Good evening to all,

For

, I am considering the mapping:

,

I would like to show that

actually maps

to

.

I have done the following: First I notice that:

Hence to show the intended I must show that when the operator

is applied on a function

then the outcome is a function which is also in

.

Using rules of calculus I can argue that the function

is continous. Now I just need to show that this function has compact support and my job is done. The support of a function

is the smallest closed set outside which the function is equal to zero, i.e.

. Because I don't have a specific function to work with I'm finding it a bit difficult to show the case of compact support. How do I show that the function

has compact support?

Thanks.