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Proceedings of the American Mathematical Society

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Pseudo-uniform convexity of $ H\sp{1}$ in several variables

Author: Laurence D. Hoffmann
Journal: Proc. Amer. Math. Soc. 26 (1970), 609-614
MSC: Primary 46.30; Secondary 32.00
MathSciNet review: 0268656
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Abstract: A convergence theorem of D. J. Newman for the Hardy space $ {H^1}$ is generalized to several complex variables. Specifically, in both $ {H^1}$ of the polydisc and $ {H^1}$ of the ball, weak convergence, together with convergence of norms, is shown to imply norm convergence. As in Newman's work, approximation of $ {L^1}$ by $ {H^1}$ is also considered. It is shown that every function in $ {L^1}$ of the torus, (or in $ {L^1}$ of the boundary of the ball), has a best $ {H^1}$-approximation which, in several variables, need not be unique.

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Keywords: Several complex variables, polydisc, torus, Hardy space $ {H^1}$, uniform convexity, pseudo-uniform convexity, weak convergence, norm convergence, best approximation, $ {H^1}$-approximation of $ {L^1}$
Article copyright: © Copyright 1970 American Mathematical Society

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