Full Text
V. P. ZAKHARYUTA
THE PRODUCT OF BASES IN ARBITRARY DOMAINS
(Presented by Academician V. I. Smirnov on 31 I 1963)
By the product of two systems
\[ p_n^{(s)}(z)=\sum_{k=0}^{\infty} p_{n,k}^{(s)} z^k \quad (s=1,2) \]
of functions analytic in some neighborhood of zero \(|z|<\delta\), we shall mean the system
\[ p_n(z)=\sum_{i=0}^{\infty} p_{n,i}^{(1)} p_i^{(2)}(z), \]
where uniform convergence of the series in some neighborhood of zero \(|z|<\delta_1\) is assumed \(({}^{1,2})\). The question of the product of bases in the spaces \(A_R\) of functions analytic in the disk \(|z|<R\) has been considered in detail in papers \(({}^{2-4})\).* Recently a number of papers \(({}^{5-7})\) have appeared in which the product of bases in an arbitrary domain is studied, but for the special case of simple unit bases, i.e., bases of the form
\[ p_n(z)=\sum_{k=0}^{n} p_{n,k} z^k,\qquad p_{n,n}=1 \quad (n=0,1,2,\ldots). \tag{1} \]
In the present paper the question of the product of bases in an arbitrary domain is studied for a broad class of bases, and it is possible to obtain a number of more precise results than in the papers mentioned. In addition, for arbitrary domains we consider questions concerning inverse bases and the quotient of two bases.
Let \(D\) be an open simply connected domain of the complex plane. In what follows we shall consider the following spaces of analytic functions: 1) \(\mathfrak A(D)\)—the space of functions analytic in \(D\), with the topology of uniform convergence inside \(D\); 2) \(\mathfrak A(\overline D)\)—the space of functions analytic in \(\overline D\), with the topology of the union of spaces \(({}^{9},\) p. 89):
\[ \mathfrak A(\overline D)=\bigcup_\lambda \mathfrak A(D_\lambda), \]
where \(D_\lambda\) are open domains, with \(D_\lambda \supset \overline{D}_{\lambda+1}\) and
\[ \bigcap_\lambda \overline{D}_\lambda=\overline D,\quad \lambda=1,2,\ldots . \]
A sequence \(x_n(z)\) converges to \(x(z)\) in \(\mathfrak A(\overline D)\) if and only if there exists \(\lambda_0\) such that \(x_n,x\in \mathfrak A(D_{\lambda_0})\) and \(x_n\to x\) in \(\mathfrak A(D_{\lambda_0})\). If the boundary of the domain \(D\) is a rectifiable Jordan curve, we shall also consider one more space: 3) \(\mathfrak A C(D)\)—the space of functions analytic in \(D\) and continuous in \(\overline D\), with norm
\[ \|x\|=\max_{z\in \overline D}|x(z)|. \]
In the case when \(D=K_r=\{|z|<r\}\), preserving the notation adopted in \(({}^{8})\), we shall denote \(\mathfrak A(D)\), \(\mathfrak A(\overline D)\), \(\mathfrak A C(D)\) respectively by \(A_s,\ \overline A_r,\ A_r C\).
Definition 1. Let \(E\) be a linear topological space and let \(E_1\subset E\). Then we shall call \(x_n\in E\) a basis in the topology \(E\) for \(E_1\) if every element \(x\in E_1\) can be uniquely expanded in the series
\[ x=\sum \xi_n x_n, \tag{2} \]
convergent in the topology of the space \(E\). In the case \(E_1=E\) we obtain the usual Schauder basis in \(E\).
Definition 2. The system \(x_n\) will be called a pseudobasis in the topology \(E\) for \(E_1\) if every function from \(E_1\) can be expanded in the series (2) (not necessarily uniquely).
\[ \text{* Here and below, the results of the cited papers are stated in the terms and notation adopted in }({}^{8}). \]
Let \(\Gamma\) be a closed Jordan curve and
\[ t=\Phi(z)=z+a_0+a_1/z+\cdots,\qquad \Phi'_{\infty}(z)\ne z \tag{3} \]
a conformal mapping of the exterior of the curve \(\Gamma\) onto the exterior of the disk
\(K_\gamma=\{|t|\le \gamma\}\). The images of the circles \(\{|t|=r\}\) \((r>\gamma)\) under this mapping will be denoted by \(\Gamma_r\), and the domains interior to \(\Gamma\) and \(\Gamma_r\) will be denoted respectively by \(D\) and \(D_r\). We shall also denote
\[ \beta=\inf\{r:D_r\supset K_\gamma\},\qquad \alpha=\max_{|z|\in\Gamma_\beta}|z|. \tag{4} \]
Lemma 1. The inequality \(\gamma<\beta<\alpha\)* holds.
Consider the Faber polynomials \(\Phi_n(z)\) for the contour \(\Gamma\) \((^{10})\). Obviously, \(\Phi_n(z)\) is a basis in \(\mathfrak A(\overline D)\), \(\mathfrak A(D_r)\), \(\mathfrak A(\overline{D}_r)\) \((r>\gamma)\).
Definition 3. A basis \(p_n(z)\) in \(\mathfrak A(\overline D)\), or in \(\mathfrak A(D_r)\), \(\mathfrak A(\overline{D}_r)\), will be called quasipower if the series \(\sum_n \xi_n p_n(z)\) and \(\sum_n \xi_n\Phi_n(z)\) simultaneously converge or diverge in \(\mathfrak A(\overline D)\) (respectively in \(\mathfrak A(D_r)\) and \(\mathfrak A(\overline{D}_r)\)) for every numerical sequence \(\xi_n\)**.
Definition 4. A basis \(p_n(z)\) in \(\mathfrak A(\overline D)\) will be called (quasipower-) continuable to \(\overline{D'}\) if \(p_n(z)\) is a (quasipower) basis in \(\mathfrak A(\overline{D'})\). If, moreover, \(D'\subset D\) \((D\subset D')\), then \(p_n(z)\) will be called an internally (externally) continuable basis (cf. \((^{11})\)).
The product of the systems \(p_n^{(1)}(z)\) and \(p_n^{(2)}(z)\) will henceforth be denoted by
\(\{p_n(z)\}=\{p_n^{(1)}(z)\}\{p_n^{(2)}(z)\}\). We formulate the main theorem.
Theorem 1. Let \(p_n^{(1)}(z)\) be a basis in \(\mathfrak A(\overline D)\), continuable to \(\overline{D}_\beta\), and let \(p_n^{(2)}(z)\) be a quasipower basis in \(\mathfrak A(\overline D)\), quasipower-continuable to \(\overline{D}_\alpha\)***. Then the product
\[
\{p_n(z)\}=\{p_n^{(1)}(z)\}\{p_n^{(2)}(z)\}
\]
is a basis in the topology \(\mathfrak A(\overline D)\) for \(\mathfrak A(\overline{D}_a)\) (for the definition of \(\alpha\) and \(\beta\) see (4)).
Before passing to the proof of the theorem, we consider lemmas.
Lemma 2. In order that a basis \(p_n(z)\) in \(\overline A_\gamma\) be quasipower, it is necessary and sufficient that, for every \(r\) satisfying \(\gamma<r<r_0\), there exist numbers \(C'(r)\), \(C''(r)\), \(\rho_1(r)\), \(\rho_2(r)\) such that
\[ C'(r)\rho_1^n(r)\le \max_{|z|<r}|p_n(z)|\le C''(r)\rho_2^n(r), \tag{5} \]
where \(\rho_i(r)\downarrow \gamma\) as \(r\downarrow\gamma\).
For \(A_\gamma\) an analogous assertion was proved by Yu. F. Korobeinik.
Lemma 3. Let
\[
x(z)=\sum_{n=0}^{\infty}\xi_n z^n
\]
be a function analytic in \(|z|\le r+\varepsilon\). Then
\[ \sum_{n=0}^{\infty}|\xi_n|\,r^n \le \frac{r+\varepsilon}{\varepsilon} \max_{|z|\le r+\varepsilon}|x(z)|. \tag{6} \]
Lemma 4. Let
\[
x(z)=\sum \xi_n z^n\in \overline A_\gamma.
\]
Then the mapping \(\mathfrak R:\ A_\gamma\to \mathfrak A(\overline D)\) such that
\[
y(z)=\mathfrak R x(z)=\sum \xi_n\Phi_n(z)
\]
is an isomorphism of \(\overline A_\gamma\) \((A_r,\overline A_r,\ r>\gamma)\) onto \(\mathfrak A(\overline D)\) (respectively onto \(\mathfrak A(\overline{D}_r)\), \(\mathfrak A(D_r)\), \(r>\gamma\)).
This assertion is obtained by applying the estimates for Faber polynomials \(((^{10}),\ p. 421)\) and Lemma 3.
* It is assumed that \(\Gamma\) is different from a circle with center at the origin.
* For the disk this definition coincides with that adopted in \((^{8,11})\).
** Since, by (4), \(D_\alpha\supset D_\beta\supset K_\gamma\), the \(p_n^{(s)}(z)\) are analytic in \(K_\gamma=\{|z|<\gamma\}\), \(s=1,2\).
Corollary 1. If
\[ p_n(z)=\sum q_{n,k}\Phi_k(z) \tag{7} \]
is a (quasi-power) basis in \(\mathfrak A(\overline D)\) (or in \(\mathfrak A(D_r), \mathfrak A(\overline D_r)\)), then
\[ q_n(z)=\sum q_{n,k}z^k \tag{8} \]
is a (quasi-power) basis in \(\overline A_\gamma\) (respectively in \(A_r,\overline A_r\)), and conversely.
Corollary 2. If (7) is a basis in the topology \(\mathfrak A(\overline D)\) for \(\mathfrak A(\overline D_r)\), then (8) is a basis in the topology \(\overline A_\gamma\) for \(\overline A_r\), and conversely.
Let us proceed to the proof of the theorem. Let (7),
\[ p_n^{(1)}(z)=\sum p_{n,k}^{(1)}z^k,\qquad p_n^{(2)}(z)=\sum q_{n,k}^{(2)}\Phi_k(z) \tag{9} \]
be the bases given by the conditions of the theorem, and let \(q_n^{(2)}(z)=\sum q_{n,k}^{(2)}z^k\). In view of (7) and (9),
\[ p_n(z)=\sum_i p_{n,i}^{(1)}p_i^{(2)}(z) =\sum_k\left(\sum_i p_{n,i}^{(1)}q_{i,k}^{(2)}\right)z^k =\sum_k q_{n,k}\Phi_k(z). \tag{10} \]
The order of summation in (10) may be interchanged, since, using the estimates for Faber polynomials and (5), (6), we have
\[ \sum_i |p_{n,i}^{(1)}|\sum_k |q_{i,k}^{(2)}|\max_{z\in\Gamma_r}|\Phi_k(z)| \le M\sum_i |p_{n,i}^{(1)}|\sum_k |q_{i,k}^{(2)}|r^k \le \]
\[ \le M\sum_i |p_{n,i}^{(1)}|\frac{r+\varepsilon}{\varepsilon} \max_{|z|\le r+\varepsilon}|q_i^{(2)}(z)| \le M\frac{r+\varepsilon}{\varepsilon}C''(r+\varepsilon)\sum_i |p_{n,i}^{(1)}|\rho_2^i(r+\varepsilon). \]
Taking into account that \(p_n^{(1)}(z)\) are functions analytic in \(\overline D_\beta\), and hence analytic in \(A_{\gamma+\delta}\) for some \(\delta>0\), for \(r\) and \(\varepsilon\) such that \(\rho_2(r+\varepsilon)\le \gamma+\delta\), we obtain the inequality
\[ \sum_i |p_{n,i}^{(1)}|\sum_k |q_{i,k}^{(2)}|\max_{z\in\Gamma_r}|\Phi_k(z)|<M_n<\infty, \]
which proves the legitimacy of the interchanges in (10). From (10),
\[ q_n(z)=\sum_k\left(\sum_i p_{n,i}^{(1)}q_{i,k}^{(2)}\right)z^k =\sum_i p_{n,i}^{(1)}q_i^{(2)}(z). \tag{11} \]
By Corollary 2, in order to prove the theorem it is enough to show that \(q_n(z)\) is a basis in the topology \(\overline A_\gamma\) for \(\overline A_\alpha\). We first show that \(p_n^{(1)}(z)\) is a basis in the topology \(\overline A_\gamma\) for \(\overline A_\alpha\). By assumption, \(p_n^{(1)}(z)\) is a basis in \(\mathfrak A(\overline D_\beta)\), and therefore every function from \(\mathfrak A(\overline D_\beta)\) can be expanded in a series converging in \(\mathfrak A(\overline D_\beta)\); since, by (3), \(\overline K_\gamma\subset \overline D_\beta\subset \overline K_\alpha\), it is all the more true that every function from \(\overline A_\alpha\) can be expanded in a series in \(p_n^{(1)}(z)\), converging in \(\overline A_\gamma\). The expansion will be unique. Indeed, suppose
\[ \sum \xi_n p_n^{(1)}(z)=0,\qquad \xi_n\ne 0, \tag{12} \]
where the series converges in \(\overline A_\gamma\), i.e. converges in the disk \(|z|<r_0\) \((r_0>\gamma)\). From the continuation of \(p_n^{(1)}(z)\) from \(\overline D\) into \(\overline D_\beta\) it follows that \(p_n^{(1)}(z)\) is a basis in \(\mathfrak A(D_\beta)\), moreover internally continuable (see (1)). But then, by (11), from the convergence of the series (12) inside the domain \(|z|<r_0\), which has a boundary point on the boundary of the domain \(D_\beta\), it follows that the series (12) converges in \(\mathfrak A(D_\beta)\). The latter contradicts the fact that \(p_n^{(1)}(z)\) is a basis in \(\mathfrak A(D_\beta)\). Thus, \(p_n^{(1)}(z)\) is a basis in the topology \(\overline A_\gamma\) for \(\overline A_\alpha\).
By Corollary 1, \(q_n^{2}(z)\) is a quasi-power basis both in \(\overline A_\gamma\) and in \(\overline A_\alpha\). Then the mapping \(\mathfrak K_1\), defined as follows: if \(x(z)=\sum \xi_n z^n\), then \(\mathfrak K_1 x=\sum \xi_n q_n^{(2)}(z)\), is an isomorphism of \(\overline A_r\) onto itself \((\gamma\le r\le \alpha)\). Therefore, by the preceding remark and (11), \(q_n(z)\) is a basis in the topology \(\overline A_\gamma\) for \(\overline A_\alpha\). The theorem is proved.
Theorem 1 contains Newns’s result (⁷), if the notation and terminology are brought into agreement. Namely, in (⁷) the assertion of Theorem 1 is proved for bases of the form (1). But such bases are quasipower-extendable into any \(\mathfrak A(\overline D_r)\), \(r>\gamma\), and hence certainly satisfy the conditions of Theorem 1.
The following theorem shows that the result of Theorem 1 is sharp for any pair of bases, in contrast to (⁶, ⁷), where it is proved that the result is sharp only for the whole class of simple bases.
Theorem 2. The constant \(\alpha\) in the conditions of Theorem 1 is sharp for every pair of bases \(p_n^{(1)}(z)\) and \(p_n^{(2)}(z)\), i.e. it cannot be decreased (and thereby the class of expandable functions enlarged).
In what follows we shall need the following.
Theorem 3. Let \(\Gamma\) be a regular curve. If \(p_n(z)\) is a simple pseudobasis in the topology \(\mathfrak A C(D)\) for \(\mathfrak A(\overline D)\), then \(p_n(z)\) is a simple basis in \(\mathfrak A(\overline D)\), and conversely.
The theorem is true when \(D\) is a disk with center at the origin ((²), p. 462). Since the mapping \(\mathfrak R\), considered in Lemma 4, is an isomorphism of \(\overline A_\gamma\) onto \(\mathfrak A(\overline D)\) and, obviously, takes a simple basis into a simple one, the assertion of the theorem follows from the following lemma.
Lemma 5. In the case of a regular contour \(\Gamma\), the mapping \(\mathfrak R\), considered in Lemma 4, is an isomorphism of \(A_\gamma C\) onto \(\mathfrak A C(D)\).
From Theorems 1 and 3 there follows the result of (⁶), which we formulate differently from the cited paper.
Theorem 4. Let \(\Gamma\) be a regular curve and let \(p_n^{(i)}(z)\), \(i=1,2\), be simple unit pseudobases in the topology \(\mathfrak A C(D)\) for \(\mathfrak A(\overline D)\); then the system
\[ \{p_n(z)\}=\{p_n^{(1)}(z)\}\{p_n^{(2)}(z)\} \tag{13} \]
is a pseudobasis in the topology \(\mathfrak A C(D)\) for \(\mathfrak A(\overline D_\alpha)\).
Remark 1. We obtain here even more: the system (13) is a basis in the topology \(\mathfrak A(\overline D)\) for \(\mathfrak A(\overline D_\alpha)\). This shows that the result of Barsoum and Nassif is in fact identical with Newns’s result. In particular, it follows from what has been set forth that the constants \(\gamma/\alpha\) of Newns (⁷) and \(\beta/S(\beta)\) of Nassif–Barsoum (⁶) coincide.
Let \(p_n(z)\) be a basis in \(\mathfrak A(\overline D)\). We shall call the system \(\{\pi_n(z)\}=\{p_n(z)\}^{-1}\) inverse to \(p_n(z)\) if
\[
\{\pi_n(z)\}\{p_n(z)\}=\{p_n(z)\}\{\pi_n(z)\}=\{z^n\}.
\]
Let \(z=\Psi(t)\) be the mapping inverse to the mapping (3), \(t=\Phi(z)\). Put
\[
\tau=\sup_{|t|=\gamma}|\Psi(t)|=\sup_{t\in\Gamma'}|t|.
\]
Let \(\Gamma'\) be the preimage of the circle \(|z|=\tau\) under the mapping \(\Psi(t)\), and
\[
\mu=\sup_{|z|=\tau}|\Phi(z)|=\sup_{t\in\Gamma'}|t|.
\]
Theorem 5. Let \(p_n(z)\) be a quasipower basis in \(\mathfrak A(\overline D)\), quasipower-extendable into \(\overline D_\mu\). Then the inverse system \(\pi_n(z)\) is a basis in the topology \(\overline A_\gamma\) for \(\overline A_\mu\).
In conclusion we give a theorem for the quotient of two bases in \(\mathfrak A(\overline D)\).
Theorem 6. Let \(p_n^{(1)}(z)\) be a basis in \(\mathfrak A(\overline D)\), and let \(p_n^{(2)}(z)\) be a quasipower basis in \(\mathfrak A(\overline D)\). Then the quotient \(\{p_n(z)\}=\{p_n^{(1)}(z)\}\{p_n^{(2)}(z)\}^{-1}\) is a basis in \(\overline A_\gamma\).
Rostov-on-Don
State University
Received
27 I 1963
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