Local Linear Dependence and the Vanishing of the Wronskian

Abstract

on the interval I. However, it has also long been known that for n functions which are only (n 1)-times differentiable (so that their Wronskian is defined) the sufficiency part of the above statement no longer holds. Peano [12] seems to have been the first to point this out, and Bocher [3] has given an example which shows that even if the functions involved are infinitely differentiable on I, the identical vanishing of the Wronskian is still not sufficient to imply their linear dependence on I. It was then recognized that the functions involved must satisfy other conditions, supplementary to the vanishing of their Wronskian, in order to guarantee their linear Peano [13] and Bocher [5] have given such conditions (for example, Lemma 3 of Sec. 2 of this paper) and Bocher has shown that his conditions include those of Peano. In this paper we look at this problem from a slightly different point of view. Namely, instead of placing linear dependence in the forefront and looking for a condition to supplement the identical vanishing of the Wronskian, rather, we shall put the vanishing of the Wronskian in the forefront and ask for a kind of generalized dependence (necessarily weaker than linear dependence) which is equivalent to it. This point of view has led the author to define a new type of dependencef relation for functions of a real variable which he has called linear dependence. It is shown that local linear dependence and the identical