MATHEMATICS

A. F. LEONT’EV

ON THE COMPLETENESS OF THE SYSTEM \(\{e^{\lambda_n z}\}\) IN A CLOSED STRIP

(Presented by Academician P. S. Novikov on 3 IV 1963)

Let \(\lambda_n > 0\) \((n = 1, 2, \ldots)\), \(\lambda_n \uparrow \infty\), and
\[ \tau=\lim_{R\to\infty}\sum_{\lambda_i<R}\lambda_i^{-1}. \]
Carleman proved \((^1)\) that the system \(\{e^{\lambda_n z}\}\) is complete in the open strip \(|\operatorname{Im} z|<\pi\tau\) (any function analytic in this strip can be approximated with arbitrary accuracy on each bounded closed set \(F\) of this strip by finite linear combinations of functions from the system under consideration). In the case when the limit
\[ \sigma=\lim_{n\to\infty}\frac{n}{\lambda_n} \]
exists (in this case \(\tau=\sigma\)), the system \(\{e^{\lambda_n z}\}\), \(z=x+iy\), is complete \(( (^2)\) and \((^3)\), p. 286) in any open curvilinear strip \(\varphi(x)<y<\varphi(x)+2\pi\sigma\), \(-\infty<x<\infty\), of width \(2\pi\sigma\) (in the vertical direction), and is not complete in a strip of greater width. In this note the question is the completeness of the system \(\{e^{\lambda_n z}\}\) in a closed strip.

Theorem 1. Let \(\lambda_n>0\), \(\lambda_n\uparrow\infty\), and suppose the sequence \(\{\lambda_n\}\) can be represented as the union of two subsequences: \(\{\lambda_n\}=\{\lambda'_n\}+\{\lambda''_n\}\), possessing the following properties: 1) the numbers \(\lambda'_n\) \((n=1,2,\ldots)\) are zeros of an entire function of exponential type \(L(z)\), with \(|L(iy)|\ge B\exp(A|y|)\), \(B\ne0\), \(z=x+iy\), and \(L(z)\) has no zeros in the right half-plane \(x\ge0\) other than \(\lambda'_n\); 2) the numbers \(\lambda''_n\) \((n=1,2,\ldots)\) satisfy the condition \(\sum(\lambda''_n)^{-1}=\infty\). Then any function \(f(z)\), analytic in the open strip \(|\operatorname{Im}z|<A\) and continuous at all finite points of the closed strip \(|\operatorname{Im}z|\le A\), can be approximated with arbitrary accuracy on each bounded set \(F\) belonging to the closed strip \(|\operatorname{Im}z|\le A\) by finite linear combinations of the functions of the system \(\{e^{\lambda_n z}\}\).

Theorem 2. Let \(\lambda_n>0\), \(\lambda_n\uparrow\infty\), and \(\{\lambda_n\}=\{\lambda'_n\}+\{\lambda''_n\}\), where: 1) \(\lambda'_n\) \((n=1,2,\ldots)\) are zeros of an entire function of exponential type \(L(z)\not\equiv0\) such that \(|L(iy)|\le B\exp(A|y|)\); 2) the sequence \(\{\lambda''_n\}\) is either empty, finite, or infinite, and in the last case \(\sum(\lambda''_n)^{-1}<\infty\). Let, further, \(P_n(z)\) \((n=1,2,\ldots)\) be finite linear combinations of functions from the system \(\{e^{\lambda_n z}\}\). If the sequence \(\{P_n(z)\}\) converges uniformly on any bounded set \(F\) belonging to the closed strip \(|\operatorname{Im}z|\le A\), then it converges uniformly in every bounded domain of the plane.

It follows, of course, from Theorem 2 that under the conditions of this theorem the system \(\{e^{\lambda_n z}\}\) is not complete in the closed strip \(|\operatorname{Im}z|\le A\).

Theorem 3. Let \(\lambda_n>0\), \(\lambda_n\uparrow\infty\), suppose the limit
\[ \lim_{n\to\infty}\frac{n}{\lambda_n}=\frac{A}{\pi} \]
exists and \(\{\lambda_n\}=\{\lambda'_n\}+\{\lambda''_n\}\), where \(\sum(\lambda''_n)^{-1}<\infty\) and
\[ \prod_{n=1}^{\infty}\left(1+\frac{r^2}{\lambda_n'^2}\right)\le Be^{Ar}. \]

Let \(P_{n1}(z)\), \(P_{n2}(z)\) be finite linear combinations of functions respectively from the systems \(\{e^{\lambda'_n z}\}\), \(\{e^{-\lambda_n z}\}\), and \(P_n(z)=P_{n1}(z)+P_{n2}(z)\). If the sequen-

the sequence \(\{P_n(z)\}\) converges uniformly in the rectangle \(|\operatorname{Im} z|\le A\), \(|\operatorname{Re} z|\le \delta\), where \(\delta>0\), then the subsequences \(\{P_{n1}(z)\}\) and \(\{P_{n2}(z)\}\) converge inside the half-planes \(\operatorname{Re} z<x_1\), \(\operatorname{Re} z>x_2\), respectively, with \(x_2<0<x_1\).

Let us note the main stages of the proofs of the theorems.

For the proof of Theorem 1, put

\[ \Phi_n(s)=L(\beta)q_n(\beta)\cdot \frac{1}{2\pi i} \int_{-\infty i}^{\infty i} \frac{e^{sz}\,dz}{(\beta-z)L(z)q_n(z)},\qquad q_n(z)=\prod_{\nu=1}^{n}\left(1-\frac{z}{\lambda_\nu}\right); \tag{1} \]

where \(\beta>0\). The function \(\Phi_n(s)\), by condition 1) of the theorem, is regular in the strip \(|\operatorname{Im} s|<A\), continuous in the closed strip \(|\operatorname{Im} s|\le A\), and

\[ |\Phi_n(s)|\le C|L(\beta)|\,|q_n(\beta)|,\qquad |\operatorname{Im} s|\le A, \tag{2} \]

where \(C\) does not depend on \(n\). There is a system of circles \(|z|=\rho_k\), \(\rho_k\uparrow\infty\), on which \(|L(z)|>\exp(-K|z|)\). The same inequality (with the same \(K\), we may assume) holds for large \(|z|\) also on the boundary \(\Gamma\) of the angle \(|\arg z|<\pi/4\). In view of this, the integration along the imaginary axis in the integral (1) may be replaced by integration over the contour \(\Gamma\), after which, according to the residue theorem, we obtain in the region \(\sigma\le-(\sqrt2K+|t|+1)\), \(s=\sigma+it\),

\[ \Phi_n(s)=e^{\beta s}-\lim_{k\to\infty}\sum_{\lambda_\nu<\rho_k} a_\nu^{(n)}e^{\lambda_\nu s}, \tag{3} \]

and the convergence in the indicated region is uniform. From condition 2) of the theorem it follows that \(q_n(\beta)\to0\) as \(n\to\infty\). Therefore from (2)—(3) we conclude that for every \(\varepsilon>0\) there are such \(n\) and \(k\) that we shall have

\[ \left|e^{\beta s}-\sum_{\lambda_\nu<\rho_k}a_\nu^{(n)}e^{\lambda_\nu s}\right|<\varepsilon,\qquad |t|\le A,\quad \sigma\le\sigma_0=-(\sqrt2K+A+1). \tag{4} \]

By replacing \(s\) by \(s-h\), one can ensure that an inequality of the form (4) will hold in the half-strip \(|t|\le A\), \(\sigma\le\sigma_0\), where \(\sigma_0\) is arbitrary. It remains to add that the functions \(e^{\beta_n s}\), where, for example, \(\beta_n=n\pi/2A\) \((n=1,2,\ldots)\), form a system complete in the strip \(|\operatorname{Im}s|<2A\) of width \(4A\).

The proofs of Theorems 2 and 3 are based on the following lemma.

Lemma. Suppose \(\lambda_n'>0\) and \(\sum(\lambda_n')^{-1}<\infty\). For any \(\beta>0\) there exists a function \(\Phi(z)\not\equiv0\) of the form

\[ \Phi(z)=\int_0^\beta \psi(t)e^{zt}\,dt,\qquad z=x+iy, \tag{5} \]

satisfying the condition

\[ |\Phi(iy)|\le \frac{1}{T(|y|)},\qquad T(r)>\frac{1}{3r}M(r),\quad r>r_0,\qquad M(r)=\prod_{\nu=1}^{\infty}\left(1+\frac{r^2}{\lambda_\nu'^2}\right). \]

Let us prove the lemma. From the convergence of the series it follows \(\left(^{4}\right)\), see also \(\left(^{5}\right)\), p. 35, that

\[ \int_1^\infty \frac{\ln M(r)}{r^2}\,dr<\infty. \tag{6} \]

Let \(M(r)=\sum_{k=0}^{\infty}\frac{r^k}{m_k}\), \(T(r)=\max_{k\ge0}\frac{r^k}{m_k}\). We have \(M(r)\le T(r)\). For \(T(r)\) a condition of the form (6) is fulfilled. From this condition it follows, by Carleman’s fundamental theorem on quasianalytic functions (see, for example, \(\left(^{6}\right)\), p. 31), that on \([0,\beta]\) there is a function \(\psi(t)\not\equiv0\) with the properties: \(\psi^{(n)}(0)=\psi^{(n)}(\beta)=0\) \((n=0,1,2,\ldots)\), \(|\psi^{(n)}(t)|\le m_n\) \((n=1,2,\ldots)\). In the inte-

in (5) we shall take precisely this function as \(\psi(t)\). We have

\[ \Phi(z)=\frac{(-1)^n}{z^n}\int_0^\beta \psi^{(n)}(t)e^{zt}\,dt, \]

whence, for any \(n\), we obtain
\[ |\Phi(iy)|\le m_n |y|^{-n} \]
(let \(\beta<1\)) and, consequently,
\[ |\Phi(iy)|\le [T(|y|)]^{-1}. \]
The function \(M(r)\) grows no faster than an entire function of order one and minimal type; therefore, for large \(n\), for example, \(m_n^{-1}<n^{-n}\), whence

\[ M(r)\le \sum_{n=1}^{[2r]}\frac{r^n}{m_n}+\sum_{[2r]+1}^{\infty}\left(\frac{r}{n}\right)^n<2rT(r)+1<3rT(r). \]

The lemma is proved.

We pass to the proof of Theorem 2. Put

\[ L_1(z)=L(z)\Phi(z)\prod_{n=1}^{\infty}\left(1-\frac{z}{\lambda_n'}\right), \]

where \(\Phi(z)\) is the function (5). By the lemma and condition 1) of the theorem, we have
\[ |L_1(iy)|=O\bigl(|y|^{-2}e^{A|y|}\bigr). \]
Therefore there exists a function \(\gamma(\xi)\), analytic outside a certain rectangle
\[ |\operatorname{Im}\xi|\le A,\qquad |\operatorname{Re}\xi|\le d, \]
with \(\gamma(\infty)=0\), continuous up to the boundary \(\Gamma\) of this rectangle, and such that

\[ L_1(z)=\frac{1}{2\pi i}\int_\Gamma \gamma(\xi)e^{z\xi}\,d\xi. \]

Put

\[ \omega(\mu,\alpha,P_n)=e^{-\alpha\mu}\frac{1}{2\pi i} \int_\Gamma\left[\int_0^\xi P_n(\xi-\eta+\alpha)e^{-\mu\eta}\,d\eta\right]\gamma(\xi)\,d\xi, \]

where \(\alpha\) is a real parameter. It is verified directly that the quotient of the function

\[ \frac{\omega(\mu,\alpha,f)}{L_1(\mu)},\qquad f(z)=e^{\lambda z},\qquad L_1(\lambda)=0, \]

as a function of \(\mu\), is equal to \(e^{\lambda z}\) at the point \(\mu=\lambda\) and is equal to zero at the points \(\mu\ne\lambda\). Hence we obtain

\[ P_n(z)=\frac{1}{2\pi i}\int_C \frac{\omega(\mu,\alpha,P_n)}{L_1(\mu)}e^{\mu z}\,d\mu, \tag{7} \]

where \(C\) is the boundary of the angle \(|\arg\mu|<\pi/4\). The singular points of the integrand can only be the points from \(\{\lambda_n\}\). From the uniform convergence of \(\{P_n(z+\alpha)\}\) on \(\Gamma\) it follows that on \(C\)

\[ \left|\frac{\omega(\mu,\alpha,P_n)}{L_1(\mu)}\right| <Ne^{q|\mu|}|e^{-\alpha\mu}|, \]

where \(N\) and \(q\) do not depend on \(n\). From this estimate and representation (7) it follows that \(\{P_n(z)\}\) converges uniformly in every bounded domain.

Theorem 3 is proved essentially by the same scheme.

Received
23 III 1963

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\[ {}^{1} \]
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\[ {}^{2} \]
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\[ {}^{3} \]
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\[ {}^{4} \]
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\[ {}^{5} \]
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\[ {}^{6} \]
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