Mollifiers for a function on $[0,T]\times R$

Question

Do you have any precise and comprehensive reference for how to build a sequence $\phi^\epsilon(t,x)$ of $C^\infty([0,T]\times R)$ functions that converge uniformly on $(0,T)\times R$ to a given function $\phi:[0,T]\times R$? I know in $x$ the standard mollifier works when we have no time dependence, but I am stuck on how to get the "standard mollifier" and the convolution in time. I read something about partitions of unity but, as mentioned, I would appreciate a concise approach, not an "argument by hand waving". Thanks!