Definition of distribution, pseudofunction, and tempered distribution

Question

As a physicist, I do not know the difference

What is a pseudofunction? How is it different from a distribution? What is a tempered distribution?

If possible could you give examples and tell what the difference is?

Answer 1

Like Henning, I have also never seen "pseudofunction" as a technical term. Check your source.

A tempered distribution is a continuous linear functional on the set of Schwarz functions or "rapidly decreasing functions" in the same way as a regular distribution is a continuous linear functional on $C^\infty_c$, the set of smooth compactly supported functions.

Since there are more Schwarz functions than functions in $C^\infty_c$, the tempered distributions are a subset of the usual ones. The nice thing is about tempered distributions, $\mathcal S'$, is that the Fourier transform is a linear isomorphism $\mathcal S' \to \mathcal S'$, but in general you can't compute the Fourier transform of a usual distribution (at least the result won't be a distribution in general).

A typical example of a distribution that is not tempered is $f(x) = e^{x^2}$.