can anyone please explain to me why convolution in does not have an identity element?
there is no such that
If you want a more mathematical argument, then you could try something like this. let and define f by Then . When |x|>ε that gives . You can probably believe (and you may even be able to prove) that this implies that u(t)=0 whenever |x|>ε. Since ε is arbitrary, it follows that u is zero everywhere except at 0.
However, does have what is called an approximate identity. That is a sequence of elements of norm 1 such that (in the -norm) as k→∞, for all . For example, when n=1 you can take . This gives a sequence of functions that have increasingly large sharp spikes at the origin and are very small elsewhere.
both explanations are pretty convincing.
I've been trying with characteristic functions but haven't managed to prove the claim.
By the way, if
how could one prove that if
I've been trying with Holder but can't seem to get anywhere, I really should need a good functional analysis book...
Thanks again for your help.