## Pseudo-uniform convexity of $H^{1}$ in several variables

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- by Laurence D. Hoffmann
- Proc. Amer. Math. Soc.
**26**(1970), 609-614 - DOI: https://doi.org/10.1090/S0002-9939-1970-0268656-5
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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.## References

- Lars Gȧrding and Lars Hörmander,
*Strongly subharmonic functions*, Math. Scand.**15**(1964), 93–96. MR**179373**, DOI 10.7146/math.scand.a-10732 - C. N. Kellogg,
*Pseudo-uniform convexity in $H^{1}$*, Proc. Amer. Math. Soc.**23**(1969), 190–192. MR**250050**, DOI 10.1090/S0002-9939-1969-0250050-6 - V. P. Havin,
*Spaces of analytic functions*, Math. Analysis 1964 (Russian), Akad. Nauk SSSR Inst. Naučn. Informacii, Moscow, 1966, pp. 76–164 (Russian). MR**0206694** - D. J. Newman,
*Pseudo-uniform convexity in $H^{1}$*, Proc. Amer. Math. Soc.**14**(1963), 676–679. MR**151834**, DOI 10.1090/S0002-9939-1963-0151834-X - W. W. Rogosinski and H. S. Shapiro,
*On certain extremum problems for analytic functions*, Acta Math.**90**(1953), 287–318. MR**59354**, DOI 10.1007/BF02392438 - Walter Rudin,
*Function theory in polydiscs*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0255841**

## Bibliographic Information

- © Copyright 1970 American Mathematical Society
- 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