MATHEMATICS

A. F. LEONT'EV

ON THE COMPLETENESS OF THE SYSTEM \(\{z^{\lambda_k}\}\) ON CURVES IN THE COMPLEX PLANE

(Presented by Academician I. N. Vekua, 10 IV 1958)

Let \(L\) be an unbounded curve with a finite number of branches going off to infinity (\(L\) may consist of several (a finite number of) connected pieces), having no loops and dividing the plane into a finite number of simply connected infinite domains \(G_1, G_2,\ldots,G_m\). Let \(L\) be rectifiable in every finite part of the plane and, if \(\sigma(z)\) is the length of the arc of a connected piece, counted from some point of the piece to its point with affix \(z\) (far from the initial point we regard \(\sigma(z)\) as a single-valued function of \(|z|\)), then \(d\sigma(z)\le M\,d|z|\), where \(M\) is a constant. Suppose that on \(L\) a continuous real function \(p(z)\) is given such that, for large \(|z|\),

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

where \(\omega(t)\ge 0\) and \(\omega(t)\uparrow +\infty\). Assume, finally, that each domain \(G_i\) \((i=1,2,\ldots,m)\) contains within itself an angle \(\Delta_i\) of opening \(\pi/\alpha_i\), \(1/2<\alpha_i<\infty\) (with vertex not necessarily at the origin).

M. M. Dzhrbashian \((^{1,2})\) showed that if under these conditions

\[ \int^\infty \frac{p_0(r)\,dr}{r^{1+\omega}}=+\infty,\qquad \omega=(\alpha_1,\alpha_2,\ldots,\alpha_m), \]

then on \(L\), in the class \(L_2[p(z)]\) of functions \(f(z)\) defined and measurable on \(L\) and such that

\[ \int_L e^{-p(z)}|f(z)|^2\,d\sigma<\infty, \tag{1} \]

the system of polynomials is complete in the sense that

\[ \inf_{\{Q\}}\int_L e^{-p(z)}|f(z)-Q(z)|^2\,d\sigma=0, \tag{2} \]

where \(Q(z)\) are arbitrary polynomials.

In the present note we are concerned with deriving conditions under which (2) holds for functions \(f(z)\) from \(L_2[p(z)]\), if in (2) the functions \(Q(z)\) are not arbitrary polynomials, but arbitrary finite linear combinations of functions of the system \(\{z^{\lambda_k}\}\), where \(\lambda_1,\lambda_2,\ldots\) are given whole nonnegative numbers. In short, we are concerned with completeness, in the indicated sense, of the system \(\{z^{\lambda_k}\}\).

M. M. Dzhrbashian and I. O. Khachatryan \((^6)\) considered this question in the case when \(L\) is a pair of rays, \(\arg z=\pm \pi/2\alpha\), and the \(\lambda_n\) are not necessarily integers.

Theorem. Let \(\{\mu_n\}\) be the sequence of all positive integers not included in \(\{\lambda_k\}\), and let

\[ \lim_{n\to\infty}\frac{n}{\mu_n}=\sigma<1 . \tag{3} \]

Suppose further that each angle \(\Delta_i\) \((i=1,2,\ldots,m)\) has aperture \(\pi/\alpha_i>2\pi\sigma\), and one of the domains \(G_i\), for example \(G_1\), contains a curvilinear angle \(P\) (far from the origin the angles \(P\) and \(\Delta_1\) coincide) with vertex at the origin (the origin, in general, does not belong to the domain \(G_1\)), intersecting every circle \(|z|=r\), \(b<r<\infty\), in an arc of length \(>2\pi\sigma r\) (in this sense the aperture of the angle \(P\) is greater than \(2\pi\sigma\)). If, for some \(\varepsilon_0>0\),

\[ \int^\infty \frac{p_0(r)\,dr}{r^{1+\omega_1+\varepsilon_0}}=\infty,\qquad \omega_1=\max(\beta_1,\beta_2,\ldots,\beta_m),\qquad \frac{\pi}{\beta_j}=\frac{\pi}{\alpha_j}-2\pi\sigma, \tag{4} \]

then on \(L\) the system \(\{z^{\lambda_k}\}\) is complete in the sense of (2) (\(Q(z)\) being combinations of powers from \(\{z^{\lambda_k}\}\) (in the class \(L_2[p(z)]\)).

The proof of the theorem is based on the simultaneous application of the method of the Cauchy integral transform, used by Dzhrbashian in \((^2)\), and the method of using differential equations of infinite order.

To prove the theorem, it is necessary to prove that from the conditions

\[ \int_L e^{-p(z)}\overline{f(z)}z^{\lambda_k}\,d\sigma=0,\qquad f(z)\in L_2[p(z)] \quad (k=1,2,\ldots) \tag{5} \]

it follows that \(f(z)=0\) almost everywhere on \(L\).

Thus, suppose that (5) holds. As in \((^2)\), consider the function

\[ F(w)=\frac{1}{2\pi i}\int_L \frac{e^{-p(z)}\overline{f(z)}\,d\sigma}{z-w}. \tag{6} \]

In each domain \(G_j\) it represents some analytic function \(F_j(w)\). Since

\[ \frac{1}{z-w}=-\frac{1}{w}-\frac{z}{w^2}-\ldots-\frac{z^{\,n-1}}{w^n}-\frac{z^n}{w^n(z-w)}, \]

then, by virtue of (5), in \(G_j\)

\[ F_j(w)= \sum_{\mu_k<n}\frac{a_k}{w^{\mu_k+1}} -\frac{1}{w^n}\frac{1}{2\pi i}\int_L \frac{e^{-p(z)}\overline{f(z)}\,z^n\,d\sigma}{z-w} =\varphi_n(w)+r_n(w). \tag{7} \]

Let \(w=e^t\). In the \(t\)-plane the function \(F_j(e^t)\) is regular, in particular, in the domain \(D_j\) (it is obtained from the angle \(\Delta_j\)), which for large \(\operatorname{Re}(t)\) asymptotically approaches a horizontal strip of width \(\pi/\alpha_j>2\pi\sigma\). It should be noted that \(F_1(e^t)\) is regular in the domain \(Q\) (it is obtained from the angle \(P\)), which contains an entire curvilinear strip of width (in the vertical direction) \(>2\pi\sigma\). Put

\[ \prod_{n=1}^{\infty}\left(1-\frac{t^2}{(\mu_n+1)^2}\right) =\sum_0^\infty c_n t^n,\qquad M(y)=\sum_0^\infty c_n y^{(n)}(t). \]

By virtue of (3), the operator \(M(y)\) has the properties \((^3)\):

1) The operator \(M(y)\) is defined at every point \(t_0\) which is the center of a vertical segment of length \(>2\pi\sigma\), on which the function \(y(t)\) is regular; moreover, if \(y(t)\) is regular in the rectangle \(R:\ |\operatorname{Re}(t-t_0)|\leqslant \varepsilon\),

\(|\operatorname{Im}(t-t_0)| \leqslant \pi \omega+\varepsilon\), then there exists a constant \(N(\varepsilon)\), independent of \(y(t)\), such that at the point \(t_0\)

\[ |M(y)|<N(\varepsilon)\max_{t\in R}|y(t)|. \tag{8} \]

2) \(M\left(e^{\pm(\mu_n+1)t}\right)=0\), whence \(M[\varphi_n(e^t)]=0\).

3) If \(y(t)\) is regular in the rectangle \(R\) and \(M(y)=0\), then in a neighborhood of the point \(t_0\) the function \(y(t)\) is represented by the absolutely convergent series

\[ y(t)=\sum c_{\pm n}e^{\pm(\mu_n+1)t}; \]

under the additional assumption that \(y(t)\) is regular in a domain of type \(D_j\), it follows from this that \(y(t)\) is regular in some half-plane \(\operatorname{Re}(t)>\alpha\), and under the assumption that \(y(t)\) is regular in a domain of type \(Q\), it is regular in the whole plane.

Let \(E_j\) be the half-strip \(\operatorname{Re}(t)\geqslant\alpha,\ |\operatorname{Im}(t-t_0)|<\pi/2(\omega_1+\varepsilon_0)\) (the quantities \(\omega_1\) and \(\varepsilon_0\) occur in (4)), contained in \(D_j\). Since for large \(\operatorname{Re}(t)\) the domain \(D_j\) approaches a strip of width \(\pi/\alpha_j\), the distance from the boundary of \(E_j\) to the boundary of \(D_j\) (in the limit as \(\operatorname{Re}(t)\to+\infty\), equal to

\[ \frac{\pi}{2\alpha_j}-\frac{\pi}{2(\omega_1+\varepsilon_0)}>\pi\varepsilon \]

) may be regarded throughout as \(>\pi\omega+\varepsilon,\ \varepsilon>0\). Taking (7) into account, in \(E_j\) we obtain

\[ \Phi_j(t)\equiv M[F_j(e^t)]=M[r_n(e^t)], \]

whence, by virtue of (8), in \(E_j\)

\[ |\Phi_j(t)|<N(\varepsilon)\max |r_n(e^\xi)|,\qquad \xi\in D_j;\qquad \operatorname{Re}(t)-\varepsilon\leqslant \operatorname{Re}(\xi)\leqslant \operatorname{Re}(t)+\varepsilon. \]

Passing again from \(t\) to \(w\) and putting \(\Phi_j(t)=\psi_j(w)\), we obtain that in a certain angle \(P_j\) of opening \(\dfrac{\pi}{\omega_1+\varepsilon_0}\)

\[ |\psi_j(w)|<N(\varepsilon)\max |r_n(\eta)|,\qquad \eta\in\Delta_j;\qquad \frac{|w|}{a}\leqslant|\eta|\leqslant a|w|,\quad a=e^\varepsilon. \]

From this point, relying on (4), we can repeat verbatim the arguments on pp. 362–363 of the article \((^2)\) and become convinced that \(\psi_j(w)\equiv0\). Thus, \(M[F_j(e^t)]\equiv0\). Hence the function \(F_j(e^t)\) is regular in some half-plane \(\operatorname{Re}(t)>\alpha\) and is represented there by a series in the functions \(e^{\pm(\mu_n+1)t}\), while the function \(F_j(w)\) is regular for \(|w|>\mathrm{const}\) and is represented by a Laurent series in powers \(w^{\pm(\mu_n+1)}\). Since \(F_j(w)\) is bounded (this is clear from (6)) in an angle of opening \(>2\pi\sigma\) (for this angle one may take any angle internal to \(\Delta_j\) with sides parallel to the sides of \(\Delta_j\)), it follows, by virtue of (3), that the Laurent series for \(F_j(w)\) contains no positive powers and, consequently, the function \(F_j(w)\) is regular at \(\infty\). As for the function \(F_1(w)\), since \(F_1(e^t)\) is regular in \(Q\) and, consequently, is an entire function, it is regular everywhere for \(|w|>0\). We shall verify that \(F_1(w)=F_2(w)=\cdots=F_m(w)\).

Let \(\mu(z)\) be the angle formed by the tangent to \(L\) at the point \(z\) with the positive direction of the real axis. We have

\[ F(w)=\frac{1}{2\pi i}\int_L \frac{e^{-p(z)}\overline{f(z)}e^{-i\mu(z)}\,dz}{z-w}. \]

If \(\Gamma\) is a part of \(L\) to one side of which the domain \(G_k\) adjoins, and to the other side the domain \(G_s\), then, by a known property of the Cauchy-type integral, almost everywhere on \(\Gamma\)

\[ e^{-p(z)}\overline{f(z)}e^{-i\mu(z)}=\pm(F_k(z)-F_s(z)), \]

whence

\[ \overline{f(z)}=\pm(F_k(z)-F_s(z))e^{p(z)}e^{i\mu(z)}. \tag{9} \]

If it were the case that \(F_k(w)-F_s(w)\ne 0\) and, consequently, for large \(|w|\)

\[ F_k(w)-F_s(w)=\frac{\mathrm{const}}{w^p}, \]

then a function \(f(z)\) of the form (9) would not satisfy condition (1). Hence \(F_1(w)=\ldots=F_m(w)\) and, by virtue of (4), \(f(z)=0\) almost everywhere on \(L\), as was required to prove.

In \((^2)\), from completeness in the sense of (2), there is derived as a consequence (under the additional assumption that \(L\) consists of one connected piece) the completeness of polynomials on \(L\) in the sense

\[ \inf_{\{Q\}}\max_{z\in L} e^{-p(z)}|f(z)-Q(z)|=0 \tag{10} \]

in the class \(C[p(z)]\) of functions \(f(z)\) continuous on \(L\) with the property \(e^{-p(z)}f(z)\to 0\) as \(z\to\infty\), \(z\in L\). In exactly the same way, under the conditions of our theorem one can derive the completeness of \(\{1,z^{\lambda_k}\}\) on \(L\) in the class \(C[p(z)]\) in the sense of (10), where \(Q(z)\) denotes all possible linear combinations of the functions from \(\{1,z^{\lambda_k}\}\).

The completeness of the system \(\{1,z^{\lambda_k}\}\) in this sense in the cases when \(L\) is either the whole axis \((-\infty,\infty)\), or the half-axis \((0,+\infty)\) (in the latter case the \(\lambda_k\) are not necessarily integers) was considered by S. Mandelbrojt \((^5)\). M. M. Dzhrbashyan \((^4)\) considered the same question in the case when \(p(z)=|z|^p\) and \(L\) is topologically equivalent to the axis \((-\infty,\infty)\) and is situated between two angles of definite openings with vertices at the points \(0\) and \(\alpha>0\).

In conclusion we give an example indicating the essential nature in the theorem of the requirement that the domain \(G_j\) contain an angle of opening precisely \(>2\pi\sigma\), and the domain \(G_1\) a curvilinear angle of opening \(>2\pi\sigma\) with vertex precisely at the origin. Let \(\mu_k=kp+p-1\) \((k=1,2,\ldots)\), where \(p\) is an integer \(\ge 2\). Then \(\sigma=\frac1p\), and \(\lambda_k\) has the form \(mp+j\), \(j=0,1,\ldots,p-2\).

It is not difficult to see that if on the rays \(\arg z=\frac{2\pi}{p}s\) \((s=0,1,\ldots,p-1)\) one takes points \(\alpha_s\) at equal distances from the origin, then the function \(z^{p-1}\) cannot be approximated simultaneously at all the points \(\alpha_s\) \((s=0,1,\ldots,p-1)\), with arbitrary accuracy, by linear combinations of the functions \(z^{\lambda_k}\) with the indicated \(\lambda_k\). Hence it follows that:

1) The theorem is not true if, as \(L\), one takes a system (connected with one another in some way) of rays \(\arg z=\frac{2\pi}{p}s\), \(|z|\ge r_s\), with arbitrary \(r_s\). This shows the essential nature of the requirement that the angle \(\Delta_j\) have opening \(>2\pi\sigma\), since in our case \(\Delta_j\) has opening \(\frac{2\pi}{p}=2\pi\sigma\).

2) The theorem is not true if \(L\) consists of the ray \(\arg z=0\) and an arc of the circle \(|z|=r\), \(0\le \arg z\le \frac{2\pi}{p}(p-1)\). This shows the essential nature of the requirement that \(G_1\) contain an angle \(P\) of opening \(>2\pi\sigma\), since in our case the angle \(P\) has opening \(2\pi\sigma\).

Moscow Power Engineering
Institute

Received
7 IV 1958

CITED LITERATURE

\(^1\) M. M. Dzhrbashyan, Dokl. AN ArmSSR, 7, No. 2 (1947).
\(^2\) M. M. Dzhrbashyan, Matem. sborn., 36 (78), 3 (1955).
\(^3\) A. F. Leont’ev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 39 (1951).
\(^4\) M. M. Dzhrbashyan, DAN, 67, No. 1 (1949).
\(^5\) S. Mandelbrojt, Contiguous series, regularization of sequences, applications, 1955.
\(^6\) M. M. Dzhrbashyan, I. O. Khachatryan, DAN, 110, No. 6 (1956).