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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On bounded functions satisfying averaging conditions. I

Author: Rotraut Goubau Cahill
Journal: Trans. Amer. Math. Soc. 206 (1975), 163-174
MSC: Primary 30A76; Secondary 31A05
MathSciNet review: 0367208
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Abstract: Let $ R(T)$ be the space of real valued $ {L^\infty }$ functions defined on the unit circle $ C$ consisting of those functions $ f$ for which $ li{m_{h \to 0}}(1/h)\int_\theta ^{\theta + h} {f({e^{it}})dt = f({e^{i\theta }})} $ for every $ {e^{i\theta }}$ in $ C$. The extreme points of the unit ball of $ R(T)$ are found and the extreme points of the unit ball of the space of all bounded harmonic functions in the unit disc which have non-tangential limit at each point of the unit circle are characterized. We show that if $ g$ is a real valued function in $ {L^\infty }(C)$ and if $ K$ is a closed subset of $ \{ {e^{i\theta }}\vert li{m_{h \to 0}}(1/h)\int_\theta ^{\theta + h} {g({e^{it}})dt = g({e^{i\theta }})\} } $, then there is a function in $ R(T)$ whose restriction to $ K$ is $ g$. If $ E$ is a $ {G_\delta }$ subset of $ C$ of measure 0 and if $ F$ is a closed subset of $ C$ disjoint from $ E$, there is a function of norm 1 in $ R(T)$ which is on $ E$ and 1 on $ F$. Finally, we show that if $ E$ and $ F$ are as in the preceding result, then there is a function of norm 1 in $ {H^\infty }$ (unit disc) the modulus of which has radial limit along every radius, which has radial limit of modulus 1 at each point of $ F$ and radial limit 0 at each point of $ E$.

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Keywords: Zahorski, extreme points, harmonic functions, $ {H^\infty }$ (unit disc)
Article copyright: © Copyright 1975 American Mathematical Society

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