I'm looking to evaluate the following integral:
$\int_{- \infty}^{\infty} f(x) \delta(x-a) \delta(x-a)dx$
If instead of two identical Dirac deltas we had:
$\int_{- \infty}^{\infty} f(x) \delta(x-a) \delta(x-b)dx$
I believe we would have
$\int_{- \infty}^{\infty} f(x) \delta(x-a) \delta(x-b)dx = f(a) \delta(a-b)$
which with $a=b$ leaves the result undefined. I'm tempted to say the result of the integral is just $f(a)$, but I feel that that may not exactly be rigorous.
Edit:
I should note that this problem came up in computing the power spectrum of a Langevin equation, so the integral is actually:
$\int_{- \infty}^{\infty} f(x) \delta^*(x-a) \delta(x-a)dx$
where * denotes the complex conjugate. I'm not sure if the conjugate of the Dirac delta has any meaning.