Why distributions can not be multiplied ?

[zetafunction]

Why distributions can not be multiplied ??

why in general can not give a meaningful expression for

$$\delta (x) \delta ^{m} (x)$$ or $$H(x) \delta (x)$$

for example the Fourier transform (with respect to 'x') of the expression (theoretically)

$$\int_{-\infty}^{\infty}dt (x-t)^{m}t^{n} =g(x)$$

 

[CompuChip]

Remember that distributions are defined by integrating them with test functions, for example, $\delta$ is defined by
$$\int_{-\infty}^{\infty} \delta(x - a) f(x) \, \mathrm dx := f(a)$$
for test functions f.

So what do you propose that
$$\int_{-\infty}^{\infty} \delta(x) \delta(x - 1) f(x) \, \mathrm dx$$
evaluates to?
Zero? f(1/2)? f(0)f(1) ?