A. F. LEONT’EV

ON THE REPRESENTATION OF ARBITRARY FUNCTIONS BY DIRICHLET SERIES

(Presented by Academician Yu. V. Linnik on 3 III 1965)

MATHEMATICS

We consider Dirichlet series

\[ f(z)=\sum_{n=1}^{\infty} a_n e^{\lambda_n z}, \qquad \lambda_1 \ne 0,\quad |\lambda_n|\uparrow\infty, \tag{1} \]

with, in general, complex exponents \(\lambda_n\), which satisfy the condition
\[ \lim_{n\to\infty}\ln n/\lambda_n=0. \]
Under this condition, as is known, the open domain of convergence of the series coincides with the open domain of absolute convergence of the series. Let us pose the question: does there exist a domain \(D\) in which an arbitrary function \(f(z)\), analytic in \(D\) or \(\overline D\), for example \(f(z)\equiv 1\), could be expanded in the series (1)?

If the \(\lambda_n\) are located on one or two rays, then the answer to this question is negative. Indeed, suppose first that the \(\lambda_n\) are located on one ray, for example, all \(\arg \lambda_n=0\). Assume that in \(D\) there is an expansion (1). Then the series (1) converges in some half-plane \(\operatorname{Re} z<a\) containing the domain \(D\), and the sum of the series tends to zero as \(\operatorname{Re} z\to -\infty\). Consequently, the function \(f(z)\equiv 1\) cannot be represented by the series (1). Suppose now that the \(\lambda_n\) are located on two rays, for example
\(\arg \lambda_n'=0\), \(\arg \lambda_n''=\varphi_0\), \(0<\varphi_0\le \pi\), \(\{\lambda_n'\}\cup\{\lambda_n''\}=\{\lambda_n\}\). Suppose that in some domain \(D\) the expansion (1) is valid for \(f(z)\equiv 1\). In the domain \(D\), since the series converges absolutely, we have

\[ 1=\sum_{n=1}^{\infty} a_n e^{\lambda_n z} =\sum a_n' e^{\lambda_n' z}+\sum a_n'' e^{\lambda_n'' z} =\Sigma_1+\Sigma_2. \tag{2} \]

The series \(\Sigma_1\) converges in the half-plane \(\operatorname{Re} z<a\), containing the domain \(D\); the series \(\Sigma_2\) converges in the half-plane \(\operatorname{Re}(z e^{i\varphi_0})<b\), also containing the domain \(D\). Both converge simultaneously in the common part \(G\) of these half-planes. For \(\varphi_0<\pi\) the domain \(G\) is an angle; along the bisector of this angle, when \(z\to\infty\), the sum of the series \(f(z)\equiv 1\) must tend to zero, which is impossible. For \(\varphi_0=\pi\) the domain \(G\) is a vertical strip. From equality (2) it follows that in this case the functions \(f_1(z)\) and \(f_2(z)\)—the sums, respectively, of the series \(\Sigma_1\) and \(\Sigma_2\)—are regular in the whole plane and bounded in modulus. Therefore \(f_1(z)\) and \(f_2(z)\) are constants: \(f_1(z)=\alpha\), \(f_2(z)=\beta\), where, since \(\alpha+\beta=1\), one of the numbers \(\alpha\) and \(\beta\) is nonzero. We arrive at the preceding case. Consequently, in the case under consideration as well, \(f(z)\equiv 1\) cannot have a representation (1).

The purpose of this article is to note that for some \(\{\lambda_n\}\) one can indicate a domain \(D\) in which an arbitrary function \(f(z)\), analytic in \(\overline D\), admits an expansion (1). In particular, such a domain \(D\) can be indicated in the case when the \(\lambda_n\) are chosen in a definite way on three rays. We formulate the result obtained in this direction.

Let \(L(\lambda)\) be an entire function of exponential type, and let \(h(\varphi)\) be its growth indicator. Suppose that \(h(\varphi)>0,\ 0\leq \varphi<2\pi\), and that the following condition is satisfied: there is a system of circles \(|\lambda|=r_k,\ r_k\uparrow\infty\), such that

\[ \ln |L(re^{i\varphi})|>[h(\varphi)-\varepsilon]r,\qquad r=r_k,\quad k>K(\varepsilon). \]

Denote by \(\gamma(t)\) the function associated with \(L(\lambda)\) in the sense of Borel. Let \(D\) be the smallest convex closed set containing all singularities of the function \(\gamma(t)\); \(D\) is the set of interior points (the open part) of \(\overline D\). From the condition \(h(\varphi)>0,\ 0\leq \varphi<2\pi\), it follows that \(D\) is not empty; it contains, for example, the origin. Take an arbitrary function \(f(z)\), analytic on the closed set \(\overline D\). Let \(C\) be a convex closed contour satisfying the following conditions: the function \(f(z)\) is regular on \(C\) and inside \(C\), and the set \(\overline D\) lies inside \(C\). Put

\[ \omega(\mu)=\frac{1}{2\pi i}\int_C\left[\int_0^\xi f(\xi-\eta)e^{\mu\eta}\,d\eta\right]\gamma(\xi)\,d\xi . \]

We assume that in the inner integral the variable \(\eta\) varies from \(0\) to \(\xi\) along a rectilinear segment. Under this condition the point \((\xi-\eta)\) does not go outside the contour \(C\). Let \(\lambda_1,\lambda_2,\ldots\) be the distinct zeros of the function \(L(\lambda)\), arranged in nondecreasing order of their moduli, and let \(p_1,p_2,\ldots\) be their respective multiplicities. Put further

\[ P_\nu(z)e^{\lambda_\nu z} = \frac{1}{2\pi i}\int_{C_\nu}\frac{\omega(\mu)e^{\mu z}}{L(\mu)}\,d\mu \qquad (\nu=1,2,\ldots). \]

Here \(C_\nu\) is a closed contour inside which lies \(\lambda_\nu\) and there are no other zeros of the function \(L(\lambda)\); \(P_\nu(z)\) is a polynomial of degree \(<p_\nu\).

Theorem. In the domain \(D\) the representation

\[ f(z)=\sum_{k=1}^{\infty} f_k(z),\qquad f_1(z)=\sum_{|\lambda_\nu|<r_1} P_\nu(z)e^{\lambda_\nu z}, \]

\[ f_k(z)=\sum_{r_{k-1}<|\lambda_\nu|<r_k} P_\nu(z)e^{\lambda_\nu z}\qquad (k\geq 2), \tag{3} \]

holds, the series converging absolutely and uniformly inside \(D\).

In the case when the zeros \(\lambda_n\) are all simple and in each annulus \(r_{k-1}<|\lambda|<r_k\) there is only one zero, representation (3) takes the form

\[ f(z)=\sum_{n=1}^{\infty} a_n e^{\lambda_n z},\qquad a_n=\frac{\omega(\lambda_n)}{L'(\lambda_n)},\qquad z\in D. \tag{4} \]

We indicate two simple examples where representation (4) holds. In the first example,

\[ L(z)=\prod_{n=1}^{\infty}\left(1-\frac{z^3}{n^3}\right),\qquad \{\lambda_n\}=\bigcup_{k=0}^{2}\{ne^{2k\pi i/3}\} \]

and the domain \(D\) is a regular triangle with vertices at the points \(ae^{\pi i/3},\ -a,\ ae^{-\pi i/3}\), \(a=(2\sqrt3/3)\pi\). Here the \(\lambda_n\) are situated on three rays.

In the second example

\[ L(z)=\sin k_1z\,\sin ik_2z/z^2,\qquad k_1>0,\qquad k_2>0, \]

\[ \{\lambda_n\}=\{\pm k\pi/k_1\}\cup\{\pm k\pi i/k_2\}. \]

and the domain \(D\) is the rectangle \(|\operatorname{Re} z|<k_2,\ |\operatorname{Im} z|<k_1\). Here the \(\lambda_n\) are located on four rays.

Let us note that the representation of the function \(f(z)\) in the above-mentioned domain \(D\) by a Dirichlet series with exponents \(\lambda_n\) is not unique. Indeed, suppose in (4), for example, that \(a_1\ne0\). Put \(L_1(z)=(z-\lambda_1)^{-1}L(z)\). The function \(L_1(z)\) has, like \(L(z)\), all the necessary properties, but \(L_1(\lambda_1)\ne0\). Therefore in the domain \(D\) we have

\[ f(z)=\sum_{n=2}^{\infty} b_n e^{\lambda_n z}. \]

The absence of uniqueness leads to the following conclusion: if, concerning a Dirichlet series with the exponents \(\lambda_n\) under consideration, it is known only that it converges in the domain \(D\) or in some part of it, then in principle it is impossible to indicate a formula for determining the coefficients of this series.

Let us note one more circumstance. The representation (3) cannot in general hold in a domain \(G\supset\overline{D}\). Indeed, it is not difficult to show that if the series (3) converges uniformly inside some domain \(G\supset\overline{D}\), then the function \(f(z)\) must satisfy, in some neighborhood of the origin, the equation

\[ M(f)=\frac{1}{2\pi i}\int_C \gamma(\xi) f(z+\xi)\,d\xi=0, \]

But not every, even entire, function satisfies this equation; for example, if \(\lambda\ne\lambda_n\) \((n=1,2,\ldots)\), then \(M(e^{\lambda z})=e^{\lambda z}L(\lambda)\ne0\).

Let us indicate one possible application of the theorem. Suppose it is required to find at least one solution \(F(z)\) of the nonhomogeneous equation

\[ \frac{1}{2\pi i}\int_{C_1}\gamma_1(z+t)F(t)\,dt=f(z) \tag{5} \]

with a known right-hand side \(f(z)\). Here \(\gamma_1(u)\) is regular outside some convex closed set \(G\). The closed contour \(C_1\) encloses the set \(G\). Suppose that the function \(f(z)\) is regular in the convex domain \(\overline{D}\) and in \(D\) is represented by the series (4). A formal solution of equation (5) will be

\[ F(z)=\sum_{n=1}^{\infty}\frac{a_n e^{\lambda_n z}}{L_1(\lambda_n)},\qquad L_1(\mu)=\frac{1}{2\pi i}\int_{C_1}\gamma_1(t)e^{\mu t}\,dt. \tag{6} \]

The question reduces to where the series (6) will converge. Consider, for example, the case when \(D\) is a rectangle: \(|\operatorname{Im} z|<k_1,\ |\operatorname{Re} z|<k_2\). In this case, as we saw above, \(f(z)\) can indeed be represented by the series (4), with
\(\{\lambda_n\}=\{\pm n\pi/k_1\}\cup\{\pm n\pi i/k_2\}\). Suppose that, whatever \(\varepsilon>0\) may be, for sufficiently large \(n\) the expression \(|L_1(\lambda_n)|\) is greater than \(\exp[(\alpha-\varepsilon)|\lambda_n|]\) for
\(\lambda_n\in\{\pm n\pi/k_1\}\) and greater than \(\exp[(\beta-\varepsilon)|\lambda_n|]\) for
\(\lambda_n\in\{\pm n\pi i/k_2\}\), where \(\alpha\ge0,\ \beta\ge0\). Then the series (6) will converge uniformly inside the rectangle \(D_1:\ |\operatorname{Im} z|<k_1+\beta,\ |\operatorname{Re} z|<k_2+\alpha\). If it is additionally assumed that \(|L_1(z)|\), for sufficiently large \(|z|\), is less than \(\exp[(\alpha+\varepsilon)|z|]\) on the real axis and less than \(\exp[(\beta+\varepsilon)|z|]\) on the imaginary axis, then one may assert that the function (6), analytic in the rectangle \(D_1\), is a solution of equation (5) in the rectangle \(D\).

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
18 II 1965