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Proceedings of the American Mathematical Society

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Support points of the unit ball of $ H\sp{p}$ $ (1\leq p\leq \infty )$


Author: Yusuf Abu-Muhanna
Journal: Proc. Amer. Math. Soc. 89 (1983), 229-235
MSC: Primary 30D55; Secondary 30D50, 46J15
DOI: https://doi.org/10.1090/S0002-9939-1983-0712628-X
MathSciNet review: 712628
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Abstract: The following results are obtained for the $ {H^p}$ class, over the open unit disc, whenever $ 1 \leqslant p \leqslant \infty $.

(1) $ f$ is a support point of the unit ball of $ {H^p}$, whenever $ 1 \leqslant p < \infty $, if and only if $ {\vert\vert f \vert\vert _p} = 1$ and $ f$ is of the form $ f(z) = {[Q(z)]^{2/p}} \cdot W(z)$ where $ W(z)$ is a function analytic in the closed unit disc and nonvanishing on its boundary and $ Q(z)$ is either a nonzero constant or a polynomial with all of its zeros on the boundary of the unit disc.

(2) $ f$ is a support point of the unit ball of $ {H^\infty }$ if and only if $ f$ is a finite Blaschke product.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0712628-X
Keywords: Analytic function, continuous linear functional, $ {H^p}$ classes, support point, extreme point
Article copyright: © Copyright 1983 American Mathematical Society