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.
The first inequality is: for every , (I bound by its maximum value, and because is positive (?)), hence we have an upper bound independent of . This gives what I wrote.
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 .
I guess you mean .
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.