Can there be a full Fourier series on range $0<x<L$?

Question

I am stuck at this problem:

Compute the Fourier Series for the function $x^2$ on the interval $0<x<L$ using as a basis of function with boundary conditions $u'(0)=0$ and $u'(L)=0$. Sketch the partial sums of the series for $1,2,3$ terms.

Answer 1

If you find a Fourier series for $x \mapsto x^2$ on $[-L,L]$ then since the function is even, the series will only contain terms of the form $\cos (n \pi {x \over L})$ terms.

The derivatives of the $\cos$ terms are $-n \pi { 1\over L} \sin (n \pi {x \over L})$ which are zero at $x=0,L$ as required.

The series will converge pointwise everywhere (since the function is Lipschitz).