UDC 513.88 : 517.544

MATHEMATICS

S. A. VINOGRADOV

PALEY SINGULARITIES AND THE RUDIN–CARLESON INTERPOLATION THEOREMS FOR CERTAIN CLASSES OF ANALYTIC FUNCTIONS

(Presented by Academician V. I. Smirnov on 17 III 1967)

As is well known, the Fourier coefficients of a function continuous on the circle \(\{e^{it}: |t|\leqslant \pi\}\) may tend to zero very slowly even in the case when this function admits a continuous extension to a function holomorphic in the open unit disk
\[ D=\{re^{it}: 0\leqslant r<1,\ |t|\leqslant \pi\}. \]
This circumstance manifests itself, in particular, in the so-called Paley singularities, which were the subject of investigation in papers \((^{1,5-8})\). In this connection let us also mention Banach’s theorem \((^{2})\), which asserts, roughly speaking, that the Fourier coefficients of a continuous function, taken with indices forming a lacunary sequence, are indistinguishable from the Fourier coefficients of an arbitrary function of class \(L^2\). On the other hand, Rudin \((^{3})\) and Carleson \((^{4})\) showed that from certain infinite subsets of the closed unit disk \(\overline D\), every continuous function can be extended to a function holomorphic in \(D\) and continuous in \(\overline D\).

In the present note we formulate assertions which generalize and refine some results of the papers of R. Paley \((^{1})\), S. B. Stechkin \((^{5})\), V. P. Khavin \((^{6})\), and A. S. Makhmudov \((^{8})\) on singularities of Paley type (Theorem 1 and its corollaries). It turns out that the singularities studied by these authors already occur for functions analytic in the whole complex plane, except for the point \(z=1\), and continuous in the closed disk \(\overline D\). In addition, a theorem is formulated which simultaneously generalizes the above-mentioned theorems of Banach and Rudin–Carleson (Theorem 3). In Theorems 4 and 5 we are concerned with certain singularities of power series uniformly convergent in \(\overline D\).

Let us introduce some notation.

\(1^\circ.\) \(\hat C\) is the extended complex plane,
\[ D=\{z\in \hat C: |z|<1\},\qquad \partial D=\{z\in C: |z|=1\},\qquad \overline D=D\cup \partial D. \]

\(2^\circ.\) \(G_\alpha\) is a domain in the extended complex plane such that:
a) \(D\subset G_\alpha\), the point at infinity belongs to the domain \(G_\alpha\); b) the boundary \(\partial G_\alpha\) of the domain \(G_\alpha\) is a closed rectifiable Jordan curve containing the arc of the circle
\[ \Gamma_\alpha=\{e^{i\theta}: |\theta|<\alpha\}, \]
where \(0<\alpha<\pi\).

\(3^\circ.\) \(A(G_\alpha)\) is the space of all functions \(f\) holomorphic in the domain \(G_\alpha\) and continuous in \(\overline{G_\alpha}\) (\(\overline{G_\alpha}\) is the closure of \(G_\alpha\)), with norm
\[ \|f\|_{A(G_\alpha)}=\sup_{z\in G_\alpha}|f(z)|. \]

\(4^\circ.\) \(T\) is a closed set lying on the unit circle \(\partial D\) (\(T\ne \partial D\)), and \(A_C(T)\) is the space of all functions holomorphic everywhere outside the set \(T\) (including the point at infinity) and continuous in the closed disk \(\overline D\). In particular, if \(T=\{1\}\), then \(A_0(\{1\})\) is the space of all functions holomorphic everywhere except \(z=1\) and continuous in \(\overline D\).

\(5^\circ.\) \(E\) is an arbitrary infinite set consisting of nonnegative integers,
\[ E_n=\{k\in E: k\leqslant n\}, \]
where \(n\) is a natural number.

\(6^\circ.\) \(d=\{d_k\}_{k\in E}\) is a family consisting of positive numbers.

\(7^\circ.\)
\[ M_n(p,d)=\sup_{\|f\|_{A(G_\alpha)}\leqslant 1} \left(\sum_{k\in E_n}|\hat f(k)|^p d_k^{\,2-p}\right)^{1/p}, \quad 0<p<2,\quad n=1,2,\ldots, \]
where
\[ \hat f(k)=\frac{1}{2\pi i}\int_{|z|=1} f(z)z^{-(k+1)}\,dz, \quad k=0,1,2,\ldots . \]

\(8^\circ.\) \(l^2(E)\) is the space of all families of complex numbers \(x=\{x_k\}_{k\in E}\) such that the norm
\[ \|x\|_2=\left(\sum_{k\in E}|x_k|^2\right)^{1/2} \]
is finite.

\(9^\circ.\) \(R\) is the linear operator acting from \(A_C(T)\) into \(l^2(E)\) as follows:
\(Rf=\{\hat f(k)\}_{k\in E}\) (\(\hat f(k)\) is defined in \(7^\circ\)).

\(10^\circ.\) \(S\) is the restriction operator, defined on the space of functions \(A_C(T)\) on the set \(Q\) \((Q\subset \bar D)\), i.e.
\[ (Sf)(z)=f(z),\quad z\in Q,\quad f\in A_C(T). \]

Theorem 1. Let \(G_\alpha\) be the domain defined in \(2^\circ\). Then there exists a finite positive constant \(B(\alpha,p)\) such that
\[ B(\alpha,p)\left(\sum_{k\in E_n} d_k^2\right)^{1/p-1/2} \leqslant M_n(p,d)\leqslant \left(\sum_{k\in E_n} d_k^2\right)^{1/p-1/2}, \quad p\in(0,2),\quad n=1,2,\ldots . \]

This theorem is a generalization of the well-known Paley theorem (see \((^1)\)).

Corollary 1. Let the family \(d=\{d_k\}_{k\in E}\) satisfy the condition
\[ \sum_{k\in E} d_k^2=+\infty . \]
Then there exists a function \(f\), holomorphic everywhere except at the point \(z=1\), continuous in \(\bar D\) (i.e. \(f\in A_C(\{1\})\)) and such that
\[ \sum_{k\in E}|\hat f(k)|^p d_k^{\,2-p}=+\infty \]
for all \(p\in(0,2)\).

Corollary 2. Let \(\{t_k\}_{k=0}^{\infty}\) be a sequence of nonnegative numbers such that
\[ \sum_{k=0}^{\infty} t_k^r=+\infty \]
for every \(r>0\). Then there exists a function \(f\) such that \(f\in A_C(\{1\})\) and
\[ \sum_{k=0}^{\infty} t_k|\hat f(k)|^{2-\varepsilon}=+\infty \quad\text{for all }\varepsilon\in(0,2). \]

Corollaries 1 and 2 are a generalization of certain results of V. P. Khavin contained in \((^6)\).

Remark. All results from \((^{7,8})\) concerning the space of all functions holomorphic outside the ray \([1,+\infty)\) and continuous in \(\bar D\) remain valid if this space is replaced by the space \(A_C(\{1\})\).

Definition. We shall say that the set \(E\) satisfies the Hadamard condition (or is lacunary) if
\[ \gamma=\inf_{\substack{0<k<m\\ k,m\in E}} m/k>1. \]

Theorem 2. Let the set \(E\) satisfy the Hadamard condition. Then
\[ R(A_C(\{1\}))=l^2(E). \]

This theorem is a strengthening of the well-known theorem of Banach \((^2)\).

Remark. The Hadamard condition can be considerably weakened.

Theorem 3. Let the sets \(T\) and \(Q\), appearing in \(4^\circ\) and \(10^\circ\), coincide \((Q=T)\), and let \(\operatorname{mes}T=0\) (\(\operatorname{mes}\) denotes Lebesgue measure on

circle \(\partial D\), and the set \(E\) satisfies Adamyan’s condition. Then for every function \(\psi\), continuous on the set \(T\), and for every family \(x=\{x_k\}_{k\in E}\in l^2(E)\), there exists a function \(f\in A_C(T)\) such that \(Sf=\psi\) and \(Rf=x\).

In particular, this theorem is a strengthening of the well-known theorem of Rudin–Carleson (see \((^3,^4)\)).

Denote by \(AU(\{1\})\) the space of all functions \(f\), holomorphic everywhere except at the point \(z=1\) and having a Maclaurin series uniformly convergent on the unit circle \(\partial D\).

Theorem 4. Let \(\{d_k\}_{k=0}^{\infty}\) be a sequence of nonnegative numbers and \(p\in(0,2)\). Then, if

\[ \frac{1}{\ln(n+2)} \left(\sum_{k=0}^{n} d_k^2\right)^{1/p-1/2} \longrightarrow +\infty, \qquad n\to\infty, \]

there exists a function \(f\) such that:

1. \(f\in AU(\{1\})\).

2. \[ \sum_{k=0}^{\infty} |\hat f(k)|^p d_k^{\,2-p}=+\infty . \]

Corollary. There exists a function \(f\) such that:

1. \(f\in AU(\{1\})\).

2. \[ \sum_{k=0}^{\infty} |\hat f(k)|^{2-\varepsilon}=+\infty \quad \text{for all } \varepsilon\in(0,2). \]

Denote by the symbol \(AU_0(\{1\})\) the space of all functions \(f\) such that \(f\in AU(\{1\})\) and

\[ \hat f(n)=o\left(\frac1n\right),\qquad n\to\infty \]

(\(\hat f(n)\) is defined in \(7^\circ\)), and by the symbol \(C(Q)\) the space of all functions uniformly continuous on the set \(Q\) \((Q\subset D)\).

Theorem 5. Let the set \(Q\) \((Q\subset D)\) satisfy the condition: there exists a number \(\lambda\in[1,+\infty)\) such that

\[ |1-\eta|\le \lambda(1-|\eta|) \tag{I} \]

for all \(\eta\in Q\).

Then the following assertions are equivalent:

1. \(S(AU_0(\{1\}))=C(Q)\).

2. The set \(Q\) satisfies Carleson’s condition (see \((^{10})\)), i.e.

\[ \delta=\inf_{\xi\in Q}\prod_{\eta\in Q_\xi} \left|\frac{\xi-\eta}{1-\xi\bar\eta}\right|>0, \]

where \(Q_\xi=\{\eta\in Q:\eta\ne\xi\}\).

Remark. It is interesting to compare the assertion of Theorem 5 with Theorem 1 from \((^9)\), which is formulated as follows:

Let \(l_A^1\) denote the space of all functions \(f\) such that

\[ f(z)=\sum_{k=0}^{\infty} \hat f(k) z^k,\qquad z\in D,\qquad \sum_{k=0}^{\infty} |\hat f(k)|<+\infty; \]

\(S\) is the restriction operator:

\[ (Sf)(z)=f(z),\qquad z\in Q,\qquad f\in l_A^1 . \]

Then \(S(l_A^1)=C(Q)\) if and only if \(Q\) is a finite set.

At the same time, there exist infinite sets satisfying both condition (I) and Carleson’s condition (see \((^{10})\)). For example, \(Q=\{1-2^{-k}: k=1,2,\ldots\}\).

In conclusion, the author expresses his sincere gratitude to V. P. Havin for posing the problem and for his attention to the work.

Leningrad State University
named after A. A. Zhdanov

Received
15 III 1957

REFERENCES

¹ R. E. A. C. Paley, J. London Math. Soc., 7, 122 (1932).
² S. Banach, Studia Math., 2, 207 (1930).
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⁴ L. Carleson, Math. Zs., 66, 447 (1957).
⁵ S. B. Stechkin, Izv. AN SSSR, ser. matem., 20, 765 (1956).
⁶ V. P. Havin, Vestn. Leningrad. Univ., No. 19, ser. matem., mekh. i astr., issue 4 (1959).
⁷ A. S. Makhmudov, Collection: Some Questions of Functional Analysis and Its Applications, Baku, 1965, p. 103.
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⁹ S. A. Vinogradov, DAN, 160, No. 2 (1965).
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