I'm studying for analysis and I have the following question (given by a professor), which does not seem correct the way it is stated
be a function, so that
. Show that for any continuous function
we have that the sequence
on the compact sets of IR.
First, the way it is written doesn't even seem to be make sense because it is integrating with respect to x so I believe that the integral should actually be with resepect to y. But that doesn't seem to be true, for instance if I define
and 0 otherwise. This integrates to 1. Moreoever, If I pick a function like f(x)= x^2 which is continuous on R, the function g_n doesn't even converge.
I think the only way this problem could make sense is if the function
is defined to have compact support.
No! The question is essentially correct (and is how you construct approximations to Dirac delta).
is supported on
, so the first equality in the last equation should be
which then gives
pointwise, and uniformly on compacts.
The only typo in the question is integrate with respect to y, not x.
shouldn't be in the support. Also, spoke too soon about the question being OK. I think there will be a problem if f grows way too fast, e.g.
, which means the convolution
blows up dramatically for each fixed n. You want something like
rather than just
. Of course, it is easiest if
has compact support....