MATHEMATICS

S. V. VARGANOVA and Yu. F. KOROBEINIK

ON SUPERCONVERGENCE AND NONCONTINUABILITY OF FUNCTIONAL SERIES

(Presented by Academician A. N. Kolmogorov, 22 X 1960)

Studying lacunary power series, Ostrovskii showed that these series superconverge in a neighborhood of every regular point on the boundary of the circle of convergence \((^1)\). He also proved the converse theorem, according to which every power series possessing a partial sequence uniformly convergent in a neighborhood of some regular point on the boundary of the circle of convergence is a series of lacunary structure. Thus, the existence of a superconvergent partial sequence completely determines the lacunary structure of the series. It should be noted that Ostrovskii’s converse theorem is proved much more difficultly than the direct one. Ostrovskii’s works were developed in Burion’s papers \((^2)\), which obtained important results on superconvergent power series. Recently there have appeared papers \((^{3–6})\) in which the superconvergence and noncontinuability of functional series were studied (mainly, series in polynomials).

In the present note Ostrovskii’s direct theorem is carried over to the case of functional series of a rather general nature; the result obtained includes, as special cases, some results from \((^{3,5,6})\). In addition, with the aid of certain characteristics introduced by M. A. Evgrafov \((^7)\), an assertion is proved which generalizes Ostrovskii’s converse theorem.

For the formulation of the results obtained we shall need certain concepts and definitions.

Let \(u(z) \ge 0\) be a continuous function of the variable \(z\) such that the equations \(u(z)=\rho\) represent rectifiable curves \(C_\rho\), surrounding the origin, which, as \(\rho\) decreases, contract to the point \(z=0\); moreover the curve \(C_{\rho_1}\) lies inside the curve \(C_{\rho_2}\) if \(\rho_1<\rho_2\). As in \((^7)\), by \(D_\rho\) we denote the simply connected domain lying inside \(C_\rho\), and by the domain \(R_1<u(z)<R_2\) we shall mean the ring-shaped domain between \(C_{R_1}\) and \(C_{R_2}\).

We shall say that a system of functions \(\{\varphi_n(z)\}\) possesses property \((S)\) in the domain \(R_1<u(z)<R_2\), if for every \(\varepsilon>0\) and any \(r_1\) and \(r_2\), \(R_1<r_1<r_2<R_2\), one can indicate a constant \(A\) and a number \(N\) such that for all \(n>N\) and all \(z\) in the domain \(r_1 \le u(z) \le r_2\)

\[ |\varphi_n(z)|<A\,(u(z)+\varepsilon)^n . \tag{1} \]

It is obvious that if the system of functions \(\{\varphi_n(z)\}\) satisfies condition \((S)\), then for every \(\rho\) from \((R_1,R_2)\)

\[ \overline{\lim}_{n\to\infty}\sqrt[n]{\max_{z\in C_\rho}|\varphi_n(z)|}\le \rho . \tag{2} \]

It is easy to show that condition \((S)\) is satisfied by the regular systems of M. A. Evgrafov \((^7)\), almost regular systems \((^6)\), and also by the quasi-power-like internally continuable bases of M. G. Khaplanov \((^8)\).

Functional series

\[ f(z)=\sum_{n=0}^{\infty} a_n\varphi_n(z) \tag{3} \]

with coefficients satisfying the condition

\[ \varlimsup_{n\to\infty}\sqrt[n]{|a_n|}=\frac1R \qquad (R_1<R<R_2), \tag{4} \]

converges uniformly in every closed domain lying inside the domain \(R_1<u(z)<R\).

The series (3) is called a series of lacunary structure if it can be split into two series in such a way that one series has gaps of relative length bounded below, while the other has a larger radius of convergence.

It follows from the definition that \(a_\nu=a'_\nu+a''_\nu\), where

\[ \varlimsup_{\nu\to\infty}\sqrt[\nu]{|a'_\nu|}=\frac1R, \]

\[ \varlimsup_{\nu\to\infty}\sqrt[\nu]{|a''_\nu|}=\frac1r \qquad (r>R) \tag{5} \]

and the coefficients \(a'_\nu=0\) for \(m_k<\nu<n_k\) \((k=1,2,\ldots)\), \(n_k>\lambda m_k\), \(\lambda>1\).

In the paper [7], M. A. Evgrafov introduced the concept of a first reduced system. Namely, he calls a system of functions \(\{\varphi_n(z)\}\), analytic in some neighborhood of the origin, a first reduced system if \(\varphi_n(z)=\sum_{k=n}^{\infty} a_{n,k}z^k\) \((a_{n,n}\ne0,\ n=0,1,2,\ldots)\). If the first reduced system is regular and forms a basis in \(D_\rho\) for all \(\rho\) from \((R_1,R_2)\), then we shall call such a system a regular first reduced basis.

Theorem 1. Let the functions \(\{\varphi_n(z)\}\) be analytic in the domain \(R_1<u(z)<R_2\) and satisfy condition (S) there. In order that the series (3), under condition (4), be a series of lacunary structure, it is necessary, and in the case where \(\ln u(z)\) is a superharmonic function in \(R_1<u(z)<R_2\) and \(\{\varphi_n(z)\}\) is a regular first reduced basis, also sufficient, that there exist a subsequence \(S_{n_k}(z)\) of the series (3) possessing the following property: in some bounded domain \(G\), exterior to \(C_R\), the inequalities

\[ \frac1{n_k}\ln |S_{n_k}(z)|<\psi(z)+\varepsilon_{n_k}(z), \qquad k=1,2,\ldots, \]

hold, where \(\psi(z)\) is a continuous function in \(G\) satisfying there the condition

\[ \psi(z)<\ln\frac{u(z)}{R}, \]

and \(\varepsilon_{n_k}(z)\to0\) uniformly in this domain.

Putting, in particular, \(\varphi_n(z)=z^n\), and \(u(z)=|z|\), we obtain Bourion’s theorem [2].

Theorem 2 (an analogue of Ostrowski’s direct theorem). If the series (3), under condition (4), is a series of lacunary structure, i.e. the relations (5) hold, the system of functions \(\{\varphi_n(z)\}\) satisfies condition (S), and \(\ln u(z)\) is a superharmonic function in the domain \(R_1<u(z)<R_2\), then the subsequence of partial sums

\[ S_{m_k}(z)=\sum_{n=0}^{m_k} a_n\varphi_n(z) \]

converges uniformly in a neighborhood of each point of regularity of \(f(z)\) lying on \(C_R\).

Theorem 2 contains the result of paper \((^3)\), as well as Theorem 2 from \((^6)\). From Theorem 2 there follows an assertion generalizing the well-known theorem of Alamar \((^1)\):

Theorem 3. Let the functions \(\{\varphi_n(z)\}\) form a regular system in \(D_{R_2}\), or an almost regular system in the domain \(R_1<u(z)<R_2\), and let \(\ln u(z)\) be a superharmonic function in this domain. Then, if the indices of the nonzero coefficients of the series (3), under condition (4), form a sequence \(\{n_m\}\) in which \(n_{k+1}>(1+\theta)n_k,\ \theta>0\), then the curve \(C_R\) is a cut for the function \(f(z)\).

Theorem 4. Suppose that the functions \(\varphi_n(z)\) are analytic in \(D_{R_2}\) and constitute a regular first reduced basis. Suppose further that \(\ln u(z)\) is a superharmonic function in \(R_1<u(z)<R_2\). Then every series (3), under condition (4), having a subsequence \(S_{n_k}(z)\) of partial sums that is overconvergent in a neighborhood of some point of \(C_R\), is a series of lacunary structure.

Remark. Theorem 4 will remain true if (with the other indicated conditions satisfied) the series (3) has not an overconvergent partial subsequence \(S_{n_k}(z)\), but a partial subsequence \(\{S_{n_k}(z)\}\) bounded in some sufficiently small neighborhood of a point \(z_0\in C_R\).

We shall call a point \(z_0\) on the contour \(C_R\) a generalized singular point for the function \(f(z)\), defined by equality (3) under condition (4), if no subsequence of the series
\[ \sum_{n=0}^{\infty} a_n\varphi_n(z) \]
converges uniformly in a neighborhood of the point \(z_0\). Then the following result, which follows from Theorem 4, is valid:

Theorem 5. If the series (3), under condition (4), in functions of a regular first reduced basis is not a series of lacunary structure, and \(\ln u(z)\) is a superharmonic function in \(R_1<u(z)<R_2\), then all points on \(C_R\) are generalized singular points of the function \(f(z)\).

Rostov-on-Don
State University

Received
21 X 1960

CITED LITERATURE

\(^1\) E. Titchmarsh, Theory of Functions, Moscow–Leningrad, 1951.
\(^2\) G. Bourion, Ann. Écol. Norm., 50, 245 (1933).
\(^3\) S. Ya. Al’per, DAN, 59, No. 41, 625 (1948).
\(^4\) E. Z. Shul’man, Matem. sborn., 31 (73), 76 (1952).
\(^5\) T. I. Krasnoshchekova, DAN, 77, No. 5 (1951).
\(^6\) Yu. F. Korobeinik, UMN, 15, issue 4 (94), 149 (1960).
\(^7\) M. A. Evgrafov, Tr. Mosk. matem. obshch., 5, 89 (1956).
\(^8\) M. G. Khaplanov, DAN, 80, No. 1, 21 (1951).