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#### TaylorM0192

##### New member

- Mar 6, 2012

- 5

Let me first just say, I posted this thread on mathhelpforum.com - but I read a post by Plato somewhere or another recommending here instead, since apparently the other site had some bad customer service issues...

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 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 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.

Thanks!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.

Thanks!