Is seems to me that it makes sense to presume some relations between derivatives of Dirac delta functions and its powers. I wonder, whether someone proposed a similar theory?

Particularly, it could make sense that

$$\frac{\pi^3 \delta(0) ^3}{6}+\frac{\pi \delta(0) }{12}=-\pi \delta''(0)$$

In such formalism, the more "inner" peaks of the distribution (when we approximate it with a smooth function) would be represented by higher powers of $\pi\delta(0)$.

I wonder whether similar relations ever arised as a formal method or in any other form.

**UPDATE**

This paper seemingly makes an attempt, but their results are quite different. Particularly, following their paper

$$\delta''(t)=16\pi^2\delta^3(t)$$

which is different from the above expression. To my view, their approach has many weaknessess, particularly, in my view $\delta''(0)$ should have the opposite sign than $\delta^3(0)$ (which is the case if we approximate delta function with a smooth curve).

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