S. A. Vinogradov

ON INTERPOLATION AND ZEROS OF POWER SERIES WITH A SEQUENCE OF COEFFICIENTS FROM \(l^p\)

(Presented by Academician V. I. Smirnov, 3 VII 1964)

I. Let \(U\) be the open, \(\overline U\) the closed unit disk of the complex plane, \(T \subset \overline U\); let \(l_A^p\) be the space of all functions \(f\), regular in \(U\) and such that the norm
\[ \|f\|_p=\left(\sum_{k=0}^{\infty}\left|\frac{f^{(k)}(0)}{k!}\right|^p\right)^{1/p}<+\infty \qquad (p\geqslant 1). \]
Denote by the symbol \(I\) the embedding operator from the space \(l_A^1\) into the space of all finite complex functions defined on \(T\): \(If(z)=f(z)\) for all \(f\in l_A^1,\ z\in T\). The aim of the first part of this note is, for some sets \(T\subset U\), to describe the set \(I(l_A^1)\). An analogous problem for other classes of analytic functions was solved in papers \((^{1-4})\).

Obviously, all elements of \(I(l_A^1)\) are functions continuous on \(T\). Following \((^7)\), we shall call a set \(T\subset \overline U\) a Carleson set if \(I(l_A^1)=C_T\), where \(C_T\) is the space of all functions continuous on \(T\). In \((^7)\) the existence of infinite and even perfect Carleson sets lying on the unit circle was proved.

Theorem 1. Every Carleson set contained in \(U\) is finite.

Let us introduce some notation:

\(1^\circ.\) \(V_A\) is the space of all functions \(f\), analytic in \(U\) and such that
\[ \sup_{0<r<1}\int_0^{2\pi}\left|f'(re^{it})\right|\,dt<+\infty. \]
In \(V_A\) the norm
\[ \|f\|_{V_A}=\frac1{2\pi}|f(1)|+\frac1{2\pi}\int_{|\zeta|=1}|f'(\zeta)|\,|d\zeta|, \]
is introduced, where \(f'(\zeta)\) is the boundary value of the function \(f'\).

Let us note that \(V_A\subset l_A^1\) (see \((^6)\)).

\(2^\circ.\)
\[ T=\{z_k\}_{k=1}^{\infty},\quad |z_k|<1\quad (k=1,2,\ldots),\quad z_k\ne z_m,\ \text{if } k\ne m. \]
The sequence \(\{z_k\}_{k=1}^{\infty}\) is numbered so that
\[ |z_1|\leqslant |z_2|\leqslant \cdots \leqslant |z_k|\leqslant\cdots . \]

\(3^\circ.\) \(bv_T\) is the space of all functions \(\psi\), defined on \(T\) and such that the norm
\[ \|\psi\|_{bv_T}=|\alpha_0|+\sum_{k=1}^{\infty}|\psi(z_{k+1})-\psi(z_k)|<+\infty, \]
where
\[ \alpha_0=\lim_{k\to\infty}\psi(z_k). \]

\(4^\circ.\)
\[ B_n(z)=\prod_{\substack{k=1\\ k\ne n}}^{\infty}\frac{|z_k|}{z_k}\,\frac{z_k-z}{1-\overline{z_k}z},\qquad b_n(z)=\prod_{k=n+1}^{\infty}\frac{|z_k|}{z_k}\,\frac{z_k-z}{1-\overline{z_k}z}, \]
where \(|z_k|<1\) \((k=1,2,\ldots)\) and
\[ \sum_{k=1}^{\infty}(1-|z_k|)<+\infty. \]

In the case when
\[ T=\{z_k\}_{k=1}^{\infty}\subset[0,1],\qquad z_k<z_{k+1}\quad (k=1,2,\ldots), \]
we have
\[ I(l_A^1)\subset bv_T. \]
Indeed, let \(f\in l_A^1,\ f(z)=\sum_{k=0}^{\infty}a_k z^k\); then
\[ \sum_{n=1}^{\infty}|f(z_n)-f(z_{n+1})| \leqslant \sum_{n=1}^{\infty}\sum_{k=0}^{\infty}|a_k|\,|z_{n+1}^k-z_n^k| \leqslant \]
\[ \leqslant \sum_{k=0}^{\infty}|a_k|\sum_{n=1}^{\infty}(z_{n+1}^k-z_n^k) \leqslant \sum_{k=0}^{\infty}|a_k|<+\infty. \]

Let the set \(T=\{z_k\}_{k=1}^{\infty}\subset U\) satisfy the following condition: \(z_k\to 1\) as \(k\to\infty\) along a nontangential path, i.e., there exists \(\lambda\geqslant 1\) such that

\[ |1-z_k|\leqslant \lambda(1-|z_k|)\quad (k=1,2,\ldots). \tag{*} \]

Theorem 2. Let the set \(T=\{z_k\}_{k=1}^{\infty}\subset U\) satisfy conditions \(2^0\), \((*)\). Then the following assertions are equivalent:

1. \(I(l_A^1)=bv_T\).
2. \(I(V_A)=bv_T\).
3. \[ \inf_n \prod_{\substack{k=1\\ k\ne n}}^{\infty} \left|\frac{z_k-z_n}{1-\bar z_k z_n}\right|>0. \tag{**} \]

Let us note that condition \((**)\) is a criterion for the solvability of a number of other interpolation problems (see \((^1,^2,^3)\)).

The proof of Theorems 1 and 2 is based on the following fundamental lemmas.

Lemma 1. Let \(T=\{z_k\}_{k=1}^{\infty}\) satisfy conditions \(2^0\), \((*)\), \((**)\). Then the sequence of functions

\[ \Phi_n(z)=\frac{(1-z_n^2)(1-z^2)}{(1-zz_n)^2}\, \frac{B_n(z)}{B_n(z_n)} \quad (n=1,2,\ldots) \]

satisfies the conditions:

1. \(\{\Phi_n\}_{n=1}^{\infty}\subset V_A\).
2. \(\Phi_n(z_k)=\delta_{nk}\) (\(\delta_{nk}\) is the Kronecker symbol, \(n,k=1,2,\ldots\)).
3. \(\|\Phi_n\|_V<M_1<+\infty\) \((n=1,2,\ldots)\), \(M_1\) independent of \(n\).

Lemma 2. Under the conditions of Lemma 1, the sequence of functions

\[ F_n(z)=\frac{1-z}{1-zz_n}\,b_n(z) +\sum_{k=1}^{n}\left(1-\frac{1-z_k}{1-z_k z_n}\,b_n(z_k)\right)\Phi_k(z) \quad (n=1,2,\ldots), \]

where \(\Phi_k(z)\) is from Lemma 1, satisfies the conditions:

1. \(\{F_n\}_{n=1}^{\infty}\subset V_A\).
2. \(F_n(z_k)=1\), if \(k\leq n\); \(F_n(z_k)=0\), if \(k>n\).
3. \(\|F_n\|_V<M_2<+\infty\) \((n=1,2,\ldots)\), \(M_2\) independent of \(n\).

Corollary. Under the conditions of Lemma 1 we have: to every function \(\psi\in bv_T\) there corresponds a function

\[ f(z)=\alpha_0+\sum_{k=1}^{\infty}\bigl(\psi(z_k)-\psi(z_{k+1})\bigr)F_k(z) \]

(the functions \(F_k(z)\) are from Lemma 2, and \(\alpha_0=\lim_{k\to\infty}\psi(z_k)\)) such that:

1. \(f\in V_A\).
2. \(f(z_n)=\psi(z_n)\) \((n=1,2,\ldots)\).

II. If \(p>2\), then the description of \(I(l_A^p)\) is difficult already for the reason that the structure of the sets \(T=\{z_k\}_{k=1}^{\infty}\subset U\) on which functions of the class \(l_A^p\) \((p>2)\) can vanish has not been studied. We prove in this direction the following theorem.

Theorem 3. There exists a function \(G\ne 0\) such that \(G\in \bigcap_{p>2} l_A^p\),

\[ G(z_k)=0,\quad |z_k|<1\quad (k=1,2,\ldots) \]

and

\[ \sum_{k=1}^{\infty}(1-|z_k|)=+\infty. \]

For the proof of the theorem we shall need

Lemma 3. Let

\[ \varphi(z)=\sum_{k=0}^{\infty}a_k z^k,\quad \varphi\in l_A^1;\qquad \psi(z)=\sum_{k=0}^{\infty}b_k z^k,\quad \psi\in l_A^p\ (p\geqslant 1). \]

Then for any \(\varepsilon>0\) there exists \(N=N(\varepsilon,\varphi)\) such that, for \(n>N\),

\[ \|\varphi\psi_n\|_p\leqslant \|\varphi\|_p\|\psi\|_p(1+\varepsilon)\quad (p\geqslant 1), \]

where \(\psi_n(z)=\psi(z^n)\).

Proof. Take an arbitrary \(\varepsilon>0\). Let \(N=N(\varepsilon,\varphi)\) be such that

\[ \sum_{k=N}^{\infty}|a_k|<\varepsilon\|\varphi\|_p, \]

and let

\[ S_n(z)=\sum_{k=0}^{n-1}a_k z^k,\qquad R_n(z)=\sum_{k=n}^{\infty}a_k z^k. \]

Then, for \(n<N\), we have

\[ \|\varphi\psi_n\|_p=\|S_n\psi_n+R_n\psi_n\|_p\le \|S_n\psi_n\|_p+\|R_n\psi_n\|_p; \]

since \(\|S_n\psi_n\|_p=\|S_n\|_p\|\psi_n\|_p\) for every \(n\), and \(\|\Phi F\|_p\le \|\Phi\|_1\|F\|_p\) for all \(\Phi\in l_A^1\) and \(F\in l_A^p\) \((p\ge 1)\), it follows that

\[ \|\varphi\psi_n\|_p\le \|S_n\|_p\|\psi_n\|_p+\|R_n\|_1\|\psi_n\|_p <\|\varphi\|_p\|\psi_n\|_p+\varepsilon\|\varphi\|_p\|\psi_n\|_p =\|\varphi\|_p\|\psi\|_p(1+\varepsilon). \]

The lemma is proved.

Proof of the theorem. For the proof of the theorem we shall need the following obvious assertions:

\[ xe^{-x}\le \frac{2}{l}e^{-x/2}\quad \text{for all } x\in[0,+\infty), \tag{1} \]

\[ \text{the function } y=\frac{x}{1-e^{-x}} \text{ is increasing on } (0,+\infty). \tag{2} \]

Let

\[ g_n(z)=1-\frac{(1-e^{-2/n})^2}{(1-ze^{-1/n})^2} =1-(1-e^{-2/n})^2-(1-e^{-2/n})^2\sum_{k=1}^{\infty}(k+1)e^{-k/n}z^k \]

\[ (n=1,2,\ldots). \]

We have the estimates:

\[ (n=1,2,\ldots). \]

\[ (1-e^{-2/n})^2(k+1)e^{-k/n} <\left(\frac{2}{n}\right)^2(2k)e^{-k/n} =\frac{8}{n}\left(\frac{k}{n}\right)e^{-k/n} <\frac{8}{n}e^{-k/2n}\quad (n,k=1,2,\ldots)\quad \text{(see (1));} \]

\[ \|g_n\|_p< \left(1+\sum_{k=1}^{\infty}\left(\frac{8}{n}\right)^p(e^{-p/2n})^k\right)^{1/p} < \left(1+\left(\frac{8}{n}\right)^p\frac{1}{1-e^{-p/2n}}\right)^{1/p}, \]

and, applying (2) \(\left(x=\frac{p}{2n}\right)\), we obtain

\[ \|g_n\|_p< \left(1+\left(\frac{8}{n}\right)^p\frac{n}{1-e^{-p/2}}\right)^{1/p} = \left(1+\frac{B_p}{n^{p-1}}\right)^{1/p} \quad (p\ge 1;\ n=1,2,\ldots), \tag{3} \]

where

\[ B_p=\frac{8^p}{1-e^{-p/2}}. \]

Consider a sequence of positive numbers \(\{\varepsilon_n\}_{n=1}^{\infty}\) such that

\[ \sum_{k=1}^{\infty}\varepsilon_k<+\infty. \]

We construct by induction a sequence of natural numbers \(\{\lambda_n\}_{n=1}^{\infty}\) satisfying the condition

\[ \|G_n\|_p\le \|G_{n-1}\|_p\cdot \|g_n\|_p\cdot(1+\varepsilon_n) \quad (p\ge 1;\ n=1,2,\ldots), \tag{4} \]

where

\[ G_0\equiv 1,\qquad G_n(z)=\prod_{k=1}^{n}g_k(z^{\lambda_k})\quad (n=1,2,\ldots). \]

Set \(\lambda_1=1\). Suppose the numbers \(\lambda_1,\ldots,\lambda_{n-1}\) have been constructed. The corresponding function \(G_{n-1}\in l_A^1\). Applying the lemma to \(G_{n-1}\), \(g_n\), \(\varepsilon_n\), we find \(\lambda_n\) such that

\[ \|G_n\|_p\le \|G_{n-1}\|_p\cdot \|g_n\|_p\cdot(1+\varepsilon_n). \]

The infinite product

\[ G(z)=\prod_{k=1}^{\infty}g_k(z^{\lambda_k}) \]

converges uniformly in every disk \(|z|\le r<1\). Consequently, \(G(z)\) is a function analytic in the disk \(|z|<1\), \(G(z)\not\equiv 0\)

\[ \left(0<G(0)=\prod_{n=1}^{\infty}\left(1-\left(1-e^{-2/n}\right)^2\right)<+\infty\right). \]
For any \(p>2\) and any \(n=1,2,\ldots\), applying (4), (3), we obtain
\[ \|G_n\|_p \leq \|G_{n-1}\|_p\|g_n\|_p(1+\varepsilon_n)\leq \cdots \leq \prod_{k=1}^n\left(\|g_k\|_p(1+\varepsilon_k)\right)< \]
\[ <\prod_{k=1}^{\infty}\left(1+\frac{B_p}{k^{p-1}}\right)^{1/p} \prod_{k=1}^{\infty}(1+\varepsilon_k)=D_p<+\infty . \tag{5} \]

From the fact that \(G_n(z)\to G(z)\) uniformly in every disk \(|z|\leq r<1\), and from (5) it follows that
\[ \|G\|_p\leq D_p<+\infty\quad (p>2). \]

Since \(g_n(e^{-1/n})=0\) \((n=1,2,\ldots)\), the numbers
\[ z_{n,j}=\zeta_j e^{-1/n\lambda_n}\quad (j=1,\ldots,\lambda_n;\ n=1,2,\ldots), \]
where \(\zeta_j\) are the roots of the equation \(\zeta^{\lambda_n}-1=0\) \((j=1,\ldots,\lambda_n)\), are zeros of the function \(G(z)\). Consider the sum
\[ \sigma=\sum_{n=1}^{\infty}\sum_{j=1}^{\lambda_n}(1-|z_{n,j}|) =\sum_{n=1}^{\infty}\lambda_n(1-e^{-1/n\lambda_n}); \]
since
\[ \lambda_n(1-e^{-1/n\lambda_n})\sim_{n\to\infty}\frac{1}{n}, \]
we have \(\sigma=+\infty\). Thus, \(G(z)\) satisfies the conditions of the theorem.

Remark 1. The assertion of the theorem ceases to be true for \(G\in l_A^2=H^2\) (see \((^5)\)).

Remark 2. Let \(G\in l_A^p\) \((p>2)\), \(G\not\equiv0\), and \(G(z_k)=0\), \(|z_k|<1\) \((k=1,2,\ldots)\). Then, since for all \(z\in U\)
\[ |G(z)|<\frac{M}{1-|z|}, \]
where \(M\) does not depend on \(z\), it follows (see \((^8)\)) that
\[ \sum_{k=1}^{\infty}(1-|z_k|)^{1+\varepsilon}<+\infty \]
for every \(\varepsilon>0\).

Remark 3. Let \(\{z_k\}_{k=1}^{\infty}\) be a sequence of numbers \((|z_k|<1,\ k=1,2,\ldots)\) satisfying condition (*) of item I. In order that there exist a function \(G\in l_A^p\) \((p>2)\) such that \(G\not\equiv0\) and \(G(z_k)=0\) \((k=1,2,\ldots)\), it is necessary and sufficient that (see \((^8)\))
\[ \sum_{k=1}^{\infty}(1-|z_k|)<+\infty . \]

In conclusion I express my sincere gratitude to V. P. Khavin for posing the problem and for his attention to the work.

Leningrad State University
named after A. A. Zhdanov

Received
11 VI 1964

References

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