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?