UDC 517.521.8

MATHEMATICS

I. I. VOLKOV

LINEAR TRANSFORMATIONS OF UNBOUNDED COMPLEX SEQUENCES

(Presented by Academician A. N. Kolmogorov, 21 V 1966)

Let \(G\) be an unbounded domain of the complex plane. A sequence \(\{s_n\}\) tends to \(\infty\) in the domain \(G\), which is denoted by

\[ s_n \to \infty G, \]

if: 1) \(s_n \in G\) for \(n>N\), where \(N\) in general depends on \(\{s_n\}\) and \(G\); 2) \(\lim\limits_{n\to\infty}|s_n|=\infty\). The domain \(G\) itself in this case will be called the domain of convergence of the sequence \(\{s_n\}\) to \(\infty\).

Let \((a_{mn})\) \((m,n=1,2,\ldots)\) be a regular matrix with complex entries, and let \(\{\sigma_m\}\) be the sequence obtained as the result of the linear transformation of the sequence \(\{s_n\}\) by the matrix \((a_{mn})\):

\[ \sigma_m=\sum_{n=1}^{\infty} a_{mn}s_n \quad (m=1,2,\ldots). \tag{1} \]

The following question is posed: what conditions must the matrix \((a_{mn})\) satisfy so that from the convergence \(s_n\to \infty G\) there follows the convergence \(\sigma_m\to \infty G'\), where \(G\) and \(G'\) are certain unbounded domains. In this formulation of the question, the concept of a fully regular transformation, i.e. a transformation for which from \(s_n\to +\infty\) there follows \(\sigma_m\to +\infty\), is in a certain sense carried over to complex sequences and matrices. Obviously, in our case the conditions on the matrix \((a_{mn})\) will depend on the form of the domains \(G\) and \(G'\) (or on the parameters determining these domains). Some cases in which the domains of convergence of the sequences \(\{s_n\}\) and \(\{\sigma_m\}\) to \(\infty\) were angles and half-strips were considered in \((^1)\). In the present note more general results are given, pertaining to domains of the indicated types, as well as to arbitrary convex domains.

We first recall some definitions and notation. By \(\Phi=\Phi(z,\alpha,\varphi)\) is denoted an angle with vertex at the point \(z\), direction \(\alpha^*\), and magnitude \(\varphi\) \((0\leq \varphi\leq 2\pi)\). By \(\Pi\equiv \Pi(z,\alpha,h)\) is denoted a half-strip of width (base) \(h\), direction \(\alpha\), and with the midpoint of its base at the point \(z\). A ray is regarded as an angle for \(\varphi=0\) or a half-strip for \(h=0\). By \(G_r\) is denoted the domain consisting of the domain \(G\) with the addition to it of all points whose distance from \(G\) is not greater than \(r\). All domains are regarded as closed. If for the sequence \(\{s_n\}\) the convergence \(s_n\to \infty G_r\) takes place for every \(r>0\), then we shall say that \(\{s_n\}\) converges asymptotically to \(\infty\) in the domain \(G\) and denote this by \(s_n\to \infty\ \mathrm{as}\ G\). The matrix \((a_{mn})\) will everywhere be assumed regular and row-finite.

We shall first give results showing that the linear transformation (1) cannot contract or shift the domain of convergence of each sequence. This follows from the following theorem:

* As the direction of the angle \(\Phi\) we take the direction of the bisector of the angle; as the direction of the half-strip \(\Pi\), the direction of the rays bounding the half-strip.

Theorem 1. Let \(G\) be an arbitrary unbounded domain. Whatever regular matrix \((a_{mn})\) is taken, it is always possible to construct a sequence \(\{s_n\}\), \(s_n \to \infty\, G\), such that for the sequence \(\{\sigma_m\}\) convergence \(\sigma_m \to \infty\) as \(G\) over this same domain \(G\) will take place.

From this theorem there follows

Corollary. Let \(G\) and \(G'\) be two arbitrary unbounded closed domains, and suppose that outside \(G'\) there lies an infinite part of \(G\). There is no regular matrix \((a_{mn})\) which would transform every sequence \(\{s_n\}\) converging to \(\infty\) in the domain \(G\) into a sequence \(\{\sigma_m\}\) converging to \(\infty\) in the domain \(G'\).

From the results stated it follows that, in studying conditions under which from the convergence \(s_n \to \infty\, G\) there always follows the convergence \(\sigma_m \to \infty\, G'\), it is of interest to consider the case when \(G \subset G'\).

In the following theorems, conditions are established on the transformation matrix for certain kinds of domains. By \(\widehat{a_0a_1}\) is denoted the magnitude of the (smallest) angle through which a ray with direction \(a_0\) must be turned in order to coincide with the ray having direction \(a_1\); the angle \(\widehat{a_0a_1}\) is considered positive if this rotation is made counterclockwise, and negative in the opposite case. By \(\arg z\) is denoted the principal value of the argument of the complex number \(z\): \(-\pi < \arg z \leq \pi\), and, by definition, \(\arg 0 = 0\).

Theorem 2. Let \(\Phi(z_0,\alpha_0,\varphi_0)\) and \(\Phi(z_1,\alpha_1,\varphi_1)\) be fixed angles, \(0 \leq \varphi_0 \leq \varphi_1 < 2\pi\), \(\beta=\widehat{\alpha_0\alpha_1}\). The condition
\[ |\arg a_{mn}-\beta| \leq (\varphi_1-\varphi_0)/2 \qquad (m>m_0,\ n>n_0)\ * \tag{2} \]
is necessary in order that from the convergence
\[ s_n \to \infty\, \Phi(z_0,\alpha_0,\varphi_0) \tag{3} \]
there should always follow the convergence
\[ \sigma_m \to \infty\, \Phi_r(z_1,\alpha_1,\varphi_1) \tag{4} \]
at least for one \(r \geq 0\), in general depending on \(\{s_n\}\); for \(0 \leq \varphi_0 \leq \varphi_1 < \pi\) condition (2) is also sufficient in order that from convergence (3) there should follow convergence (4) for every \(r>|z_1-z_0|\).

Let us note that condition (2) for \(0 \leq \varphi_0 \leq \varphi_1 < 2\pi\) is no longer sufficient for (4) to follow from (3) for some \(r\) (see Theorem 6 below).

Let us mention one corollary of this theorem.

Corollary. Let \(\Phi(z_0,\alpha_0,\varphi_0)\) and \(\Phi(z_0,\alpha_0,\varphi_1)\) \((0 \leq \varphi_0 \leq \varphi_1 < \pi)\) be fixed angles with common vertex \(z_0\) and one and the same direction \(\alpha_0\). In order that from the convergence \(s_n \to \infty\, \Phi(z_0,\alpha_0,\varphi_0)\) there should always follow the convergence \(\sigma_m \to \infty\) as \(\Phi(z_0,\alpha_0,\varphi_1)\), it is necessary and sufficient that the matrix \((a_{mn})\) satisfy the condition:
\[ |\arg a_{mn}| \leq \frac{\varphi_1-\varphi_0}{2} \qquad (m>m_0,\ n>n_0). \]

For \(\varphi_0=\varphi_1\) we obtain Theorem 6 from (1), and for \(z_0=\alpha_0=\varphi_0=\varphi_1=0\), the conditions of “weakened” complete regularity.

In the following theorem the position of the angle of convergence of the sequence \(\{\sigma_m\}\) to \(\infty\) is not fixed in advance, but only it is required that its magnitude not exceed the given value \(\varphi_1\).

Theorem 3. Let \(\Phi(z_0,\alpha_0,\varphi_0)\) be a fixed angle and \(0 \leq \varphi_0 \leq \varphi_1 < 2\pi\). If the matrix \((a_{mn})\) has the property that from the converg—

* That is, there exist natural numbers \(m_0\) and \(n_0\) such that for \(m>m_0,\ n>n_0\) the indicated inequality holds; this is meant everywhere below.

of the convergence \(s_n\to\infty\,\Phi(z_0,\alpha_0,\varphi_0)\) there always follows the convergence \(\sigma_m\to\infty\,\Phi(z,\alpha,\varphi)\) for some \(z,\alpha\) and \(\varphi,\ \varphi\leq\varphi_1\), in general depending on \(\{s_n\}\), then there necessarily exists such a direction \(\alpha_1\) that, for any sequence \(\{s_n\}\) satisfying the condition \(s_n\to\infty\,\Phi(z_0,\alpha_0,\varphi_0)\), there will hold the convergence \(\sigma_m\to\infty\,\Phi_r(z_0,\alpha_1,\varphi_1)\) for some \(r>0\), in general depending on \(\{s_n\}\).

From Theorems 2 and 3 it follows that

Corollary. Let \(\Phi(z_0,\alpha_0,\varphi_0)\) be a fixed angle and let \(0\leq\varphi_0\leq \varphi_1<2\pi\). In order that from the convergence \(s_n\to\infty\,\Phi(z_0,\alpha_0,\varphi_0)\) there always follow the convergence \(\sigma_m\to\infty\,\Phi(z,\alpha,\varphi)\) for some \(z,\alpha\) and \(\varphi,\ \varphi\leq\varphi_1\), in general depending on \(\{s_n\}\), it is necessary that there exist a real number \(\beta\) such that

\[ |\arg a_{mn}-\beta|\leq(\varphi_1-\varphi_0)/2\qquad (m>m_0,\ n>n_0). \tag{5} \]

For \(0\leq\varphi_0\leq\varphi_1<\pi\) this condition is also sufficient for the convergence \(\sigma_m\to\infty\) as \(\Phi(z_0,\alpha_1,\varphi_1)\) to take place, where \(\alpha_1=\alpha_0+\beta\).

If, for the magnitude \(\varphi\) of the angle of convergence of the sequence \(\{\sigma_m\}\), the restriction \(\varphi\leq\varphi_1<\pi\) is replaced simply by the inequality \(\varphi<\pi\), then the conditions on the transformation matrix change, as the following theorem shows.

Theorem 4. Let \(\Phi(z_0,\alpha_0,\varphi_0)\) \((\varphi_0<\pi)\) be a fixed angle. In order that from the convergence \(s_n\to\infty\,\Phi(z_0,\alpha_0,\varphi_0)\) there always follow the convergence \(\sigma_m\to\infty\,\Phi(z,\alpha,\varphi)\) for some \(z,\alpha,\varphi\) \((\varphi<\pi)\), in general depending on \(\{s_n\}\), it is necessary and sufficient that there exist a real number \(\beta\) such that

\[ \varlimsup_{\substack{m\to\infty\\ n\to\infty}}|\arg a_{mn}-\beta|=C<(\pi-\varphi_0)/2. \]

In the following theorem, conditions on the matrix are established in the case when, for the magnitude of the angle of convergence of the sequence \(\{\sigma_m\}\) to \(\infty\), an \(\varepsilon\)-enlargement relative to a fixed value \(\varphi_1\) is allowed.

Theorem 5. Let \(\Phi(z_0,\alpha_0,\varphi_0)\) and \(\Phi(z_1,\alpha_1,\varphi_1)\) \((0\leq\varphi_0\leq\varphi_1<\pi)\) be fixed angles, \(\beta=\widehat{\alpha_0\alpha_1}\). In order that from the convergence \(s_n\to\infty\,\Phi(z_0,\alpha_0,\varphi_0)\) there always follow the convergence \(\sigma_m\to\infty\,\Phi(z_1,\alpha_1,\varphi_1+\varepsilon)\) for every \(\varepsilon>0\) \((\varphi_1+\varepsilon<\pi)\), it is necessary and sufficient that

\[ \varlimsup_{\substack{n\to\infty\\ m\to\infty}}|\arg a_{mn}-\beta|\leq(\varphi_1-\varphi_0)/2. \]

Hence, as a special case, for \(z_0=z_1,\ \alpha_0=\alpha_1,\ \varphi_0=\varphi_1\), Theorem 7 from [1] follows.

For angles whose magnitudes \(\varphi_0\) and \(\varphi_1\) are subject to the condition \(\pi\leq\varphi_0\leq\varphi_1<2\pi\), the following result holds. By \((a_{mn})_p\) is denoted the matrix obtained from the matrix \((a_{mn})\) by replacing with zeros all elements for which either \(m<p\) or \(n<p\).

Theorem 6. Let \(\Phi(z_0,\alpha_0,\varphi_0)\) be a fixed angle and let \(\pi\leq\varphi_0\leq\varphi_1<2\pi\). If the matrix \((a_{mn})\) has the property that from the convergence \(s_n\to\infty\,\Phi(z_0,\alpha_0,\varphi_0)\) there always follows the convergence \(\sigma_m\to\infty\,\Phi(z,\alpha,\varphi)\) for some \(z,\alpha,\varphi,\ \varphi\leq\varphi_1\), in general depending on \(\{s_n\}\), then it necessarily satisfies condition (5) for some \(\beta\), and, moreover, there exists such a \(p\) that the matrix \((a_{mn})_p\) contains in each row no more than one element different from zero.

It follows from this theorem that if a regular matrix \((a_{mn})\) has the property formulated in the theorem, then in fact in this case from the convergence \(s_n\to\infty\,\Phi(z_0,\alpha_0,\varphi_0)\) there will always follow the convergence \(\sigma_m\to\infty\,\Phi(z_0,\alpha_0,\varphi_0+\varepsilon)\) for every \(\varepsilon>0\), and, moreover, for \(\varphi_0=\varphi_1\) the matrix \((a_{mn})\) must satisfy the condition \(\arg a_{mn}=0\) \((m>m_0,\ n>n_0)\), and in this case the convergence \(\sigma_m\to\infty\) as \(\Phi(z_0,\alpha_0,\varphi_0)\) will take place.

The following result concerns the transformation of sequences converging to \(\infty\) in a half-strip into sequences converging to \(\infty\) in an angle (with an \(r\)-extension).

Theorem 7. Let \(\Pi(z_0,\alpha_0,h_0)\) and \(\Phi(z_1,\alpha_1,\varphi_1)\) \((\varphi_1<\pi)\) be a fixed half-strip and angle. The condition

\[ \left|\arg a_{mn}-\beta\right|\leq \varphi_1/2 \qquad (m>m_0,\ n>n_0), \]

where \(\beta=\widehat{\alpha_0\alpha_1}\), is necessary in order that from the convergence

\[ s_n\to\infty\ \Pi(z_0,\alpha_0,h_0) \tag{6} \]

there should always follow the convergence

\[ \sigma_m\to\infty\ \Phi_r(z_1,\alpha_1,\varphi_1) \tag{7} \]

for at least one \(r\geq 0\), and is sufficient in order that (7) follow from (6) for every \(r>r_1\), where

\[ r_1=|z_1-z;\,|+\frac{h_0}{2}\sup_m\sum_{n=1}^{\infty}|a_{mn}|. \]

For \(\varphi_1=0\), this theorem yields conditions on the matrix for transforming a sequence \(\{s_n\}\), converging to \(\infty\) in a half-strip, into a sequence \(\{\sigma_m\}\) also converging to \(\infty\) in a half-strip. By virtue of the regularity of the matrix \((a_{mn})\), in this case it is necessary that \(\beta=0\).

The following theorem concerns transformations of sequences converging to \(\infty\) in a convex domain.

Theorem 8. Let \(G\) be an arbitrary convex domain contained in some angle smaller than \(\pi\). If for every such domain the convergence

\[ s_n\to\infty\ G \tag{8} \]

always implies the convergence

\[ \sigma_m\to\infty\ G_r \tag{9} \]

for at least one \(r\geq 0\), in general depending on \(\{s_n\}\), then the matrix \((a_{mn})\) must satisfy the conditions:

1) \(\arg a_{mn}=0\) \((m>m_0,\ n>n_0)\),

2) there exist numbers \(M\) and \(N\) such that

\[ \sum_{n=N}^{\infty} a_{mn}\leq 1 \quad \text{for all } m>M. \]

These conditions are sufficient in order that (9) follow from (8) for every \(r>0\) (i.e., in order that the convergence \(\sigma_m\to\infty\) as \(G\) hold).

Let us note that the necessity of conditions 1) and 2) in this theorem is understood in the sense that, if either of them is violated, then there exists a convex domain \(G\), contained in some angle smaller than \(\pi\), and a sequence \(\{s_n\}\) satisfying condition (8), for which convergence (9) does not occur for any \(r\).

Moscow Institute of Engineers
of Agricultural Production

Received
17 V 1966

CITED LITERATURE

1. I. I. Volkov, DAN, 165, No. 4, 742 (1965).