Let me make this simpler. Can we at least show that f_n*\phi_n is pointwise convergent to f?
I'm trying to show that if a sequence of functions, (f_n) is uniformly convergent to a function f, and \phi_n (do people use LaTeX notation here?) is an approximation to the dirac delta (an approximate convolution identity), then f_n*\phi_n, i.e. the sequence of convolutions, is also convergent uniformly to f (for my purposes we can restrain ourselves to showing this on a compact set luckily!). This intuitively makes sense, but I'm having a doggone hard time showing it rigorously. Anyone have any ideas? Is this the right place for this post?
So I'm assuming we can use because on a compact set the function is necessarily bounded? For the only condition I have is that it is continuous (I could use locally integrable instead). Also, is there any way you might clarify the algebra in creating that first inequality you have? I understand rewriting the convolution (very clever!) but I have trouble seeing how we then immediately go to the inequality with the norms in it. Thanks so much.
Since there is uniform convergence, by definition is finite and converges to 0. The compacity is involved in the justification of the second term: why converges uniformly to .
Since this is the result in the case when for all , I supposed you already knew that converges uniformly to . (So that you were asked for a generalization of this fact)
This is a usual property of approximations of delta, a quick one but not a trivial one. On a compact set, if is continuous, . The idea for the proof is to write , and to use the fact that is 0 away from 0 (precisions depend on your own definition) so that the integral reduces to a segment around of small width. Using the uniform continuity of (that's were compacity is involved), we conclude. (Use an for the proof) I guess you already met this proof in your lecture, didn't you?
Thankyou for your help. I have not had any sort of lecture or class on this material. I am attempting to prove some properties of generalized functions for an undergraduate research thesis, and I'm having to figure much out my own (i.e. measure theory, distributions, a lot of functional analysis). I'm trying to show that a class of generalized functions form a sheaf (you know, those things Grothendieck liked) over locally compact spaces, and am starting with . Right now I'm working on a glueing property. Thanks again for all the help.