Hilbert-Schmidt Norm: Calculation & Solution

[dirk_mec1]

Homework Statement

http://img523.imageshack.us/img523/4456/56166304yr3.png

Homework Equations

http://img356.imageshack.us/img356/2793/40249940is8.png

The Attempt at a Solution

I defined K:[a,b] --> [a,b] with $$k(s,t) = \frac{(t-s)^{n-1}}{(n-1)!}$$

I found for the norm:

$$\int_a^b \int_a^b \frac{(t-s)^{2n-2}}{(n-1)!^2}\ \mbox{d}s\ \mbox{d}t =0$$

Is this correct?

 

[morphism]

If the norm of blah is zero, then blah is zero. Is blah zero in this case?

 

[dirk_mec1]

If the norm of blah is zero, then blah is zero. Is blah zero in this case?

You're right something is wrong.

But is the integral set up with the correct boundaries?

 

[dirk_mec1]

Ok presuming the boundaries are ok I end up with:

$$||A||_{HS} = \frac{2 (b-a)^n}{((n-1)!)^2 (2n-1)(2n)}$$

Is this correct?

 

[morphism]

Did you remember to take the square root?

 

[Pere Callahan]

You might want to include some characteristic function like $\chi_{\{s\leq t\}}$ in your kernel function.