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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
DOI: https://doi.org/10.1090/S0002-9947-1975-0367208-5
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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  • [1] T. K. Boehme, M. Rosenfeld and M. Weiss, Relations between bounded analytic functionsand their boundary functions, J. London Math.Soc. 1 (1969), 609-618. MR 40 #2870. MR 0249627 (40:2870)
  • [2] M. Denjoy, Sur une propriété des fonctions dérivées, Enseignement Math. 18 (1916), 320-328.
  • [3] M. Heins, Some remarks concerning nonnegative harmonic functions, J. Approximation Theory 5(1972), 118-121. MR 0344486 (49:9225)
  • [4] K. Hoffmann, Banach spaces of analytic functions, Prentice-Hall Ser. in Modern Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR 24 #2844. MR 0133008 (24:A2844)
  • [5] A. J. Lohwater and G. Piranian, The boundary behavior of functions analytic in a disk, Ann. Acad. Sci. Fenn. Ser. A. I. no. 239 (1957), 17 pp. MR 19, 950. MR 0091342 (19:950c)
  • [6] L. H. Loomis, The converse of the Fatou theorem for positive harmonic functions, Trans. Amer. Math. Soc. 53 (1943), 239-250. MR 4, 199. MR 0007832 (4:199d)
  • [7] S. Saks, Theory of the integral, Monografie Mat., vol 7, Warsaw, 1937.
  • [8] Z. Zahorski, Über die Menge der Punkte in welchen die Ableitung unendlich ist, Tôhoku Math. J. 48 (1941), 321-330. MR 10, 359. MR 0027825 (10:359h)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0367208-5
Keywords: Zahorski, extreme points, harmonic functions, $ {H^\infty }$ (unit disc)
Article copyright: © Copyright 1975 American Mathematical Society

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