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* *be a homeomorphism onto* . *Suppose* *exists for each a in X*, *is continuous as a function of a, and* *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 .] The paper is concluded with a few remarks on Chebyshev sets.

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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