Best simultaneous approximation in a normed linear space

Abstract

The problem we consider is the approximation of m elements (xₗ,...,xₘ) in a normed linear space by one element in a subset of that normed linear space. To each element in the ordered m-tuple (xₗ,...,xₘ) we associate corresponding continuous seminorms |·|ₗ,...,|·|ₘ. Then using these seminorms to measure the closeness xₒ has to each (xₗ,...,xₘ) we get m errors Eₖ = |xₒ-xₖ|ₖ, k = l,...,m. A best simultaneous approximation to (xₗ,...,xₘ) is the xₒ which minimizes a prescribed combination of the errors Eₗ,...,Eₘ.

Using ideas from mUltiple objective prOgrammi~ we also introduce various generalizations of Bacopoulos' definition of best vectorial approximations. These sets of approximations are shown to have close relations to simultaneous approximations.

The main results of this paper are existence and characterization theorems which generalize well known functional analytica theorems from the theory of best approximation in a normed linear space. When the approximations are from a finite dimensional space we obtain a generalization of Singer's theorem using the extreme elements of the unit ball.

Many authors have considered the above problem in several special cases: approximating a bounded function, finding the restricted Chebyshev center of a finite set, approximating a function with a restricted range, approximating a function with absolute and relative errors, approximating a function and its derivatives, approximating a random function with m possible realizations.