Pseudo-uniform convexity of 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 is generalized to several complex variables. Specifically, in both of the polydisc and of the ball, weak convergence, together with convergence of norms, is shown to imply norm convergence. As in Newman's work, approximation of by is also considered. It is shown that every function in of the torus, (or in of the boundary of the ball), has a best -approximation which, in several variables, need not be unique.

**[1]**Lars Gȧrding and Lars Hörmander,*Strongly subharmonic functions*, Math. Scand**15**(1964), 93–96. MR**0179373****[2]**C. N. Kellogg,*Pseudo-uniform convexity in 𝐻¹*, Proc. Amer. Math. Soc.**23**(1969), 190–192. MR**0250050**, 10.1090/S0002-9939-1969-0250050-6**[3]**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****[4]**D. J. Newman,*Pseudo-uniform convexity in 𝐻¹*, Proc. Amer. Math. Soc.**14**(1963), 676–679. MR**0151834**, 10.1090/S0002-9939-1963-0151834-X**[5]**W. W. Rogosinski and H. S. Shapiro,*On certain extremum problems for analytic functions*, Acta Math.**90**(1953), 287–318. MR**0059354****[6]**Walter Rudin,*Function theory in polydiscs*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0255841**

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DOI:
https://doi.org/10.1090/S0002-9939-1970-0268656-5

Keywords:
Several complex variables,
polydisc,
torus,
Hardy space ,
uniform convexity,
pseudo-uniform convexity,
weak convergence,
norm convergence,
best approximation,
-approximation of

Article copyright:
© Copyright 1970
American Mathematical Society