Does the Delta Family have compact support?

Question

If the delta function is 0 everywhere except at x=0, does this mean for a delta family {${f}_{\alpha}$} it would have compact support for any ${\alpha}$? because I'm assuming each delta distribution in the family will tend to zero as x goes to infinity?

Answer 1

Not every delta family needs to have compact support. Take for example $$f_\epsilon(x) = \frac{1}{\sqrt{\epsilon\pi}} e^{-x^2/\epsilon}\quad(\text{as } \epsilon\to 0).$$

That this is a delta family is shown by $$ \langle f_\epsilon, \varphi \rangle = \int \frac{1}{\sqrt{\epsilon\pi}} e^{-x^2/\epsilon} \, \varphi(x) \, dx = \{ y = x/\sqrt{\epsilon} \} = \int \frac{1}{\sqrt{\epsilon\pi}} e^{-y^2} \, \varphi(\sqrt{\epsilon} y) \, \sqrt{\epsilon}\,dy \\ = \frac{1}{\sqrt{\pi}} \int e^{-y^2} \, \varphi(\sqrt{\epsilon} y) \, dy \to \frac{1}{\sqrt{\pi}} \int e^{-y^2} \, \varphi(0) \, dy = \frac{1}{\sqrt{\pi}} \int e^{-y^2} \, dy \, \varphi(0) \\ = \varphi(0) = \langle \delta, \varphi \rangle. $$