# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## On bounded functions satisfying averaging conditions. IHTML articles powered by AMS MathViewer

by Rotraut Goubau Cahill
Trans. Amer. Math. Soc. 206 (1975), 163-174 Request permission

## 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 $\{ {e^{i\theta }}|li{m_{h \to 0}}(1/h)\int _\theta ^{\theta + h} {g({e^{it}})dt = g({e^{i\theta }})\} }$, 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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