Origin of the name 'test functions'

This is a very simple question really: where did the name 'test functions', used nowadays when speaking of infinitely differentiable and compactly supported functions, come from? More to the point: is there a mathematical reason these functions are called that way?

Solution 1:

Okay, I am pretty sure I found the very first instance of this name.

In "Linear Partial Differential Equations, With Constant Coefficients" (Annals, 1946), Salomon Bochner defined a "testing function" to be a function of compact support, and spoke of "testing functions of class $C^k$", in a discussion concerning weak solutions and weak differentiation.

Salomon Bochner's review of Laurent Schwartz is perhaps the first instance where "testing functions" becomes associated with smooth functions of compact support, which space, in French, following the tradition of Sobolev's classic paper on the the Cauchy problem, is called "espace fondamental".

As evident in J. L. Lions' review of Gel'fand and Shilov's Russian text (also this), by 1960 it is already in the vocabulary of the experts that "test functions" should be identified with "fonctions fondamentales", and both meaning smooth functions of compact support.