Yesterday when I was processing with Ahlfors’** ***Complex Analysis,* I encountered the problem about *Residue.*

Problem:

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.

**Definition**:

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**.

Proof:

,

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