On a dimensional reduction method. I. The optimal selection of basis functions

Abstract

This paper is the first in a series of three, which analyze an adaptive approximate approach for solving ( n + 1 ) (n + 1) -dimensional boundary value problems by replacing them with systems of equations in n -dimensional space. In this approach the unknown functions of ( n + 1 ) (n + 1) variables are projected onto finite linear combinations of functions of just n variables. This paper shows how the coefficients of these linear combinations can be chosen optimally.