- #1
pellman
- 684
- 5
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 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 ## for all real x, 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.
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 ## for all real x, 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.