COMMENTS TO ASSIGNMENT 8


PROBLEM 2

(a) Parseval; 
(b) integrate the Fourier series (don't forget the constant!); 
(c) evaluate at a suitable point (why can you do that?).


PROBLEM 3

This is the Dirichlet kernel (not the Fejer kernel).
Also, there was a factor of 1/(2\pi) missing in the
integral.


PROBLEM 4

In other words, show that the map f -> f-hat 
that maps L^2 to the space of square summable
bi-infinite sequences is onto. (We already know 
that it is one-to-one).

For the proof, use the completeness of L^2 and 
Parseval's identity.  

Note that (c_k) need not be summable !