Convolution with Dirac delta How to solve this expression:
$$\int_{-\infty}^{\infty} \left[ \delta(k-k_0)f(k)\right]*f(k)dk=?$$
Here $\delta$ represents the Dirac delta function and $*$ represents the convolution over the $k$ variable.
What I think:
$$\int_{-\infty}^{\infty} \left[ \delta(k-k_0)f(k)\right]*f(k)dk=\int_{-\infty}^{\infty} \delta(k-k_0)*\left[f(k)f(k)\right]dk = \int_{-\infty}^{\infty} f(k-k_0)^2dk =\int_{-\infty}^{\infty}f(k)^2dk $$
However, I have doubts about the solution as the influence of the Dirac function seems to disappear?

$$\left[\delta\left(k-k_0\right)\ f(k)\ *\ f(k)\right](x)=f\left(k_0\right)\ f\left(x-k_0\right)$$
$$\int\limits_{-\infty}^{\infty}\left[\delta\left(k-k_0\right)\ f(k)\ *\ f(k)\right](x)\ dx=\int\limits_{-\infty}^{\infty}f\left(k_0\right)\ f\left(x-k_0\right)\ dx$$
