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MATHEMATICS
V. Kh. MUSOYAN
ON THE REPRESENTATION OF AN ARBITRARY ANALYTIC FUNCTION BY SPECIAL SERIES
(Presented by Academician I. M. Vinogradov, 10 V 1965)
A. F. Leont'ev, in the work (1), considered the question of representing arbitrary analytic functions in a convex domain by Dirichlet series with complex exponents. In the present note the question is considered of representing arbitrary analytic functions in a \(\rho\)-convex domain by more general series.
1. Let
\[ f(z)=\sum_{n=0}^{\infty} a_n z^n,\qquad a_n\ne 0, \tag{1} \]
be an entire function of finite order \(\rho\) and normal type \(\sigma\), and suppose that for it the following condition is fulfilled: there exists the limit
\[ \lim_{n\to\infty}\left(n^{1/\rho}\sqrt[n]{|a_n|}\right)=(\rho\sigma e)^{1/\rho}. \tag{2} \]
Let, further,
\[ g(z)=\sum_{n=0}^{\infty} b_n z^n \tag{3} \]
be an entire function of order \(\rho\) and type \(\sigma'\). Put
\[ \gamma(t)=\sum_{n=0}^{\infty}\frac{b_n}{a_n t^{n+1}}. \tag{4} \]
By virtue of condition (2), we have
\[ \overline{\lim_{n\to\infty}}\sqrt[n]{|b_n/a_n|}=(\sigma'/\sigma)^{1/\rho}. \]
Consequently, the radius of convergence of series (4) is equal to \((\sigma'/\sigma)^{1/\rho}\). It is easy to verify that the following integral representation of the function \(g(z)\) holds:
\[ g(z)=\frac{1}{2\pi i}\int_L \gamma(t) f(zt)\,dt, \tag{5} \]
where \(L\) is a closed contour enclosing all singularities of the function \(\gamma(t)\). We shall call the function \(\gamma(t)\) the function associated with the function \(g(z)\) with respect to the function \(f(z)\).
2. Let \(L(z)\) be an entire function of order \(\rho>1/2\) and normal type \(\sigma\), and let \(h(\varphi)>0\) be its indicator of growth. Suppose that the following condition is fulfilled:
There is a system of circles \(|z|=\rho_k,\ k=1,2,\ldots,\ \rho_k \uparrow \infty\), such that
\[ \frac{\ln |L(re^{i\varphi})|}{r^\rho}>h(\varphi)-\varepsilon,\qquad r=\rho_k,\quad k>K(\varepsilon). \tag{6} \]
By \(\gamma(t)\) we denote the function associated with the function \(L(z)\) by means of the function \(E_\rho(z)\), where
\[ E_\rho(z)=\sum_{n=0}^{\infty}\frac{z^n}{\Gamma(1+n\rho^{-1})}. \tag{7} \]
Consider in the \(z\)-plane the family of curves
\[ \operatorname{Re}(ze^{-i\varphi})^\rho=r^\rho\cos \rho(\theta-\varphi)=\tau,\qquad |\theta-\varphi|\le \pi/2\rho, \tag{8} \]
where \(z=re^{i\theta}\), and the parameters \(\varphi\) and \(\tau\) vary respectively in the intervals \([0,2\pi]\) and \([0,+\infty)\). In (8) we consider that branch of the function \((ze^{-i\varphi})^\rho\) which assumes positive values on the ray \(\arg z=\varphi\). Each curve of the family (8) divides the \(z\)-plane into two infinite domains. The closure of each of these domains will be called an elementary \(\rho\)-convex domain.
A point set \(M\) will be called \(\rho\)-convex (this definition is given in (2)) if it can be represented as the intersection of a finite or infinite number of elementary \(\rho\)-convex domains. The intersection of all \(\rho\)-convex domains containing the set \(M\) will be called the smallest \(\rho\)-convex hull of the set \(M\).
Let \(\overline D\) be the smallest \(\rho\)-convex hull of the set of all singular points of the function \(\gamma(t)\), and let \(D\) be the set of interior points of the set \(\overline D\).
Let \(F(z)\) be an arbitrary function holomorphic on the closed set \(\overline D\). Put
\[ \omega_F(\mu)=\frac{1}{2\pi i}\int_C \frac{F(t)}{t}\,dt\int_0^\infty \frac{L(x/t)-L(\mu)}{x/t-\mu}\,d\tau(x), \tag{9} \]
where \(\tau(x)=-e^{-x^\rho}\), \(C\) is a closed contour such that \(F(z)\) is regular on \(C\) and inside \(C\), and \(\overline D\) lies inside \(C\).
Theorem 1. On every closed subset \(F\) of the set \(D\) the estimate
\[ \left|F(z)-\frac{1}{2\pi i}\int_{|\mu|=\rho_k} \frac{\omega_F(\mu)E_\rho(\mu z)}{L(\mu)}\,d\mu\right| \le e^{-\delta\rho_k^\rho},\qquad z\in F,\quad k>K_0(F), \tag{10} \]
holds, where \(\delta>0\) depends on the set \(F\) and on the function \(F(z)\).
Remark. In any domain \(G\) containing all singularities of the function \(\gamma(t)\), the system \(\{z^k E_\rho^{(k)}(\lambda_\nu z),\ k=0,1,2,\ldots,\ \rho_\nu-1\}_{\nu=1}^\infty\), where \(\lambda_\nu\) is a zero of the function \(L(\lambda)\) of order \(\rho_\nu\), is incomplete. It follows from this that in such a domain \(G\) the estimate (10) cannot hold for every function analytic in the domain \(G\).
In particular, when in each annulus \(\rho_{k-1}<|z|<\rho_k,\ k=1,2,\ldots,\ \rho_0=0\), there is one zero of the function \(L(z)\), it follows from the estimate (10) that the function \(F(z)\) expands into a series.
Let us note the main stages of the proof of Theorem 1. Put
\[ \Phi_k(z,\lambda)=L(\lambda)\frac{1}{2\pi i} \int_{|\mu|=\rho_k}\frac{E_\rho(z\mu)}{(\mu-\lambda)L(\mu)}\,d\mu, \tag{11} \]
where \(\lambda\) lies inside the circle \(|\mu|=\rho_k\). The function \(\Phi_k(z,\lambda)\), as a function of \(\lambda\), is analytically continuable to the whole plane. Consequently, it is an entire function. It can be proved that the function \(\Phi_k(z,\lambda)\), as a function of \(\lambda\), for \(z\in \overline{D}\), has order \(\rho\) and indicator \(h(\varphi)\). Denote by \(\gamma_k(z,t)\) the function associated with the function \(\Phi_k(z,\lambda)\), as a function of \(\lambda\), with respect to the function \(E_\rho(z)\). As a function of \(t\), \(\gamma_k(z,t)\) is regular everywhere outside \(\overline{D}\). We assert that the equality
\[ F(z)-\frac{1}{2\pi i}\int_{|\mu|=\rho_k}\frac{\omega_F(\mu)E_\rho(\mu z)}{L(\mu)}\,d\mu = \frac{1}{2\pi i}\int_C \gamma_k(z,t)F(t)\,dt,\quad z\in\overline{D}. \tag{12} \]
holds.
Let us first verify this equality for functions \(F(t)\) of the form \(E_\rho(\lambda z)\), where \(\lambda\) is a fixed number, with \(|\lambda|<\rho_k\). In doing this we shall use the fact that, for \(f(z)=E_\rho(z)\), the representation (5) admits the inverse
\[ \gamma(t)=\frac{1}{t}\int_0^\infty g\left(\frac{x}{t}\right)d\tau(x), \tag{13} \]
where \(\tau(x)=-e^{-x^\rho}\). From (9) and (13) it follows that, for \(F(z)=E_\rho(\lambda z)\), we have
\[ \omega_F(\mu)=\frac{L(\lambda)-L(\mu)}{\lambda-\mu}, \]
and therefore the left-hand side of relation (12) is equal to
\[ E_\rho(\lambda z)-\frac{1}{2\pi i}\int_{|\mu|=\rho_k}\frac{E_\rho(\mu z)}{\mu-\lambda}\,d\mu + \frac{L(\lambda)}{2\pi i}\int_{|\mu|=\rho_k}\frac{E_\rho(\mu z)}{(\mu-\lambda)L(\mu)}\,d\mu = \Phi_k(z,\lambda). \]
The right-hand side of relation (12) is also equal to \(\Phi_k(z,\lambda)\). Since there exists a system of functions \(\{E_\rho(\lambda_i z)\}_{i=1}^{\infty}\), complete in the whole plane, where \(|\lambda_i|<\rho_k\), and since both sides of relation (12) depend continuously on the function \(F(z)\), relation (12) is also valid for an arbitrary function \(F(z)\) analytic on \(C\) and inside \(C\). Thus relation (12) has been established. Using the integral representation for \(\gamma_k(z,t)\), analogous to representation (13), one can show that the function \(\gamma_k(z,t)\), for \(z\in F\) and \(t\in C\), has the estimate
\[ |\gamma_k(z,t)|\le e^{-\delta\rho_k^\rho},\quad k>K_0(F), \tag{14} \]
where \(\delta>0\) depends on the set \(F\) and on the function \(F(z)\). Hence, from equality (12), Theorem 1 follows.
- Let \(L(\lambda)\) be an entire function of order \(\rho\) and type \(\sigma_1\), possessing property (6), and let \(h(\varphi)>0\) \((0\le \varphi\le 2\pi)\) be its growth indicator. Further, let \(f(z)\) be an entire function possessing properties (1) and (2). By \(\gamma(t)\) we denote the function associated with the function \(L(\lambda)\), with respect to the function \(f(z)\). The function \(\gamma(t)\) is regular outside the circle
\[ |t|\le (\sigma_1/\sigma)^{1/\rho}. \tag{15} \]
Let \(F(z)\) be an analytic function in the closed circle (15). Put
\[ \omega_F(\mu)=\frac{1}{2\pi i}\int_C F(t)\gamma(t,\mu)\,dt, \tag{16} \]
where \(C\) is a circle of radius \(a\), \(a>(\sigma_1/\sigma)^{1/\rho}\); \(\gamma(t,\mu)\) is the function associated with the function \((L(\lambda)-L(\mu))/(\lambda-\mu)\), as a function of \(\lambda\), with respect to the function \(f(z)\).
Theorem 2. In any disk of radius \(\beta\)
\[ \beta < \left[\min_{\varphi} h(\varphi)/\sigma\right]^{1/\rho} \]
the estimate
\[ \left| F(z) - \frac{1}{2\pi i} \int \frac{\omega_F(\mu) f(\mu z)}{L(\mu)}\, d\mu \right| \le e^{-\delta_0 \rho}_k,\qquad |z|\le \beta,\qquad k>K_0(\beta), \tag{17} \]
holds, where \(\delta>0\) depends on \(\beta\) and on the function \(F(z)\).
Theorem 2 is proved essentially by a scheme analogous to the proof of Theorem 1.
In conclusion I express my gratitude to Prof. A. F. Leont’ev for posing the problem and for a number of valuable suggestions.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
29 IV 1965
References
\(^{1}\) A. F. Leont’ev, DAN, 164, No. 1 (1965).
\(^{2}\) A. E. Avetisyan, DAN, 105, No. 5 (1955).
\(^{3}\) A. O. Gel’fond, Matem. sborn., 4 (46), 1, 149 (1938).