✦ LIBER ✦

On a Generalized Best Approximation Problem

✍ Scribed by F.S. De Blasi; J. Myjak

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 371 KB
Volume: 94
Category: Article
ISSN: 0021-9045
DOI: 10.1006/jath.1998.3177

No coin nor oath required. For personal study only.

✦ Synopsis

Let C be a closed bounded convex subset of a Banach space E which has the origin of E as an interior point and let p C denote the Minkowski functional with respect to C. Given a closed set X/E and a point u # E we consider a minimization problem min C (u, X ) which consists in proving the existence of a point x~# X such that p

If such a point is unique and every sequence [x n ]/X satisfying the condition lim n Ä + p C (x n &u)=* C (u, X ) converges to this point, the minimization problem min(u, X ) is called well posed. Under the assumption that the modulus of convexity with respect to p C is strictly positive, we prove that for every closed subset X of E, the set E o (X ) of all u # E for which the minimization problem min C (u, X) is well posed is a residual subset of E. In fact we show more, namely that the set E"E o (X) is _-porous in E. Moreover, we prove that for most closed bounded subsets X of E, the set E "E o (X ) is dense in E.

📜 SIMILAR VOLUMES

On Well Posed Generalized Best Approxima

On Well Posed Generalized Best Approximation Problems

✍ Chong Li 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 143 KB

Let C be a closed bounded convex subset of X with 0 being an interior point of C and p C be the Minkowski functional with respect to C. Let G be a nonempty closed, boundedly relatively weakly compact subset of a Banach space X. For a point x # X, we say the minimization problem min C (x, G) is well

On Best Simultaneous Approximation

✍ Chong Li; G.A Watson 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 258 KB

The problem is considered of best approximation of finite number of functions simultaneously. For a very general class of norms, characterization results are derived. The main part of the paper is concerned with proving uniqueness and strong uniqueness theorems. For a particular subclass, which incl

A Note on Pointwise Best Approximation

✍ Guo Tie-Xin; You Zhao-Yong 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 137 KB

The aim of this note is to fill in a gap in our previous paper in this journal. Precisely, we give a new proof of the following theorem: let (0, A, +) be a \_-finite measure space with +(0)>0, 0<p<+ , and Y a separable subspace of a Banach space X. Then Y is proximinal in X iff L p (+, Y) is proximi

Constructing Best Approximations on a Jo

Constructing Best Approximations on a Jordan Curve

✍ Douglas Bridges; Wang Yuchuan 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 249 KB

It is shown, within Bishop's constructive mathematics, that if a point is sufficiently close to a differentiable Jordan curve with suitably restricted curvature, then that point has a unique closest point on the curve.

On Best Approximation by Ridge Functions

✍ V.E Maiorov 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 208 KB

The Limit of Best Generalized Peak Norm

The Limit of Best Generalized Peak Norm Approximations

✍ Chong Li; G.A. Watson 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 94 KB

Norms referred to as generalized peak norms involve a parameter α which lies between 0 and 1. We consider relationships between the limits of solutions to approximation problems involving these norms and solutions of problems using the norms associated with the limiting values.