Proper convolution notation

Question

What would be the correct way to write down the convolution in "star" notation for these two functions; $h(t)$ and $\delta(t-x)$. $\delta$ is the Dirac delta function. The integral notation should be

$$ \int_{-\infty}^{\infty}h(t-\tau)\delta(\tau-x)d\tau$$

It feels a little awkward with the delta function in there.

Answer 1

How about $h* \delta(\cdot - x)$?

Answer 2

Maybe not as rigorous as the previous comments, but probably a bit nicer in notation: You could define an ensemble of functions $\delta_\tau(t):= \delta(t-\tau)$ and then write the convolution as $h*\delta_\tau$.