Schwarz space $S(\mathbb{R})$ is dense inside $L_p(\mathbb{R})$-spaces

Question

I was wondering why the schwarz functions $S(\mathbb{R})$ are dense inside the $L_p(\mathbb{R})$ spaces and I was reading this answer, but I don't understand why the $g_t$ are in $S(\mathbb{R})$. Could someone explain this?

Answer 1

Note the following first:

The convolution with any function with compact support is given by an integral on a compact interval.

So now you can establish the second point:

The convolution with any $C^\infty$ function given by an integral on a compact interval gives a $C^\infty$ function (basically you can take the derivatives inside).