Yesterday when I was processing with Ahlfors’ Complex Analysis, I encountered the problem about Residue.
If , is analytic and bounded, assuming , then
It is an equation, but more than an equation. It is the meaning of Reproduce Kernel.
When learning Functional Analysis, I found some interesting schemes for proofs, esp. with Riesz Theorem and its corollaries. One of them is the Reproduce Kernel, which is more of an operator in Hilbert Space.
Assuming is a Hilbert Space of complex-valiued functions on set . We will say is a Reproduce Kernel Hilbert Space if every linear map:
is continuous at every .
By Riesz Theorem, there exists a function of H s.t.
We can define the function this way:
This is called reproduce kernel.
Let’s see what is its good properties:
- , which is the inner product;
- iff every .
However if there is an orthonormal basis for H, , then
Let’s prove the problem on Ahlfors’ book, we will use Residue Theorem.