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 be the space of real valued functions defined on the unit circle consisting of those functions for which for every in . The extreme points of the unit ball of 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 is a real valued function in and if is a closed subset of , then there is a function in whose restriction to is . If is a subset of of measure 0 and if is a closed subset of disjoint from , there is a function of norm 1 in which is on and 1 on . Finally, we show that if and are as in the preceding result, then there is a function of norm 1 in (unit disc) the modulus of which has radial limit along every radius, which has radial limit of modulus 1 at each point of and radial limit 0 at each point of .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0367208-5

Keywords:
Zahorski,
extreme points,
harmonic functions,
(unit disc)

Article copyright:
© Copyright 1975
American Mathematical Society