What is the relationship between Fourier transforms and Plancherel's theorem?

[titaniumx3]

Fourier transform question

Homework Statement

http://img410.imageshack.us/img410/852/question3jh8.gif

Homework Equations

I will be using the following definitions and theorems:

http://img338.imageshack.us/img338/4173/moderatedecreasevd4.gif

http://img260.imageshack.us/img260/7461/fouriertransonmodzo5.gif

http://img338.imageshack.us/img338/3530/plancherelmodvv2.gif

The Attempt at a Solution

I've done part (a) and shown that the Fourier transform of f(x) is 4\,{\frac { \left( \sin \left( \pi \,\xi \right) \right) ^{2}}{{\xi}^<br /> {2}}} but on part (b) I am a bit lost. I know how to apply Plancherel's theorem but the function inside the modulus (i.e. {\frac { \left( \sin \left( \xi \right) \right) ^{2}}{{\xi}^{2}}}) is slightly different to the Fourier transform I got previously and I'm not sure how to relate them.

Please help!

 

[titaniumx3]

BTW, I've shown f and it's Fourier transform are both functions of "moderate decrease". I'm also assuming you have to use part (a) to answer part (b). If not, how would you calculate something like that?

 

[titaniumx3]