Mathematics

V. P. Khavin

On the Separation of Singularities of Analytic Functions

(Presented by Academician V. I. Smirnov on 12 III 1958)

In the present note we prove several theorems on the decomposition of an analytic function into a sum of functions having, as their set of singular points, a prescribed part of the set of singularities of the original function. In doing so it proves convenient to apply certain considerations from the theory of linear topological spaces.

Theorem 1. Let \(F_0, F_1, F_2\) be closed sets of the extended complex plane; let \(G_0, G_1, G_2\) be their open complements, with \(F_0 = F_1 \cup F_2\). If the function \(u_0(z)\) is analytic in \(G_0\), then there exist functions \(u_1(z)\) and \(u_2(z)\) such that in \(G_0\)

\[ u_0(z)=u_1(z)+u_2(z); \]

\(u_i(z)\) is analytic in \(G_i\) \((i=1,2)\).

Remark. Everywhere below we shall assume that the functions under consideration are equal to zero at the point at infinity, whenever it is contained in their domain of definition.

In the case when \(F_0\) is the real line, \(F_1=[-1,+1]\), \(F_2=(-\infty,-1]\cup[+1,+\infty)\), this theorem was proved by Poincaré \((^{1,2})\).

Lemma. Let \(X_0, X_1, X_2\) be locally convex Hausdorff linear topological spaces; \(X_0 \subset X_1\), \(X_0 \subset X_2\) (in the sense that \(X_0\) is algebraically isomorphic to a part of \(X_i\), \(i=1,2\)); the topology in \(X_0\) is the least upper bound of the topologies induced in \(X_0\) from \(X_1\) and \(X_2\). If \(f_0\) is an additive continuous functional given in \(X_0\), then there exist two functionals \(f_1\) and \(f_2\) such that \(f_0(x)=f_1(x)+f_2(x)\) \((x\in X_0)\), and the functional \(f_i\) is continuous in the topology induced on \(X_0\) from \(X_i\) \((i=1,2)\).

Proof. Consider the product of the spaces \(X_1\times X_2\) and in it the closed linear subspace \(\widetilde X_0\), consisting of all pairs of the form \((x,x)\), where \(x\in X_0\). Let \(\widetilde f((x,x))=f(x)\). It is easy to see that the functional \(\widetilde f\) is continuous in the topology induced on \(\widetilde X_0\) from \(X_1\times X_2\). Let \(\widetilde f\) be a linear functional extending \(\widetilde f\) from \(\widetilde X_0\) to all of \(X_1\times X_2\) with preservation of continuity \((^3)\). Put now \(f_1(x)=\widetilde f((x,0))\), \(f_2(x)=\widetilde f((0,x))\). The lemma is proved.

Now let \(X_{F_0}\) \((X_{F_1}, X_{F_2})\) be the totality of all functions analytic on \(F_0\) (respectively \(F_1, F_2\)). Let \(g_n^{(j)}\) \((j=0,1,2)\) be a decreasing sequence of open sets converging to \(F_j\) as to a kernel; let \(B_n^{(j)}\) be the normed space of functions analytic and bounded in \(g_n^{(j)}\), with norm

\[ \|f\|_n^{(j)}=\sup_{z\in g_n^{(j)}} |f(z)|. \]

Then \(B_n^{(j)}\subset B_{n+1}^{(j)}, n=1,2,\ldots,\)

\[ X_{F_j}=\bigcup_{n=1}^{\infty} B_n^{(j)}. \]

Let in \(X_{F_j}\) there be introduced the topology of the inductive limit of the sequence of spaces \(B_n^{(j)}\) \((^4)\). Then the spaces \(X_j=X_{F_j}\) turn out to be the so-called spaces of type \((LN)^*\) \((^{5,6})\).

We turn to the proof of Theorem 1. Let

\[ f_0(x)=\frac{1}{2\pi i}\int_{C_x} x(z)u_0(z)\,dz, \]

where \(x\in X_{F_0}\); \(C_x\) is a finite system of closed simple rectifiable pairwise nonintersecting contours forming the boundary of a domain containing \(F_0\) inside, and \(x(z)\) is analytic in the closure of this domain. The functional \(f_0(x)\) is continuous in \(X_{F_0}\). Put

\[ \psi_{\widetilde z}(z)=\frac{1}{z-\widetilde z}. \]

If \(\widetilde z\in F_0\), then \(\psi_{\widetilde z}\in X_{F_0}\), and

\[ u_0(\widetilde z)=f_0(\psi_{\widetilde z}). \]

Using the lemma, we find functionals \(f_1\) and \(f_2\), continuous respectively in the spaces \(X_1\) and \(X_2\), such that

\[ u(\widetilde z)=f_0(\psi_{\widetilde z})=f_1(\psi_{\widetilde z})+f_2(\psi_{\widetilde z}). \]

Let \(u_j(\widetilde z)=f_j(\psi_{\widetilde z})\), \(\widetilde z\in G_j\) \((j=1,2)\). Using the continuity of the functional \(f_j\) in the space \(X_{F_j}\), it is not difficult to prove the existence of the derivative \(u'_j(\widetilde z)\) at all points of \(G_j\). The theorem is proved.

We note a consequence of Theorem 1.

Theorem 2. Let \(G_1\) and \(G_2\) be two domains having a nonempty intersection \(G_1\cap G_2\). Suppose that the system of regular functions \(\{\theta_k^{(j)}(z)\}_{k=1}^{\infty}\) \((j=1,2)\) is complete in the domain \(G_j\) (in the sense of uniform convergence on compact subsets of \(G_j\)). Then the system \(\{\theta_k^{(1)}(z),\theta_k^{(2)}(z)\}_{k=1}^{\infty}\) is complete in \(G_1\cap G_2\).

Indeed, if the function \(u(z)\) is analytic in \(G_1\cap G_2\), then, by Theorem 1, \(u(z)=u_1(z)+u_2(z)\), where \(u_k(z)\) is analytic in \(G_k\). Hence it is clear that the linear span of the system \(\{\theta_k^{(1)}(z),\theta_k^{(2)}(z)\}_{k=1}^{\infty}\) is dense in the set of functions analytic in \(G_1\cap G_2\).

In Theorem 1 nothing is said about the behavior of the functions \(u_1\) and \(u_2\) near their sets of singularities. An idea of their properties can sometimes be obtained by knowing the behavior of the function \(u_0(z)\) itself.

Theorem 3. Let \(u_0(z)\) be a function analytic in the disk \(|z|<1\) and such that:

\[ 1)\ \int_0^\theta |u_0(re^{i\varphi})|^p\,d\varphi<C<\infty,\quad p>1;\quad \theta\text{ is some number from the interval }(0,2\pi]; \]

\[ 2)\ \text{the integrals }\int_0^1 |u_0(r)|\,dr,\quad \int_0^1 |u_0(re^{i\theta})|\,dr\text{ converge.} \]

Then there exists a function \(\Psi(\varphi)\), \(0\leq\varphi\leq\theta\), such that

\[ \int_0^\theta |\Psi(\varphi)|^p\,d\varphi<+\infty, \]

\[ u_0(z)=\int_0^\theta \frac{\Psi(\varphi)\,d\varphi}{e^{i\varphi}-z}+u_1(z),\quad |z|<1, \]

where \(u_1(z)\) is a function regular everywhere outside the arc \(z=e^{i\varphi}\), \(\theta\leq\varphi\leq 2\pi\).

Proof. Let \(X_0\) be the set of all functions analytic in the closed exterior of the unit circle and equal to zero at the infinitely distant point. Put further: \(X_1\) is the set of functions analytic on the arc \(z=e^{i\varphi}\), \(\theta\leq\varphi\leq 2\pi\); \(X_2\) is the set of all complex-valued functions summable on the arc \(z=e^{i\varphi}\), \(0\leq\varphi\leq\theta\), with exponent \(q=\frac{p}{p-1}\). Let \(B_n\) be the Banach space consisting of functions analytic in the closure of the domain whose boundary is ...

the curve \(\Gamma_n\), consisting of the arcs \(z=\left(1-\dfrac1n\right)e^{i\varphi}\), \(z=\left(1+\dfrac1n\right)e^{i\varphi}\), \(\theta-\varepsilon_n<\varphi<2\pi+\varepsilon_n\), and the segments \(z=te^{i(\theta-\varepsilon_n)}\), \(z=te^{i(2\pi+\varepsilon_n)}\), \(1-\dfrac1n\leq t\leq 1+\dfrac1n\); here \(\{\varepsilon_n\}\) is a sequence of positive numbers monotonically decreasing to zero; the norm in \(B_n\) is
\[ \|x\|_n=\max_{z\in\Gamma_n}|x(z)|. \]

In \(X_1\) we introduce the topology of the inductive limit of the sequence of spaces \(B_n\). In \(X_2\) we introduce a norm by setting
\[ \|x\|_{X_2}=\left(\int_0^\theta |x(e^{i\varphi})|^q\,d\varphi\right)^{1/q},\qquad X_0\subset X_1,\quad X_0\subset X_2, \]
in the sense that every function regular in the set \(|z|\geq 1\), regular on the arc \(z=e^{i\varphi}\), \(\theta\leq\varphi\leq 2\pi\), generates on the arc \(z=e^{i\varphi}\), \(0\leq\varphi\leq\theta\), a function summable to the power \(q\).

Introduce in \(X_0\) the topology which is the exact upper bound of the topologies induced from \(X_1\) and \(X_2\).

Define a functional on \(X_0\) by putting
\[ f_0(x)=\frac{1}{2\pi i}\int_0^{2\pi} x(re^{i\varphi})\,u_0(re^{i\varphi})\,dre^{i\varphi}\qquad (0<r<1), \]
where \(x\in X_0\), and moreover \(x(z)\) is a function analytic up to the circle \(|z|=r\). The functional \(f_0\) is continuous in the topology \(X_0\). Let us prove this.
\[ |f_0(x)|\leq \left|\frac{1}{2\pi i}\int_0^\theta x(re^{i\varphi})\,u_0(re^{i\varphi})\,dre^{i\varphi}\right| + \left|\frac{1}{2\pi i}\int_\theta^{2\pi} x(re^{i\varphi})\,u_0(re^{i\varphi})\,dre^{i\varphi}\right| \leq \]
\[ \leq \frac{1}{2\pi}C^{1/p} \left(\int_0^\theta |x(re^{i\varphi})|^q\,d\varphi\right)^{1/q} + \left|\frac{1}{2\pi i}\int_\theta^{2\pi} x(re^{i\varphi})\,u_0(re^{i\varphi})\,dre^{i\varphi}\right|. \tag{*} \]

If \(r\to1-0\), then the first term on the right-hand side tends to
\[ \frac{1}{2\pi}C^{1/p}\|x\|_{X_2}. \]
Consider the second term. Let \(1>r_1>r\). Then
\[ \int_\theta^{2\pi} x(re^{i\varphi})\,u_0(re^{i\varphi})\,dre^{i\varphi} - \int_\theta^{2\pi} x(r_1e^{i\varphi})\,u_0(r_1e^{i\varphi})\,dr_1e^{i\varphi} - \]
\[ -\int_r^{r_1} x(\zeta)\,u_0(\zeta)\,d\zeta + \int_r^{r_1} x(\zeta e^{i\theta})\,u_0(\zeta e^{i\theta})\,d\zeta e^{i\theta} =0. \]
Therefore
\[ \left| \int_\theta^{2\pi} x(re^{i\varphi})\,u_0(re^{i\varphi})\,dre^{i\varphi} - \int_\theta^{2\pi} x_i'(r_1e^{i\varphi})\,u_0(r_1e^{i\varphi})\,dr_1e^{i\varphi} \right| \leq \]
\[ \leq \max_{r<\zeta<r_1}\{\,|x(\zeta)|,\ |x(\zeta e^{i\varphi})|\,\} \left[ \int_r^{r_1}|u_0(\zeta e^{i\theta})|\,d\zeta + \int_r^{r_1}|u_0(\zeta)|\,d\zeta \right]. \]

From condition 2) of Theorem 3 we conclude that there exists the limit
\[ \lim_{r\to1-0}\int_\theta^{2\pi} x(re^{i\varphi})\,u_0(re^{i\varphi})\,dre^{i\varphi}=f^*(x). \]

Moreover, if \(x \in B_n\), then

\[ \left| \int_{0}^{2\pi} x\left(r e^{i\varphi}\right) u_0\left(r e^{i\varphi}\right)\, dr e^{i\varphi} - f^*(x) \right| \leq \|x\|_n K, \]

i.e.

\[ \left| f^*(x) \right| \leq K \|x\|_n + \left| \int_{0}^{2\pi} x\left(r e^{i\varphi}\right) u_0\left(r e^{i\varphi}\right)\, dr e^{i\varphi} \right| \leq \]

\[ \leq K \|x\|_n + \left\{ \int_{0}^{2\pi} \left| u_0\left(\left(1-\frac{1}{n}\right)e^{i\varphi}\right) \right|\, d\varphi \right\} \|x\|_n . \]

Here

\[ K=\int_{0}^{1}\left|u_0(r)\right|\,dr+\int_{0}^{1}\left|u_0\left(r e^{i\theta}\right)\right|\,dr . \]

Letting now \(r\) tend to \(1\) in inequality \((*)\), we obtain

\[ |f_0(x)| \leq \frac{C^{1/p}}{2\pi}\|x\|_{X_2} + \frac{1}{2\pi}K_1^{(n)}\|x\|_n \qquad (x\in B_n). \]

This inequality shows that the functional \(f_0\) is continuous in the topology induced on \(B_n\cap X_0\) by the topologies of \(B_n\) and \(X_2\). From a theorem of B. M. Makarov\({}^{7}\) it follows that \(f_0\) is continuous in the topology of \(X_0\). Applying the lemma, we obtain that

\[ f_0(x)=f_1(x)+f_2(x) \qquad (x\in X_0), \]

where \(f_1\) is continuous in \(X_1\), and \(f_2\)—in \(X_2\). But

\[ u_0(\tilde z)=f_0(\psi_{\tilde z})=f_1(\psi_{\tilde z})-f_2(\psi_{\tilde z}) \qquad (|\tilde z|<1). \]

It is well known that

\[ f_2(x)=\int_{0}^{\theta}\Psi(\varphi)x(e^{i\varphi})\,d\varphi, \]

where the function \(\Psi\) is such that

\[ \int_{0}^{\theta}|\Psi(\varphi)|^p\,d\varphi<+\infty . \]

At the same time, \(\psi_{\tilde z}\) is an element of \(X_1\), if \(\tilde z\ne e^{i\varphi}\), \(0\leq \varphi\leq 2\pi\). Hence it is easy to conclude, as in Theorem 1, that

\[ u_1(\tilde z)=f_1(\psi_{\tilde z}) \]

is a function regular outside the arc \(\tilde z=e^{i\varphi}\), \(0\leq \varphi\leq 2\pi\). The theorem is proved.

In conclusion, I express my heartfelt gratitude to B. M. Makarov for valuable advice.

Received
8 III 1958

CITED LITERATURE

\({}^{1}\) H. Poincaré, Am. J. Math., 14, 201 (1892).
\({}^{2}\) G. Valiron, Atti del quarto congresso del’ Unione Mat. Ital., 2, Roma, 1953.
\({}^{3}\) N. Bourbaki, Elements de Mathématique, No. 1189, Paris, 1953.
\({}^{4}\) G. Köthe, Math. Zs., 57, 13 (1952).
\({}^{5}\) J. Sebastião e Silva, Rend. Mat. e Appl., 14, 388 (1955).
\({}^{6}\) D. A. Raikov, DAN, 113, No. 5 (1957).
\({}^{7}\) B. M. Makarov, DAN, 119, No. 6 (1958).