I want to prove that if given two functions f and g (f is assumed continuous; g is assumed C^infinity with compact support on R), their convolution (f*g) is (a) well defined and (b) an element of C^infinity. The idea is to later use this result for some problems concerning "approximation to the identity."
Proving well-definedness is easy since (fg) is Riemann integrable and g is compactly supported, so the convolution does not diverge and is finite for all real x.
[i]I am aware of a result back from lower-division DE class that the derivative of the convolution can be "transferred" to either f or g; but, I don't know how to prove this. If someone could lead me in that direction, I think I would be able to prove the result from there.[i]
I also saw a proof where the Fourier transform was used; but, I want to avoid using Fourier analysis (and indeed, more sophisticated proofs involving Young/Minkowski inequalities, elements of functional analysis, measure theory, Lebesgue integration, etc.), and limit myself to just basic concepts concerning L^2 functions (i.e. mean convergence, Holder's inequality, etc.) if these concepts apply at all to any possible proofs.