In his paper, the concepts of metric curvature and folding of a Cl-representable manifold in a normed linear space are studied. With certain restrictions on the metric curvature and/or folding, one can obtain a neighborhood of unique best approximation from the manifold, and in some cases, the manifold can be shown to be Chebyshev. Several familiar examples, including some classes of T-polynomials, are given.1. Introduction. The purpose of this paper is to study unique best approximation from subsets of normed linear spaces which are Cl-manifolds with boundary. In order to study such problems from a general geometric vantage point, John R. Rice [12] introduced the concepts of folding and metric curvature (originally called curvature) in smooth, rotund, and finite-dimensional spaces. These concepts were generalized to uniformly smooth spaces by two of the authors in [13] (see also [5], [6]). In 3 and 4 of the present work, we closely examine these concepts for Cl-manifolds with boundaries and obtain results which demonstrate several connections between local uniqueness, metric curvature, and folding.In C[a, b], there are nonlinear sets, for instance, the set of rational functions of degree no greater than (m, n), that are Chebyshev. We show that the fact that Haar embedded manifolds are Chebyshev (proved in a special case by Daniel Wulbert 17] and generalized by D. Braess [2]) follows from general results on the metric curvature of the manifold (see Theorem 6.1). For example, we see in 7 that the set N is Chebyshev in C[a, b], where 0 < a < b.In L([a, b], x) it is well known [14, p. 368] that a nonconvex boundedly compact subset is not Chebyshev (i.e., there is a point which does not have a unique best approximation from the set). In 7 we exhibit for the first time some familiar subsets M of L2 ([a, b],/x) each of which is a Cl-manifold with boundary and has a neighborhood of unique best approximation for M. Thus, for points in this neighborhood of M, steepest descent methods may be attempted. Many nonlinear regression problems fall into the above category.Both of the above results are special cases of Theorem 5.1 in this paper, which basically states that every manifold M with boundary which has finite metric curvature and positive folding has a neighborhood of unique best approximation from M. * Received by the editors August 16, 1974.
Abstract.In this paper, it is proved that splines of order k (k > 2) have property SAIN. The proof of this result is based on the important properties of B-splines. Introduction.In a recent manuscript [5], Lambert proved that the twice continuously differentiable cubic splines possess property SAIN (simultaneous approximation and interpolation which is norm preserving) on C [a, b] where the interpolatory constraints are point evaluations. In this paper we establish the more general result for splines of any order greater than 1 while at the same time supplying a simple proof. More precisely, we will show
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