The definitions of distributions that I have seen (for instance https://en.wikipedia.org/wiki/Distribution_(mathematics)#Distributions ) define a distribution as a map whose domain is a set of(adsbygoogle = window.adsbygoogle || []).push({}); test functions.A defining quality of test functions is that they have compact support, which for most practical purposes means they are non-zero on only a finite interval (or intervals).

So why then are we comfortable writing something like ## \int^{\infty}_{-\infty} \delta(x-a) x^2 dx = a^2 ##? In the language of distributions this would be ## \delta_a(\phi) = a^2 ## where ## \phi(x) = x^2 ##. If we mean that ## \phi(x) = x^2 ## forallrealx, is this a problem?

Is there an applied assumption here that we don't mean ## \phi(x) = x^2 ## on the entire real line but only on some compact subspace? If so, is this potentially a problem in physics which often deals with functions whose domain is explicitly infinite? I have never seen anyone applying the Dirac delta function have any concern that the function applied to is of compact support.

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# Distributions on non-test functions

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