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Nonlinear approximation in uniformly smooth Banach spaces


Authors: Edward R. Rozema and Philip W. Smith
Journal: Trans. Amer. Math. Soc. 188 (1974), 199-211
MSC: Primary 41A65
DOI: https://doi.org/10.1090/S0002-9947-1974-0330875-5
MathSciNet review: 0330875
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Abstract: John R. Rice [Approximation of functions. Vol. II, Addison-Wesley, New York, 1969] investigated best approximation from a nonlinear manifold in a finite dimensional, smooth, and rotund space. The authors define the curvature of a manifold by comparing the manifold with the unit ball of the space and suitably define the ``folding'' of a manifold. Rice's Theorem 11 extends as follows: Theorem. Let X be a uniformly smooth Banach space, and $ F:{R^n} \to X$ be a homeomorphism onto $ M = F({R^n})$. Suppose $ \nabla F(a)$ exists for each a in X, $ \nabla F$ is continuous as a function of a, and $ \nabla F(a) \cdot {R^n}$ has dimension n. Then, if M has bounded curvature, there exists a neighborhood of M each point of which has a unique best approximation from M. A variation theorem was found and used which locates a critical point of a differentiable functional defined on a uniformly rotund space Y. [See M. S. Berger and M. S. Berger, Perspectives in nonlinearity, Benjamin, New York, 1968, p. 58ff. for a similar result when $ Y = {R^n}$.] The paper is concluded with a few remarks on Chebyshev sets.


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  • [1] V. I. Averbuh and O. G. Smoljanov, Differentiation theory in linear topological spaces, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 201-260 = Russian Math. Surveys 22 (1967), no. 6, 201-258. MR 36 #6933. MR 0223886 (36:6933)
  • [2] M. S. Berger and M. S. Berger, Perspectives in nonlinearity. An introduction to nonlinear analysis, Benjamin, New York, 1968. MR 40 #4971. MR 0251744 (40:4971)
  • [3] D. F. Cudia, The geometry of Banach spaces. Smoothness, Trans. Amer. Math. Soc. 110 (1964), 284-314. MR 29 #446. MR 0163143 (29:446)
  • [4] J. Dieudonné, Foundations of modern analysis, Pure and Appl. Math., vol. 10, Academic Press, New York, 1960. MR 22 #11074. MR 0120319 (22:11074)
  • [5] I. C. Gohberg and A. S. Markus, Two theorems on the gap between subspaces of a Banach space, Uspehi Mat. Nauk 14 (1959), no. 5 (89), 135-140. (Russian) MR 22 #5880. MR 0115077 (22:5880)
  • [6] J. R. Rice, Approximation of functions. Vol. II. Nonlinear and multivariate theory, Addison-Wesley, Reading, Mass., 1969. MR 39 #5989. MR 0244675 (39:5989)
  • [7] I. Singer, Best approximation in normed vector spaces by elements of vector subspaces, Editura Academiei Republicii Socialiste România, Bucharest, 1967; English transl., Die Grundlehren der math. Wissenschaften, Band 171, Springer-Verlag, New York and Berlin, 1970. MR 38 #3677; 42 #4937. MR 0235368 (38:3677)
  • [8] L. P. Vlasov, Čebyšev sets and some generalizations of them, Mat. Zametki 3 (1968), 59-69. (Russian) MR 37 #3329. MR 0227745 (37:3329)
  • [9] D. E. Wulbert, Continuity of metric projections, Trans. Amer. Math. Soc. 134 (1968), 335-341. MR 38 #472. MR 0232146 (38:472)
  • [10] -, Uniqueness and differential characterization of approximations from manifolds of functions, Amer. J. Math. 93 (1971), 350-366. MR 45 #4036. MR 0294968 (45:4036)
  • [11] -, Nonlinear approximation with tangential characterization, Amer. J. Math. 93 (1971), 718-730. MR 45 #4037. MR 0294969 (45:4037)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0330875-5
Keywords: Approximation, nonlinear approximation, nonlinear functional analysis, uniformly smooth Banach space
Article copyright: © Copyright 1974 American Mathematical Society

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