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Reports of the Academy of Sciences of the USSR
- Vol. 149, No. 2
MATHEMATICS
M. I. Shabunin
ON THE DETERMINATION OF THE CLASS OF CONVERGENCE OF AN INTERPOLATION SERIES FOR THE ABEL–GONCHAROV AND GELFOND PROBLEMS
(Presented by Academician I. M. Vinogradov on 29 IX 1962)
Consider a system of linear functionals \(\{L_n(F)\}\), defined on the entire class of entire functions \(F(z)\) as follows:
\[ L_n(F)=\frac{1}{2\pi i}\int_C F(z)\,u_n\!\left(\frac{1}{z}\right)\frac{dz}{z}, \]
where
\[ u_n(z)=z^n+\sum_{k=1}^{\infty} a_{n,k} z^{n+k}. \]
We shall suppose that the functions \(u_n(z/\lambda_n)\), for \(n>n_0(z)\), have in a certain domain \(G\) no zeros different from \(z=0\), and, moreover, uniformly with respect to \(z\) inside \(G\),
\[ \frac{u_{n+1}(z/\lambda_{n+1})}{u_n(z/\lambda_n)} \to u(z), \tag{1} \]
with \(u(0)=0,\ u'(0)=1\).
On the sequence \(\lambda_n\) we impose the following restrictions:
-
\(\lambda_n=\lambda(n)\), where \(\lambda(z)\) is regular in the whole plane with the cut \((-\infty,0)\), and \(\lambda(z)=w=re^{i\theta}\) maps the half-plane \(\operatorname{Re} z>0\) onto a certain domain \(\Omega\), which is bounded by curves having, in a neighborhood of \(w=\infty\), the equations \(\theta=\pm \frac12\alpha(r)\); \(\lambda(z)>0\) for \(z>0\).
-
\(\alpha(r),\ \alpha'(r),\ \alpha''(r)\) are monotone for \(r>r_1\).
-
There exists
\[ \lim_{r\to\infty}\frac{\pi}{\alpha(r)}=\rho,\qquad \frac12<\rho<\infty. \]
- \(\alpha(r)\) is a slowly increasing function, i.e.
\[ \lim_{r\to\infty} r\,\frac{\alpha'(r)}{\alpha(r)}=0. \]
Consider the entire function \(\Phi(z)\), defined by the equality
\[ \Phi(z)=\frac{1}{2\pi i}\int_L \frac{e^{h(t)}}{t-z}\,dt, \tag{2} \]
where \(h(z)\) is that branch of the function inverse to \(\lambda(z)\) which takes positive values for \(z>0\). The contour \(L\) begins at infinity below the real axis, coinciding in a neighborhood of \(t=\infty\) with the curve \(\theta=-\frac12\alpha(r)\), and ends at infinity above the real axis,
coinciding in a neighborhood of \(t=\infty\) with the curve \(\theta=-\frac12\alpha(r)\). In the finite part of the plane the contour is drawn so that the point \(t=z\) remains to the left of it.
With the aid of the function \(\Phi(z)\), the desired class of entire functions \(F(z)\) is singled out for which the interpolation series converges
\[ F(z)=\sum_{n=0}^{\infty} L_n(F)P_n(z), \tag{3} \]
where \(\{P_n(z)\}\) is a system of polynomials biorthogonal to the system of functionals, i.e. \(L_n(P_m)=\delta_{n,m};\ n,m=0,1,2,\ldots\).
Let the entire function \(F(z)\) be representable in the form
\[ F(z)=\frac{1}{2\pi i}\int_C \Phi(z\zeta) f(\zeta)\,d\zeta, \tag{4} \]
where \(f(\zeta)\) is regular outside some domain \(D\), i.e. \(f(z)\) is \(\Phi\)-associated with the entire function \(F(z)\). To prove the convergence of the series (3), it is enough to show that the system of functions
\[ v_n(\zeta)=\frac{1}{2\pi i}\int_C u_n\!\left(\frac{\zeta}{t}\right)\Phi(t)\,\frac{dt}{t} \tag{5} \]
forms a basis in the domain \(D\), since in this case \(\Phi(z\zeta)\) can be expanded in a uniformly convergent series:
\[ \Phi(z\zeta)=\sum_{n=0}^{\infty} P_n(z)v_n(\zeta). \]
The results of the papers \((^{4,5})\) make it possible to assert that the system (5) forms a basis in the domain \(D\), if the function \(v(z)\), defined by the equalities
\[ v(z)=u[\omega(z)]e^{\frac{1}{\rho}\varphi[\omega(z)]}, \qquad \varphi(t)=t\frac{u'(t)}{u(t)}, \]
\(\omega(z)\) being the solution of the equation
\[ t[\varphi(t)]^{1/\rho}=\rho^{1/\rho}z,\qquad \omega(0)=0, \]
is regular and univalent in the star-shaped domain \(D\subset G\), mapped by the function \(w=v(z)\) onto a disk. Moreover, in the domain \(D\) the condition
\[ \operatorname{Re}\left[z\frac{v'(z)}{v(z)}\right]>0 \tag{6} \]
must be satisfied.
We now turn to the Abel–Goncharov problem. For it
\[ L_n(F)=\frac{1}{2\pi i}\int_C \frac{F(z)}{(z-\lambda_n)^{n+1}}\,dz,\qquad u_n(z)=\frac{(-1)^{n+1}}{\lambda_n^{\,n+1}}\omega_n(z), \]
where
\[ \omega_n(z)=\frac{z^n}{(z-1/\lambda_n)^{n+1}}, \]
and, by virtue of the conditions on \(\lambda_n\), for \(\omega_n(z)\) (1) holds, i.e.
\[ \frac{\omega_{n+1}(z/\lambda_{n+1})}{\omega_n(z/\lambda_n)} \longrightarrow \frac{z}{z-1}. \]
Consequently, for the Abel–Goncharov problem
\[ u(z)=\frac{z}{z-1},\qquad \varphi(z)=\frac{1}{1-z}, \]
\[ v(z)=[1-\tau(z)]e^{\frac{1}{\rho}\tau(z)},\qquad \tau(z)=\frac{1}{1-\omega(z)}, \tag{7} \]
where \(\omega(z)\) is the solution of the equation
\[ \frac{t^\rho}{1-t}=\rho z^\rho,\qquad \omega(0)=0, \]
whence
\[ z=\rho^{-1/\rho}\omega(z)[1-\omega(z)]^{-1/\rho}. \tag{8} \]
To determine the domain \(D\), let us expand \(z\) in a series in powers of \(v(z)\) and determine the radius of convergence of this series. Setting \(z=\sum_{n=0}^{\infty} c_n v^n\) and using (7) and (8), we find that
\[ c_n=e^{-n/\rho}\rho^{-1/\rho}\frac{1}{2\pi i} \int \frac{(t/\rho-1)(1-t)^{1/\rho-1}e^{\frac{t}{\rho}n}}{t^n}\,dt . \]
Let us consider separately the cases \(\rho<1\), \(\rho>1\), and \(\rho=1\).
For \(\rho<1\), by the saddle-point method one can obtain the asymptotic estimate
\[ |c_n|\sim \frac{\rho^{2-1/\rho}(1-\rho)^{1/\rho-3}}{\sqrt{2\pi}}\, n^{-5/2} \left(\frac{e^{1-1/\rho}}{\rho}\right)^n, \]
whence
\[ \overline{\lim_{n\to\infty}}\, |c_n|^{1/n} = \frac{e^{1-1/\rho}}{\rho}. \]
Consequently, for \(\rho<1\), \(z(v)\) is regular in the disk
\[ |v|<\rho e^{1/\rho-1}, \tag{9} \]
and the inverse function, i.e. \(v(z)\), is univalent in the domain obtained by mapping the disk (9) by means of the function \(z(v)\).
For \(\rho>1\), \(c_n\) can be represented in the form
\[ c_n= \frac{\rho^{-1/\rho}e^{-n/\rho}}{\Gamma(1-1/\rho)} \left(\frac{n}{\rho}\right)^{n-1/\rho} \frac{1}{(n-1)!} \int_0^\infty x^{-1/\rho}(1+x)^{-1} \left[ \frac{n-1}{n(1+x)}-1 \right] e^{-\frac{n}{\rho}x+n\ln(1+x)}\,dx . \]
Estimating the integral by Laplace’s method, we obtain
\[ |c_n|\sim \frac{\rho^{-1}}{\Gamma(1-1/\rho)}(\rho-1)^{1-1/\rho}n^{-1/\rho}, \]
whence
\[ \overline{\lim_{n\to\infty}}\, |c_n|^{1/n}=1. \]
Thus, for \(\rho>1\), \(v(z)\) is univalent in the domain obtained by mapping the disk \(|v|<1\) by means of the function \(z(v)\).
The case \(\rho=1\) is especially simple; in this case \(v(z)=-ze^{1+z}\), and \(v(z)\) is univalent in the domain \(|v(z)|<1\). Condition (6), as is not hard to verify, is satisfied in the domain \(D\) in each of the cases under consideration and guarantees the starlikeness of \(D\) with respect to the point \(z=0\).
Theorem 1. Let \(v(z)\) be the function defined by conditions (7), (8), and let \(D\) be the domain consisting of points that satisfy the inequality
\(|v(z)|<\delta_\rho\) \((\delta_\rho=\rho e^{1/\rho-1}\) for \(\rho\leqslant 1\); \(\delta_\rho=1\) for \(\rho\geqslant 1)\) and form a connected set containing the point \(z=0\).
If the entire function \(F(z)\) can be represented in the form (4), where \(\Phi(z)\) is defined by formula (2), \(f(\xi)\) is regular outside \(D\), and the contour \(C\) contains all the singular points of \(f(\xi)\), then the Abel—Goncharov series
\[ \sum_{n=0}^{\infty} L_n(F) P_n(z) \]
with interpolation nodes \(\lambda_n\) converges to \(F(z)\); moreover, the latter series converges not only in every finite part of the plane, but also in the sense of the topology \(U(\Phi,D)\) \({}^{(3)}\). (In this respect the result obtained is stronger than the results set forth in (1)—(3).)
A generalization of the Abel—Goncharov problem is the Gelfond interpolation problem \({}^{(6)}\), for which
\[ L_n(F)=\frac{1}{2\pi i}\int_C \frac{F(z)}{\prod_{k=0}^{n}(z-\lambda_{n,k})}\,dz. \]
We shall require that the following conditions be satisfied:
\[ \lambda_{n,k}-\lambda_n=\lambda_n\alpha_{n,k}, \qquad \alpha_{n,k}\to 0, \tag{10} \]
\[ \sum_{k=0}^{n}|\alpha_{n+1,k}-\alpha_{n,k}|\to 0 \quad \text{as } n\to\infty, \tag{11} \]
and that \(\lambda_n\) satisfy the same conditions as in the Abel—Goncharov problem. Condition (10) means that the interpolation nodes \(\lambda_{n,k}\) lie in the disk with center \(\lambda_n\), whose radius is \(o(\lambda_n)\), while condition (11) is a smoothness condition on the quantities \(\alpha_{n,k}\).
Theorem 2. If the interpolation nodes in the Gelfond problem satisfy conditions (10), (11), where the restrictions indicated above are imposed on \(\lambda_n\), then, for the Gelfond interpolation series, Theorem 1 holds.
Received
29 IX 1962
CITED LITERATURE
\({}^{1}\) M. A. Evgrafov, The Abel—Goncharov interpolation problem, Moscow, 1954.
\({}^{2}\) M. A. Evgrafov, DAN, 101, No. 5 (1955).
\({}^{3}\) M. A. Evgrafov, Tr. Moscow Math. Soc., 5, 89 (1956).
\({}^{4}\) M. A. Evgrafov, DAN, 115, No. 1 (1957).
\({}^{5}\) M. A. Evgrafov, A. D. Solov’ev, DAN, 113, No. 3 (1957).
\({}^{6}\) A. O. Gelfond, Calculus of Finite Differences, Moscow, 1959.