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?

  • $\begingroup$ For what it's worth: The early works of Laurent Schwartz on distributions don't use that name, see here and here and if I remember correctly, his books don't use that name either (at least not in the French editions). (As far as I know test functions are called fonctions test in France). $\endgroup$ – t.b. Jun 24 '11 at 17:05
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    $\begingroup$ I was told that the name comes from the context of Razmadze's lemma, where you can guarantee a function is identically zero by "testing" it with (i.e. integrating against) such test functions. $\endgroup$ – Miha Habič Jun 24 '11 at 17:22
  • $\begingroup$ @Miha: Do you have a reference where that lemma appeared? I never heard it called this way. $\endgroup$ – t.b. Jun 24 '11 at 17:29
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    $\begingroup$ @Miha: You mean the fundamental lemma of the calculus of variations? I have not seen it called like that either. $\endgroup$ – Jose L. Lykón Jun 24 '11 at 17:34
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    $\begingroup$ @Theo: I do mean the fundamental lemma of the calculus of variations. It was referred to as such in my analysis lectures, but a Google search comes up with nothing, apart from a brief mention in a MacTutor article link $\endgroup$ – Miha Habič Jun 24 '11 at 17:45

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.

  • $\begingroup$ In the 1963 English edition of Gel'fand-Shilov, §1 of Chapter I is called "Test Functions and Generalized functions". In §1.2 we find the quotation "We shall call these functions the test functions, and we shall call $K$ the space of test functions." (original emphasis), where $K = \mathscr{D}(\mathbb{R}^n)$ in the usual notation. $\endgroup$ – t.b. Jun 25 '11 at 0:26
  • $\begingroup$ Excellent references as well, thanks! $\endgroup$ – Jose L. Lykón Jun 25 '11 at 3:25
  • $\begingroup$ @Theo: thanks for the info. And for fixing the typos. $\endgroup$ – Willie Wong Jun 25 '11 at 12:51
  • $\begingroup$ @Theo: ah, the 'info' in my previous comment refers to both that and your comment above; so yes, I saw it. I suspected the story ended tragicly, but that's of course the part I could parse the least. $\endgroup$ – Willie Wong Jun 25 '11 at 13:41

Test functions are not necessarily compactly supported functions. Test functions belonging to the function space S(R), the Schwartz space of functions, have unbounded support.


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