Given the function $$ \phi(x) =\begin{cases} 1 & 0<x\leq 1 \\ 2 & 1<x \leq 2 \\ 3 & 2<x \leq 3 \\ 4 & 3<x \leq 4 \end{cases} $$
First, I extended $\phi$ to a periodic function on $[-4, 4]$ such that $\phi(x+4) = \phi(x)$ for all $x \in [-4, 0]$.
(a) To what values does the Fourier series converge at $x=0, 1, 4, 7.4, 40$?
SOLUTION: At $x=0, 4, 40$, the Fourier series converges to $\frac{\phi(0+) + \phi(0-)}{2} = \frac{1+4}{2} = 2.5$. At $x=7.4$, the furrier series converges to $\phi(7.4 \bmod{4}) = \phi(3.4) = 4$. Finally, at $x=1$, the series converges to $\frac{1+2}{2} = 1.5$.
(b) Does the Fourier series converge uniformly to $\phi$?
SOLUTION: No, since it does not converge to $\phi$ point-wise.
(c) Find $a_0$.
SOLUTION: $$ a_0 = \frac{1}{4}\int_{-4}^4 \phi(x)\cos(0)\,\mathrm{d}{x} = \frac{1}{4}(20) = 5 $$