MATHEMATICS

A. D. SOLOV’EV

DETERMINATION OF THE CLASS OF CONVERGENCE OF AN INTERPOLATION SERIES FOR CERTAIN PROBLEMS

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

Let an entire function be given,

\[ \Phi(z)=\sum_{n=0}^{\infty}\frac{z^n}{m_n}, \]

where the numbers \(m_n\) satisfy the conditions: \(m_n\ne 0\) and \(|m_{n+1}/m_n|\) tends monotonically to infinity. By the class \([\Phi,\sigma]\) we shall mean the class of entire functions

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

for which

\[ \varlimsup_{n\to\infty}\sqrt[n]{|c_n m_n|}<\sigma<\infty. \]

It is not difficult to show that for functions of this class

\[ \varlimsup_{r\to\infty}\frac{m(r)}{m^\Phi(\sigma r)}=0, \]

where

\[ m(r)=\ln\max_{|z|=r}|F(z)|,\qquad m^\Phi(r)=\ln\max_{|z|=r}|\Phi(z)|. \]

The function

\[ f^\Phi(\zeta)=\sum_{n=0}^{\infty}\frac{c_n m_n}{\zeta^{n+1}} \]

will be called \(\Phi\)-associated with the function \(F(z)\). For entire functions of the class \([\Phi,\sigma]\), all singularities of the \(\Phi\)-associated functions lie inside the disk \(|\zeta|<\sigma\).

Consider the sequence of linear functionals

\[ A_n=A_n[F]=\frac{1}{2\pi i}\int_C \zeta^n\varphi_n^\Phi(\zeta)f^\Phi(\zeta)\,d\zeta,\qquad n=0,1,2,\ldots, \tag{1} \]

where

\[ \varphi_n^\Phi(\zeta)=\sum_{k=0}^{\infty}\frac{a_{nk}}{m_{n+k}}\zeta^k\quad (a_{n0}=1) \]

are regular for \(|z|<R\); \(F(z)\in[\Phi,R]\); the contour \(C\) encloses all singularities of \(f^\Phi(\zeta)\) and lies in the disk \(|z|<R\). Functionals (1) that are invariant with respect to the function \(\Phi(z)\) will be called moments.

Let us write the formal identity

\[ \Phi(z\zeta)=\sum_{n=0}^{\infty}p_n(z)\,\zeta^n\varphi_n^\Phi(\zeta). \]

Comparing the coefficients of equal powers of \(\zeta\), we obtain finite recurrence relations for \(p_n(z)\), from which these functions are determined successively and uniquely. It is easy to show that \(p_n(z)\) is a polynomial of degree \(n\), invariant with respect to the function \(\Phi(z)\). The functions \(p_n(z)\) are called interpolation polynomials, and the series

\[ \sum_{n=0}^{\infty} A_n p_n(z) \sim F(z) \tag{2} \]

the interpolation series of the function \(F(z)\).

We shall say that the class \([\Phi,\sigma]\) is the exact convergence class of the interpolation series if, for every entire function \(F(z)\in \mathfrak C[\Phi,\sigma]\), the series (2) converges to \(F(z)\) uniformly in every finite disk, and if for every \(\varepsilon>0\) there exists \(F_1(z)\in[\Phi,\sigma+\varepsilon]\) for which the series (2) diverges at least at one point. Denoting the exact convergence class by \(K_0\), we shall record this fact as follows: \(K_0=[\Phi,\sigma]\).

The following assertion is true \((^2)\):

Theorem 1. Let the moments (1) be given, and suppose that
\[ \lim_{n\to\infty} m_n \varphi_n^{\Phi}(\zeta)=\varphi(\zeta) \]
uniformly in every disk \(|\zeta|\le r<R_1\le R\).

If \(\varphi(\zeta)\) has zeros inside the disk \(|\zeta|<R_1\), then \(K_0=[\Phi,|\alpha_1|]\), where \(\alpha_1\) is the zero of the function \(\varphi(\zeta)\) nearest to the origin.

A generalization of this theorem is Theorem 2.

Theorem 2. Let the moments (1) be given, and suppose that
\[ \lim_{n\to\infty} m_{nk+s}\varphi_{nk+s}^{\Phi}(\zeta)=\varphi_s(\zeta), \qquad s=0,1,2,\ldots,k-1, \]
uniformly in every disk \(|\zeta|\le r<R_1\le R\).

If the function
\[ \Delta(\zeta)= \left| \begin{array}{cccc} \varphi_0(\zeta) & \varphi_1(\zeta) & \cdots & \varphi_{k-1}(\zeta)\\ \varphi_0(\varepsilon\zeta) & \varepsilon\varphi_1(\varepsilon\zeta) & \cdots & \varepsilon^{k-1}\varphi_{k-1}(\zeta)\\ \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & \cdots\\ \varphi_0(\varepsilon^{k-1}\zeta) & \cdots & \cdots & \varepsilon^{(k-1)^2}\varphi_{k-1}(\varepsilon^{k-1}\zeta) \end{array} \right|, \]
where \(\varepsilon=e^{2\pi i/k}\), has zeros in the disk \(|\zeta|<R_1\), then \(K_0=[\Phi,|\alpha_1|]\), where \(\alpha_1\) is the zero of the function \(\Delta(\zeta)\) nearest to the origin.

By virtue of the principle of duality \((^3)\), these theorems assert that the system \(\{z^n\varphi_n^{\Phi}(z)\}\) forms a basis in the disk \(|z|<|\alpha_1|\) and does not form one in any larger disk.

We shall prove a simple assertion that makes it possible to apply these theorems in a number of interesting problems.

Theorem 3. Let the moments (1) be given, and let \(\{\lambda_n\}\) be a sequence of complex numbers satisfying the conditions:
\[ \lim_{n\to\infty}\frac{\lambda_n}{\lambda_{n+1}}=q,\qquad |\lambda_n|\le |\lambda_{n+1}|,\qquad n=0,1,2,\ldots . \]

If
\[ \lim_{n\to\infty} m_n\varphi_n^{\Phi}\!\left(\frac{\zeta}{\lambda_n}\right) =\widetilde{\varphi}(\zeta)=\sum_{k=0}^{\infty} a_k \zeta^k \]
uniformly in every disk \(|\zeta|\le r<R_1\), then
\[ \lim_{n\to\infty} m_n'\varphi_n^{\Psi}(\zeta) =\varphi(\zeta)=\sum_{k=0}^{\infty} a_k q^{k(k-1)/2}\zeta^k \]
uniformly in every disk \(|\zeta|\le r<R_1\), where
\[ \Psi(z)=\sum_{n=0}^{\infty}\frac{z^n}{m_n'}, \]
\[ m_n'=m_n\lambda_1\cdots\lambda_{n-1}\quad (m_0'=m_0,\; m_1'=m_1). \]

In other words, in this case \(K_0=[\Psi,|\alpha_1|]\), where \(\alpha_1\) is the zero of the function \(\varphi(\zeta)\) closest to the origin \((|\alpha_1|<R_1)\).

Proof. We have
\[ m_n\varphi_n^\Phi\!\left(\frac{\zeta}{\lambda_n}\right) = \sum_{k=0}^{\infty}\frac{a_{nk}m_n}{m_{n+k}\lambda_n^k}\zeta^k, \]
\[ m_n'\varphi_n^\Psi(\zeta) = \sum_{k=0}^{\infty} \frac{a_{nk}m_n}{m_{n+k}\lambda_n\cdots\lambda_{n+k-1}}\zeta^k. \]

Next:
\[ \lim_{n\to\infty} \frac{a_{nk}m_n}{m_{n+k}\lambda_n\cdots\lambda_{n+k-1}} = a_kq^{k(k-1)/2} \]

and, moreover:
\[ \left| \frac{a_{nk}m_n}{m_{n+k}\lambda_n\lambda_{n+1}\cdots\lambda_{n+k-1}} \right| \leq \left| \frac{a_{nk}m_n}{m_{n+k}\lambda_n^k} \right|, \]
whence the assertion of the theorem follows.

The assertion corresponding to Theorem 2 is formulated analogously. Let us consider several examples, using the notation of the last theorem.

1. The Abel—Goncharov problem.

\[ \varphi_n^\Phi(\zeta)=\frac{1}{n!}e^{\lambda_n\zeta}, \qquad \Phi(\zeta)=e^\zeta, \qquad A_n[F]=\frac{F^{(n)}(\lambda_n)}{n!}. \]

Suppose that
\[ \lim_{n\to\infty}\frac{\lambda_n}{\lambda_{n+1}}=q\ne 1, \qquad |\lambda_n|\leq |\lambda_{n+1}|. \]

Then
\[ K_0=[\Psi,|\alpha_1|], \]
where
\[ \Psi(z)=\sum_{n=0}^{\infty} \frac{z^n}{n!\lambda_1\cdots\lambda_{n-1}}, \qquad \varphi(z)=\sum_{n=0}^{\infty} \frac{q^{n(n-1)/2}}{n!}z^n. \]

Consider special cases:

a)
\[ \lambda_n=\nu_ne^{i\alpha(-1)^n}, \qquad \nu_n\leq \nu_{n+1}, \qquad \frac{\nu_n}{\nu_{n+1}}\to 1, \]
\[ \Psi(z)=\sum_{n=0}^{\infty} \frac{z^n}{n!\nu_1\cdots\nu_{n-1}}, \qquad \Delta(z)=\cos(2z\sin\alpha). \]

Consequently,
\[ K_0=\left[\Psi,\frac{\pi}{4\sin\alpha}\right]. \]

b)
\[ \frac{\lambda_n}{\lambda_{n+1}}\to 0, \qquad \Psi(z)=\sum_{n=0}^{\infty} \frac{z^n}{n!\lambda_1\cdots\lambda_{n-1}}, \qquad \varphi(z)=1+z, \qquad |\alpha_1|=1. \]

Thus,
\[ K_0=[\Psi,1]. \]

1. A generalization of the Abel—Goncharov problem.

\[ \varphi_n^\Phi(z)=\frac{1}{n!}f(\lambda_n z), \qquad \Phi(z)=e^z, \qquad f(z)=\sum_{k=0}^{\infty}a_kz^k, \qquad a_0=1. \]

Let again

\[ \lim_{n\to\infty}\frac{\lambda_n}{\lambda_{n+1}}=q,\qquad |\lambda_n|\leqslant |\lambda_{n+1}|. \]

Then

\[ K_0=[\Psi,|\alpha_1|], \]

where

\[ \Psi(z)=\sum_{n=0}^{\infty}\frac{z^n}{n!\lambda_1\ldots\lambda_{n-1}}, \qquad \varphi(z)=\sum_{k=0}^{\infty}a_k q^{k(k-1)/2}z^k. \]

3. On A. O. Gelfond’s moment problem\(^1\).

a)

\[ \zeta^n\varphi_n^{\Phi}(\zeta)=\frac{u^n(\zeta)+v^n(\zeta)}{2n!}, \qquad \Phi(\zeta)=e^\zeta, \]

\[ u(\zeta)=\zeta+\alpha\zeta^{k+1}+\ldots, \qquad v(\zeta)=\zeta+\beta\zeta^{k+1}+\ldots, \qquad \alpha\ne\beta. \]

Then

\[ K_0=[\Psi,|\alpha_1|], \]

where

\[ \Psi(z)=\sum_{n=0}^{\infty}\frac{z^n}{(n!)^{1+1/k}}, \qquad \varphi(z)=e^{\alpha z^k}+e^{\beta z^k}, \qquad |\alpha_1|=\left(\frac{\pi}{|\alpha-\beta|}\right)^{1/k}. \]

It follows from this that the system \(\{u^n(\zeta)+v^n(\zeta)\}\), under the condition \(\alpha\ne\beta\), cannot form a basis in a neighborhood of the origin. (We note that the condition \(\alpha\ne\beta\) may be replaced by the condition \(u(\zeta)\not\equiv v(\zeta)\).)

b)

\[ \zeta^n\varphi_n^{\Phi}(\zeta)=\frac{u_{\alpha_n}^n(\zeta)}{n!}, \qquad \alpha_n= \begin{cases} 0, & n=2k,\\ 1, & n=2k+1; \end{cases} \qquad \Phi(\zeta)=e^\zeta. \]

\[ u_0(\zeta)=\zeta+\alpha\zeta^{k+1}+\ldots, \qquad u_1(\zeta)=\zeta+\beta\zeta^{k+1}+\ldots, \qquad \alpha\ne\beta. \]

Then

\[ K_0=[\Psi,|\alpha_1|], \]

\[ \Psi(z)=\sum_{n=0}^{\infty}\frac{z^n}{(n!)^{1+1/k}}, \qquad \Delta(\zeta)=\operatorname{ch}(\alpha-\beta)\zeta^k, \qquad |\alpha_1|=\left(\frac{\pi}{2|\alpha-\beta|}\right)^{1/k}. \]

Moscow State University
named after M. V. Lomonosov

Received
12 X 1956

REFERENCES

\(^1\) A. O. Gelfond, Calculus of finite differences, Moscow—Leningrad, 1952. \(^2\) M. A. Evgrafov, Izv. Akad. Nauk SSSR, Ser. Mat., 17, 421 (1953). \(^3\) A. I. Markushevich, Mat. sbornik, 17 (59), 2, 211 (1945).