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A Boundary Value Problem for $H^\infty(D)$

Rotraut Goubau Cahill
Proceedings of the American Mathematical Society
Vol. 67, No. 2 (Dec., 1977), pp. 241-247
DOI: 10.2307/2041281
Stable URL: http://www.jstor.org/stable/2041281
Page Count: 7
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A Boundary Value Problem for $H^\infty(D)$
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Abstract

Let $W = \bigcup^\infty_{n = 1} W_n$ be an $F_\sigma$ subset of the unit circle of measure 0 and let $\{q_n\}, n \geqslant 1$, be a decreasing sequence with $q_1 \leqslant 1$ and $\lim_n q_n = 0$. There exists an $H$ in $H^\infty(D)$ of norm $q_1$ whose modulus has radial limit along every radius which has radial limit of modulus $q_1$ on $W_1$ and $q_{n + 1}$ on $W_{n + 1} \backslash \bigcup^n_{k = 1}W_k$. If $W$ is simultaneously a $G_\delta$ set, $H$ may be chosen to have no zeros on $C$. It follows that for $W$ countable, say $W = \{e^{iw_n}\}, n \geqslant 1$, there is such an $H$ of norm 1 for which $\lim_{r \rightarrow 1} H(re^{iw_n}) = 1/n$. The proof of the theorem depends on the existence of a special collection of closed sets $\{S_\lambda\}, \lambda \geqslant 1$, real, for which the function $h$, defined by $h(x) = a_n + \lbrack (\inf\{\lambda\mid x \in S\}) - n \rbrack(a_{n + 1} - a_n), a_n = -\ln q_n$, is such that the function $H(w) = \exp(-1/2\pi) \int \lbrack (w + e^{iu})/(e^{iu} - w) \rbrack h(u) du$ has the required properties. Some of the techniques used are similar to those developed in an earlier paper [1].

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