On the Approximation of Continuous Functions by Linear Combinations of Continuous Functions

Abstract

1. In the present paper we propose to solve the following problem: Let there be given a sequence of arbitrary linearly independent3 functions r1 (x), 2 (x), * * So. (x), *. * *, defined and continuous in the interval a:< x < b. What is a necessary and sufficient condition that there exist linear comn binations a an) gi (x) of the i(x) which converge uniformly toward any i=1 preassignedffunctionf (x), defined and continuous in the interval a _ x < b? This problem was first proposed by E. Schmidt' in his thesis. Schmidt obtains two conditions, one necessary and the other sufficient. The necessary, but not sufficient, condition is as follows: there does not exist any continuous function b (x), not identically zero, which is orthogonal to all the functions gS (x). In other words, if the sequence of functions si (x) has the above property that there exist linear combinations of them converging uniformly to any preassigned function f(x), then there does not exist any continuous function t (x), not identically zero, satisfying the equations