MATHEMATICS

CHEN SE-CHANG

ON THE COMPLETENESS OF THE SYSTEM OF FUNCTIONS \(\{z^{\tau_n}\log^j z\}\) ON CURVES AND IN DOMAINS OF THE COMPLEX PLANE

(Presented by Academician I. M. Vinogradov on 21 I 1961)

There are numerous works devoted to the study of the question of completeness of the system of polynomials, as well as of the system \(\{z^{\tau_n}\}\), where \(\{\tau_n\}\) is a sequence of numbers, on infinite curves with a weight, in infinite domains, and in finite domains with disconnected complement. In the present paper we consider the question of completeness of the system \(\{z^{\tau_n}\log^j z\}\) on infinite curves with a weight and in domains of the complex plane. In the formulation of the problems, the methods of investigation, and the results, the work is most closely related to the papers \((^1\text{--}^7)\).

In contrast to these works, we do not require that the sequence \(\{\tau_n\}\) satisfy the condition \(\tau_{n+1}-\tau_n \ge q>0\). Moreover, our results are established under less stringent restrictions on the weight and on the curves and domains considered.

Let the sequence \(\{\nu_n\}\) of complex numbers satisfy the following conditions:

\[ 1.\quad \lim_{n\to\infty}\frac{n}{|\nu_n|}=D_\nu, \]

\[ 2.\quad \operatorname{Re}\nu_n>0,\quad |\operatorname{Im}\nu_n|<C;\quad C\text{ is a constant.} \]

Denote by \(\{\tau_n\}\) the sequence of all distinct numbers from \(\{\nu_n\}\), and by \(m_n\) \((n=1,2,\ldots)\) the number showing how many times \(\tau_n\) occurs in \(\{\nu_n\}\).

Let \(\mathscr L\) be an unbounded curve consisting of a finite number of branches going to infinity and having the following properties:

1. It contains no loops and is rectifiable in every finite part of the plane.

2. It divides the \(z\)-plane into a finite number of infinite domains \(G_i\) \((i=1,2,\ldots,m)\), each of which contains some angle \(\Delta_i\) of opening \(\pi/\alpha_i\), \(1/2 \le \alpha_i+<\infty\).

3. The domain \(G_1\) contains some curvilinear angle \(P\) with vertex at the origin. Far from the origin the angle \(P\) coincides with the angle \(\Delta_1\). Each of the sides \(l_1\) and \(l_2\) of the angle \(P\) intersects \( |z|=r,\ 0<r<+\infty\), at only one point. Tangents can be drawn at the origin to the sides \(l_1,l_2\) of the angle. The angle \(P\) has opening \(\ge \pi/\alpha_1\), and

\[ \frac{1}{\alpha_1}>2(1-D_\nu). \tag{1} \]

Here the curve \(\mathscr L\) may consist of a finite number of connected curves.

Suppose that on \(\mathscr L\) a real continuous function \(\rho(z)\) is given such that, for large \(|z|\),

\[ \rho(z)\ge p_0(|z|)=p_0(a)+\int_a^{|z|}\frac{\omega(t)}{t}\,dt, \tag{2} \]

where \(a\) is a constant, \(\omega(t)\ge0\), \(\omega(t)\uparrow+\infty\).

Theorem 1. Under the stated conditions, if for some \(\varepsilon_0>0\)

\[ \int^{\infty}\frac{p_0(r)}{r^{1+\omega}}\,dr=+\infty,\qquad \omega=\max\left(\alpha_1,\alpha_2,\ldots,\alpha_m,\frac{1}{\frac{1}{\alpha_1}-2(1-D_\nu)}+\varepsilon_0\right), \tag{3} \]

then the system \(\{z^{\tau_n}\log^j z\}\) \((n=0,1,\ldots;\ \tau_0=0;\ j=0,1,\ldots,m_n-1)\) is complete on \(\mathcal L\) in the class \(C[p(z)]\) of functions \(f(z)\) continuous on \(\mathcal L\) and such that \(\lim\limits_{\substack{z\to\infty\\ z\in\mathcal L}} e^{-p(z)}f(z)=0\), and is complete in the sense

\[ \inf_{\{Q\}}\sup_{z\in\mathcal L} e^{-p(z)}|f(z)-Q(z)|=0, \tag{4} \]

where the lower bound is taken over all possible finite linear combinations of the functions \(z^{\tau_n}\log^j z\).

If for the sequence \(\{\nu_n\}\) the limit \(\lim\limits_{n\to\infty}\dfrac{n}{|\nu_n|}\) does not exist, then we assume that the following conditions are satisfied:

\[ \begin{aligned} &1.\quad D_{*\nu}=\lim_{n\to\infty}\frac{n}{|\nu_n|}>0.\\ &2.\quad \operatorname{Re}\nu_n>0,\quad |\operatorname{Im}\nu_n|<C.\\ &3.\quad \sup_{0<\xi<1}\ \overline{\lim}_{r\to\infty}\frac{N_\nu(r)-N_\nu(r\xi)}{r-r\xi}<+\infty, \end{aligned} \]

where \(N_\nu(r)\) is the number of points from \(\{\nu_n\}\) lying in the disk \(|z|<r\).

Moreover, in the third condition of Theorem 1 concerning the curve \(\mathcal L\), the angle \(P\) will be considered rectilinear with vertex at the origin, having opening \(\pi/\alpha_1\), \(1/\alpha_1>2(1-D_{*\nu})\). The remaining conditions of Theorem 1 are retained.

Theorem 2. Under the stated conditions, if condition (3) is satisfied with \(D_\nu\) replaced by \(D_{*\nu}\), the system \(\{z^{\tau_n}\log^j z\}\) is complete in the sense (4) in the class \(C[p(z)]\).

In what follows, for simplicity, we shall formulate the results only for the case when for \(\{\nu_n\}\) the limit \(\lim\limits_{n\to\infty}\dfrac{n}{|\nu_n|}\) exists.

Under the additional assumption that, for some \(\varepsilon^*>0\),

\[ \int_{\mathcal L} e^{-p(z)\frac{1}{1+\varepsilon^*}}\,d\sigma<+\infty, \]

the following holds:

Theorem 3. Under the conditions of Theorem 1 the system \(\{z^{\tau_n}\log^j z\}\) is complete on \(\mathcal L\) in the class \(L_p[p(z)]\) \((p>0)\) of functions \(f(z)\), defined on \(\mathcal L\) and such that

\[ \int_{\mathcal L} e^{-p(z)}|f(z)|^p\,d\sigma<+\infty, \]

and is complete in the sense

\[ \inf_{\{Q\}}\int_{\mathcal L} e^{-p(z)}|f(z)-Q(z)|^p\,d\sigma=0, \]

where the lower bound is taken over all possible finite linear combinations of the functions \(z^{\tau_n}\log^j z\) \((n=1,2,\ldots;\ j=0,1,\ldots,m_n-1)\).

Now denote by \(\Delta(\alpha)\) the domain enclosed between two circles intersecting at the origin at an angle \(\pi/\alpha\).

Let the set \(B^*\) consist of a finite number of domains \(B_i^*\) \((i=1,2,\ldots,p)\) of crescent type (i.e., \(B_i^*\) is topologically equivalent to a domain bounded by two circles having internal tangency), and suppose that the domains \(B_i^*\) have only one common multiple boundary point.

at the origin. Let one of the regions complementary to \(\bar B^*\), say \(G_1^*\), contain the point at infinity. Suppose that \(G_1^*\) contains some curvilinear angle \(P\) with vertex at the origin, which has property 3 of Theorem 1, and the opening of the angle \(P\) is greater than or equal to \(\pi/\alpha_1\), \(1/\alpha_1>2(1-D_\nu)\). Here \(D_\nu\) is the density of the sequence \(\{\nu_n\}\) satisfying condition 1. Let the remaining regions \(G_i^*\) complementary to \(\bar B^*\) \((i=2,3,\ldots,p+1)\) contain, respectively, angles of type \(\Delta(\alpha_i)\) \((i=2,3,\ldots,p+1)\).

Let \(\sigma^*(r)\) be the sum of the lengths of the arcs cut off by the set \(B^*\) on \(|z|=r\), and let \(\sigma^*(r)\leq e^{-p_0^*(r)}\), where

\[ p_0^*(r)=p_0^*(r_0)+\int_r^{r_0}\frac{\omega^*(t)}{t}\,dt, \]

\(r_0\) is a constant, and \(\omega^*(t)\uparrow+\infty\) as \(t\to+0\).

Theorem 4. Under the indicated conditions, if

\[ \int_0 \frac{p_0^*(r)}{r^{1-\omega}}\,dr=+\infty, \qquad \omega=\max(\alpha_2,\alpha_3,\ldots,\alpha_{p+1}), \]

then the system \(\{z^{\tau_n}\log^j z\}\) is complete on \(B^*\) in the class \(L_2[B^*]\) of functions \(f(z)\) analytic in \(B^*\) and such that \(\iint_{B^*}|f(z)|^2\,dx\,dy<+\infty\), and is complete in the sense

\[ \inf_{\{Q\}}\iint_{B^*}|f(z)-Q(z)|^2\,dx\,dy=0. \tag{5} \]

Let the set \(B\), consisting of a finite number of simply connected infinite regions \(B_i\) \((i=1,2,\ldots,p)\), divide the plane into a finite number of simply connected regions \(G_i\) \((i=1,2,\ldots,m)\), each of which contains an angle \(\Delta_i\) with opening \(\pi/\alpha_i\), and let the region \(G_1\) have property 3 of Theorem 1 with respect to the curve \(\mathcal L\).

Suppose that the half-axis \([0,+\infty]\) can be divided into two sets \(E_1\) and \(E\) so that:

1) for \(r\in E_1\), \(\sigma(r)\leq e^{-p_0(r)}\);

2) for \(r\geq r_0\), \(E(r)\leq e^{-p_0(r)}\),

where \(\sigma(r)\) is the sum of the lengths of the arcs cut off by the set \(B\) on \(|z|=r\), \(E(r)\) is the linear measure of the intersection of \(E\) with the ray \([r,+\infty)\), and \(p_0(r)\) satisfies condition (2).

Theorem 5. Under the indicated conditions and condition (3), the system \(\{z^{\tau_n}\log^j z\}\) is complete in the sense (5) on \(B\) in the class \(L_2[B]\).

Let \(B\) be a simply connected region obtained from the whole plane by deleting a finite number of nonintersecting rectilinear angles \(\Delta_i\) \((i=1,2,\ldots,m)\) with opening \(\pi/\alpha_i\), and let one angle have vertex at the origin and \(1/\alpha_1>2(1-D_\nu)\).

Theorem 6. Under the indicated conditions, if (3) holds, the system \(\{z^{\tau_n}\log^j z\}\) is complete on \(B\) in the class \(L_2[\rho(z)]\) of functions analytic in \(B\) and such that \(\iint_B e^{-\rho(z)}|f(z)|^2\,dx\,dy<+\infty\), and is complete in the sense

\[ \inf_{\{Q\}}\iint_B e^{-\rho(z)}|f(z)-Q(z)|^2\,dx\,dy=0. \]

We note that all the conditions appearing in the preceding theorems are, generally speaking, also necessary.

The proof of these theorems is based on the simultaneous application of the theory of differential equations of infinite order, already used by A. F. Leont′ev \((^7)\) to establish the completeness of the system \(\{z^{t_n}\}\), where \(t_n\) are integers, and of a certain kernel, which in the special case is represented by a Dirichlet series and in this case was used in questions of completeness by S. Mandelbrojt \((^5)\).

The theorems listed above constitute, in one sense or another, a generalization of a number of theorems of the authors mentioned at the beginning.

In conclusion, I express my sincere gratitude to Prof. A. F. Leont′ev for posing the questions and for guiding this work.

CITED LITERATURE

\(^1\) A. L. Shaginyan, Communications of the Institute of Mathematics and Mechanics, Academy of Sciences of the Armenian SSR, vol. 1, 1 (1947).
\(^2\) W. H. Fuchs, Proc. Cambridge Phil. Soc., 42, 91 (1946).
\(^3\) M. M. Dzhrbashyan, Doctoral dissertation, Moscow State University, 1948.
\(^4\) M. M. Dzhrbashyan, Mat. sbornik, 36 (78), No. 3, 354 (1955).
\(^5\) S. Mandelbrojt, Adherent series, regularization of sequences, applications, IL, 1955.
\(^6\) M. M. Dzhrbashyan, I. O. Khachatryan, DAN, 110, No. 6, 914 (1956).
\(^7\) A. F. Leont′ev, DAN, 121, No. 5, 797 (1958).