>>12107832That has been done before and is a standard trick in analysis. You usually just do the same trick as with kernel smoothers from statistics but instead of a weighted sum you have a weighted integral.
One aplication is often when you want to show that some subset of smooth functions is dense in some Banachspace.
For example to proof that the set of smooth functions with compact support is dense in L^p you usually use an approximation of the L^p function by smoothing it with such a kernel.
