Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


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


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A65

Retrieve articles in all journals with MSC: 41A65


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0330875-5
PII: S 0002-9947(1974)0330875-5
Keywords: Approximation, nonlinear approximation, nonlinear functional analysis, uniformly smooth Banach space
Article copyright: © Copyright 1974 American Mathematical Society