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This content is available through Read Online (Free) program, which relies on page scans. 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 $H^\infty(D)$
Rotraut Goubau Cahill
Proceedings of the American Mathematical Society
Vol. 67, No. 2 (Dec., 1977), pp. 241247
Published by: American Mathematical Society
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.
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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Proceedings of the American Mathematical Society © 1977 American Mathematical Society