Best Interpolatory Approximation in Normed Linear Spaces

Abstract

A theory of best approximation with interpolatory contraints from a finite-dimensional subspaceMof a normed linear spaceXis developed. In particular, to eachx∈X, best approximations are sought from a subsetM(x) ofMwhichdependson the elementxbeing approximated. It is shown that this “parametric approximation” problem can be essentially reduced to the “usual” one involving a certainfixedsubspaceM0ofM. More detailed results can be obtained when (1)Xis a Hilbert space, or (2)Mis an “interpolating subspace” ofX(in the sense of [1]).