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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
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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Keywords: Approximation, nonlinear approximation, nonlinear functional analysis, uniformly smooth Banach space
Article copyright: © Copyright 1974 American Mathematical Society

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