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Proceedings of the American Mathematical Society

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A boundary value problem for $ H\sp{\infty }(D)$


Author: Rotraut Goubau Cahill
Journal: Proc. Amer. Math. Soc. 67 (1977), 241-247
MSC: Primary 30A72; Secondary 26A24
DOI: https://doi.org/10.1090/S0002-9939-1977-0457728-4
MathSciNet review: 0457728
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Abstract: Let $ W = \bigcup\nolimits_{n = 1}^\infty {{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\nolimits_{k = 1}^n {{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^{i{w_n}}}\} ,n \geqslant 1$, there is such an H of norm 1 for which $ {\lim _{r \to 1}}H(r{e^{i{w_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} + [(\inf \{ \lambda \vert x \in S\} ) - n]({a_{n + 1}} - {a_n}),{a_n} = - \ln \;{q_n}$, is such that the function $ H(w) = \exp ( - 1/2\pi )\smallint [(w + {e^{iu}})/({e^{iu}} - w)]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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DOI: https://doi.org/10.1090/S0002-9939-1977-0457728-4
Keywords: Zahorski, $ {H^\infty }(D)$, radial limit
Article copyright: © Copyright 1977 American Mathematical Society