Nonlinear approximation in uniformly smooth Banach spaces

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.