Characterization of Best Approximations

Abstract

We give a characterization theorem for best approximations from convex sets. This result will prove useful over and over again throughout the book. Indeed, it will be the basis for every characterization theorem that we give. The notion of a dual cone plays an essential role in this characterization. In the particular case where the convex set is a subspace, we obtain the familiar orthogonality condition, which for finite-dimensional subspaces reduces to a linear system of equations called the “normal equations.” When an orthonormal basis of a (finite or infinite-dimensional) subspace is available, the problem of finding best approximations is greatly simplified. The Gram-Schmidt orthogonalization procedure for constructing an orthonormal basis from a given basis is described. An application of the characterization theorem is given to determine best approximations from a translate of a convex cone. Finally, the first three problems stated in Chapter 1 are completely solved.