I am currently having difficulties with a problem, and require some advice. The question is:
Let $$\phi(x)=\begin{cases} 1 & 0<x\leq 1\\ 2 & 1<x\leq 2\\ 3 & 2<x\leq 3\\ 4 & 3<x\leq 4 \end{cases}$$ Extend $\phi$ by periodicity to all of $\mathbb R$, i.e. $\phi(x+4)=\phi(x)$. Consider the Full Fourier Series: $$\phi(x) = \frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos\left(\frac{n\pi x}{2}\right)+b_n\sin\left(\frac{n\pi x}{2}\right)$$ To what values will this Fourier series converge to at $x=0, x=1,x=4,x=7.4$ and $x=40$? Does the Fourier series converge uniformly to $\phi(x)$ ? Explain
Now, I know by definition we have: $$a_n = \frac{1}{2}\int_{-2}^2\phi(x)\cos\left(\frac{n\pi x}{2}\right) dx, \>\>\>\>\>\> b_n = \frac{1}{2}\int_{-2}^2\phi(x)\sin\left(\frac{ n\pi x}{2}\right) dx$$
At $x=0$, $a_n=2$ and $b_n=0$ just by plugging in the value $x=0$, and so $\phi(x)$ would converge to $\phi(0) = \frac{a_0}{2}+2$? Am I approaching this right? If so, then the first question is simple.
But I am having trouble with the second part. Is it right to say by the Uniform Convergence of periodic Fourier Series that because $\phi(-2)=\phi(2)$, we have convergence of the Fourier Series of $\phi$ on $[-2,2]$ and thus all of $\mathbb R$ by periodic extension?
For the convergence at those points you should be applying various "convergence tests" from earlier in the chapter. You know, those theorems that say that if $f$ satisfies some condition or other then the Fourier series at $x$ converges to ___. The point to the problem is to see if you understand those theorems.
Similarly, the point to the question about uniform convergence is to see if you remember the big theorem "A uniform limit of continuous functions is ___".