Pseudo-uniform convexity of $H^{1}$ in several variables
Author:
Laurence D. Hoffmann
Journal:
Proc. Amer. Math. Soc. 26 (1970), 609-614
MSC:
Primary 46.30; Secondary 32.00
DOI:
https://doi.org/10.1090/S0002-9939-1970-0268656-5
MathSciNet review:
0268656
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Abstract | References | Similar Articles | Additional Information
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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Additional Information
Keywords:
Several complex variables,
polydisc,
torus,
Hardy space <IMG WIDTH="33" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img14.gif" ALT="${H^1}$">,
uniform convexity,
pseudo-uniform convexity,
weak convergence,
norm convergence,
best approximation,
<IMG WIDTH="33" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${H^1}$">-approximation of <IMG WIDTH="28" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="${L^1}$">
Article copyright:
© Copyright 1970
American Mathematical Society