\[ d(\tau,z)= \begin{cases} (\Delta)^p, & \text{if }(\Delta)^p>|z|,\\ \Delta |z|^{1-1/p}, & \text{if }(\Delta)^p\le |z|, \end{cases} \]
where
\[ (\Delta)^p=\frac{\tau}{p!}\left(\frac{2p^p\tau}{(p-1)^{p-1}e}\right)^p; \]
3) for \(\psi(z)\equiv z\),
\[ d(\tau,z)=\frac{\tau^2}{e}. \]
Theorem 1. Let condition (5) be satisfied, and let the sequence (6) converge uniformly in some disk
\[ |z-z_0|\le d(\tau,z_0)+h. \]
Then the limits
\[ \lim_{n\to\infty} c_{k,j}^{(n)}=c_{k,j},\qquad 0\le j\le \alpha_k-1,\quad k=1,2,\ldots \]
exist.
Theorem 2. The sequence (6) and the sequence
\[ \widetilde{\mathfrak P}_n(z)= \sum_{k=1}^{\widetilde l_n}\sum_{j=0}^{\alpha_k-1} z^j f^{(j)}(z\lambda_k)\widetilde c_{k,j}^{(n)},\qquad n=1,2,\ldots, \]
satisfying the conditions of Theorem 1, converge to one and the same limiting function if and only if the limits of the corresponding coefficients are equal to one another:
\[ c_{m,j}=\widetilde c_{m,j},\qquad j=0,1,\ldots,\alpha_m-1;\quad m=1,2,\ldots . \]
Theorem 3. Let the sequence (6) converge uniformly in the disk
\[ |z-z_0|\le d(\tau,z_0)+h,\qquad h>0, \]
to a function \(P(z)\), and let \(D_\tau^{z_0}\) be the connected set on the Riemann surface of the function \(P(z)\) containing \(z_0\) and all points \(z_1\) such that \(P(z)\) is analytic in the disk
\[ |z-z_1|\le d(\tau,z_1). \]
Then uniformly inside \(D_\tau^{z_0}\),
\[ P(z)=\lim_{m\to\infty}\sum_{k=1}^{m}\sum_{j=0}^{\alpha_k-1} f^{(j)}(z\lambda_k)z^j \sum_{s=j}^{\alpha_k-1} c_{k,s}c_s^j\mathcal L_{m+1,\infty}^{(s-j)}(\lambda_k), \]
where
\[ c_{k,s}=\lim_{n\to\infty} c_{k,s}^{(n)}, \qquad \mathscr L_{m+1,\infty}(x)=\prod_{k=m+1}^{\infty}\left(1-\frac{x}{\lambda_k}\right)^{\alpha_k}. \]
In this case the domain \(D_\tau^{z_0}\) is simply connected and one-connected.
Corollary. If the numbers \(\lambda_n\) are such that
\[ \lim_{n\to\infty}\frac{n}{\mu\left(n!\sqrt[n]{\sigma_n}\right)}=0 \]
and the sequence (6) converges uniformly in some circle, then the domain of existence of the limiting function is simply connected and one-connected.
One can also obtain characteristics of certain boundary properties of the domain \(D_\tau^{z_0}\). Suppose that for all \(z_1\) and \(z_2\),
\[ |d(\tau,z_1)-d(\tau,z_2)|<|z_1-z_2|. \]
Next, let \(\alpha\) be any boundary point of \(D_\tau^{z_0}\), and let \(\beta_\alpha\) be a singular point of \(P(z)\) lying on the circle \(|\alpha-t|=d(\tau,\alpha)\); let \(\rho(\beta_\alpha)\) be the distance from \(\beta_\alpha\) to the next nearest singular point of \(P(z)\). The quantity \(\rho(\beta_\alpha)\) can be estimated from above.
Theorem 4. Under the hypotheses of Theorem 3, the inequality
\[ \rho(\beta_\alpha)\le 2d(\tau,|\beta_\alpha|+\rho(\beta_\alpha)/2) \]
holds. In addition, if the function \(d(\tau,z)\) is such that
\[ |d(\tau,z_1)-d(\tau,z_2)|\le K|z_1-z_2|,\qquad K<1, \]
where \(z_1,z_2\) are arbitrary complex numbers, then
\[ \rho(\beta_\alpha)\le \frac{2}{1-K}\,d(\tau,\beta_\alpha). \]
Under certain conditions the limiting function of the sequence (6) can be represented in the form of a series. For simplicity of formulation we restrict ourselves to the case when \(p_1=0\), \(\alpha_k=1\), \(k=1,2,\ldots\), and denote by \(\alpha\)-quantities the limits
\[ \lim_{n\to\infty} c_{s,n}, \]
which exist under the assumptions of Theorem 1.
Theorem 5. Suppose that \(f(z)\) is an entire function of the form (2), where \(\psi(z)\) is an entire function of class \([1,0]\) satisfying condition (3).
Suppose, furthermore, that the numbers \(\lambda_n\) are such that condition (5) is satisfied and one of the two conditions
\[ \text{a)}\quad \varlimsup_{n\to\infty} \frac{\left|\ln \dfrac{1}{|\mathscr L'(\lambda_n)|}\right|}{\ln|\lambda_n|}<\infty; \qquad \text{b)}\quad \varlimsup_{n\to\infty} \frac{\nu\!\left(\left|\ln \dfrac{1}{|\mathscr L'(\lambda_n)|}\right|\right)} {|\lambda_n|\left|\ln \dfrac{1}{|\mathscr L'(\lambda_n)|}\right|}<\infty. \]
Finally, suppose that the sequence (1) converges uniformly in the circle
\[ |z-z_0|\le d(\tau,z_0)+h,\qquad h>0. \]
Then in the domain \(D_\tau^{z_0}\)
\[ \mathscr P(z)=\sum_{k=1}^{\infty}\alpha_k f(\lambda_k z), \tag{7} \]
and the series (7) converges uniformly inside \(D_\tau^{z_0}\). If the domain \(D_\tau^{z_0}\) contains at least one circle
\[ |z-z_1|\le d(\tau,z_1), \]
then the expansion of \(\mathscr P(z)\) into a series of the form (7) is unique.
Under various concrete assumptions concerning the growth of the function \(\psi(z)\) of class \([1,0]\), one can obtain a number of corollaries from Theorems 1–5.*
- In the particular case when
\[ \psi(x)=\sum_{l=1}^{p}\alpha_l x^l,\qquad p>1,\qquad \psi(n)\ne0,\quad n=1,2,\ldots, \]
Theorems 1–4 were obtained earlier by A. F. Leont’ev.
- \(\psi(z)\) is a transcendental entire function of growth not exceeding order \(\rho<1\) and of finite type \(\sigma\). If the numbers \(\lambda_n\) are such that
\[ \lim_{n\to\infty}\frac{|s_n|^{\rho/(1-\rho)}}{\ln|\lambda_n|}=c<\infty, \]
where
\[ s_n=\sum_{k=1}^{n}\alpha_k, \]
then condition (5) is satisfied and
\[ \tau=(c)^{(1-\rho)/\rho}\rho(\sigma)^{1/\rho}. \]
In this
* After the article had been submitted for publication, I learned that the sequence (1) in the case (2), when \(\psi\in[1,0]\), had been considered by I. F. Lokhin, but his results were not published.
in that case Theorems 1–5 are valid; if, in particular, \(\tau<\ln 2\), then for any boundary point \(\alpha\) of the domain \(D_{0}^{z}\)
\[ \rho(\beta_\alpha)\leq \frac{2(e^\tau-1)}{(2-e^\tau)}|\beta_\alpha|. \]
- \(\psi(z)\) is a transcendental entire function of order not exceeding \(\rho\), \(\rho<1\). If
\[ \varlimsup_{n\to\infty}\frac{\ln s_n}{\ln\ln|\lambda_n|}<\frac{1-\rho}{\rho}, \]
then condition (5) is satisfied, and moreover \(\tau=0\). From Theorems 1–4 in this case it follows, in particular, that if the sequence (6) converges uniformly in some circle \(|z-z_0|\leq h\), then the Riemann surface of the limiting function is simply connected and single-sheeted. If \(\alpha_n=1\), \(n=1,2,\ldots\),
\[ \varlimsup_{n\to\infty}\frac{\ln n}{\ln\ln|\lambda_n|}<\frac{1-\rho}{\rho}, \]
and if, in addition, one of the following two conditions is satisfied:
a)
\[ \varlimsup_{n\to\infty} \frac{\ln\left|\dfrac{1}{\mathcal L'(\lambda_n)}\right|}{\ln|\lambda_n|}<\infty, \qquad \text{if } \rho\geq \frac12, \]
or
b)
\[ \varlimsup_{n\to\infty} \frac{\ln\left|\ln \dfrac{1}{|\mathcal L'(\lambda_n)|}\right|}{\ln\ln|\lambda_n|} < \frac{1-\rho}{\rho}, \qquad \text{if } \rho<\frac12, \]
then the limiting function can be represented in the form of the series (7), converging uniformly inside its entire domain of existence.
- If \(\psi(z)\) is a transcendental entire function of zero order such that
\[ \lim_{r\to\infty}\frac{\ln M_r(\psi)}{(\ln r)^\alpha}=A<\infty,\quad \alpha<1,\quad M_r(\psi)=\max_{|x|=r}|\psi(x)|, \]
and the numbers \(\lambda_n\) are such that
\[ \varlimsup_{n\to\infty}\frac{(\ln s_n)^\alpha}{\ln|\lambda_n|}<\frac{1}{A}, \]
then from the uniform convergence of the sequence (6) in some circle there follows single-sheetedness and simple connectedness of the domain of existence of the limiting function. If \(\alpha_n=1\), \(n=1,2,\ldots\),
\[ \varlimsup_{n\to\infty}\frac{(\ln n)^\alpha}{\ln|\lambda_n|}<\frac{1}{A} \]
and if
\[ \varlimsup_{n\to\infty} \frac{\ln\left|\ln \dfrac{1}{|\mathcal L'(\lambda_n)|}\right|}{\ln|\lambda_n|} <\infty, \]
then the limiting function can be represented by the series (7), converging uniformly inside its domain of existence.
Rostov-on-Don
State University
Received
4 X 1963
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