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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Simultaneous approximation and interpolation in $ l\sb{1}$

Author: Joseph M. Lambert
Journal: Proc. Amer. Math. Soc. 32 (1972), 150-152
MSC: Primary 41A65
MathSciNet review: 0291706
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Abstract: In a recent paper of R. Holmes and J. Lambert a geometrical approach was taken to the property of simultaneous approximation and interpolation which is norm preserving (SAIN), first introduced by F. Deutsch and P. Morris. An open question in both papers was if M is the subspace of $ {l_1}$ consisting of the elements having only finitely many nonzero components does the triple $ ({l_1},M,G)$ have property SAIN for all finite dimensional subspaces G contained in $ {l_\infty }$. This question is answered affirmatively by use of a generalization of Yamabe's theorem extending Helly's theorem.

References [Enhancements On Off] (What's this?)

  • [1] F. Deutsch and P. Morris, On simultaneous approximation and interpolation which preserves the norm, J. Approximation Theory 2 (1969), 355-373. MR 40 #6146. MR 0252931 (40:6146)
  • [2] R. Holmes and J. Lambert, A geometrical approach to property (SAIN) (to appear). MR 0344769 (49:9508)
  • [3] J. Lindenstrauss, On extreme points in $ {l_1}$, Israel J. Math. 4 (1966), 59-61. MR 34 #589. MR 0200701 (34:589)
  • [4] H. Yamabe, On an extenison of the Helly's theorem, Osaka Math. J. 2 (1950), 15-17. MR 12, 616. MR 0039914 (12:616a)

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Keywords: Abstract approximation, approximation and interpolation, norm preserving approximation, Helly's theorem, Yamabe's theorem
Article copyright: © Copyright 1972 American Mathematical Society

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