Check out the papers by Accardi and Boukas

[added by S. Carnahan: The relevant part of their first ArXiv paper is that they regularize powers of $\delta$, not by setting $\delta^n(x) = c_n \delta(x)$ for some real $c_n$ (which they find to work poorly for their purposes), but by using two variables and setting $\delta(t-s)^n = \delta(s)\delta(t-s)$ for all $n \geq 2$. This definition allows them to define certain representations of Lie algebras by Fock space methods. As far as I can tell, this does not yield a workable definition of $\delta^2$ for analysis on the real line.]

Check out the papers by Accardi and Boukas

[added by S. Carnahan: The relevant part of their first ArXiv paper is that they regularize powers of $\delta$, not by setting $\delta^n(x) = c_n \delta(x)$ for some real $c_n$ (which they find to work poorly for their purposes), but by using two variables and setting $\delta(t-s)^n = \delta(s)\delta(t-s)$ for all $n \geq 2$. This definition allows them to define certain representations of Lie algebras by Fock space methods. As far as I can tell, this does not yield a workable definition of $\delta^2$ for analysis on the real line.]

Check out the papers by Accardi and Boukas

Check out the papers by Accardi and Boukas

[added by S. Carnahan: The relevant part of their first ArXiv paper is that they regularize powers of $\delta$, not by setting $\delta^n(x) = c_n \delta(x)$ for some real $c_n$ (which they find to work poorly for their purposes), but by using two variables and setting $\delta(t-s)^n = \delta(s)\delta(t-s)$ for all $n \geq 2$. This definition allows them to define certain representations of Lie algebras by Fock space methods. As far as I can tell, this does not yield a workable definition of $\delta^2$ for analysis on the real line.]