= Solution
{c}
The <Fourier cosine series> converges to the continuous even $2\pi$-<periodic extension>. If $r\in[-\pi,\pi]$ represents $x$ modulo $2\pi$, its value is
$$
F_e(x)=\min(|r|,\pi/2).
$$
It has value zero at $x=2j\pi$, rises linearly with distance from these points until height $\pi/2$, and then stays flat until the next descending ramp. Thus the sketch on $[-3\pi,3\pi]$ has minima at $-2\pi,0,2\pi$, and plateaux centred at the odd multiples of $\pi$. Its large <partial sums> round the corners and closely follow the whole graph.
The <Fourier sine series> converges, away from the odd multiples of $\pi$, to the odd $2\pi$-<periodic extension>
$$
F_o(x)=\operatorname{sgn}(r)\min(|r|,\pi/2)\quad(-\pi<r<\pi).
$$
At every odd multiple of $\pi$ its value is zero, the midpoint of the jump from $\pi/2$ to $-\pi/2$. At even multiples of $\pi$ it passes continuously through zero. Thus on every interval $((2j-1)\pi,(2j+1)\pi)$ it has a negative plateau, a rising linear segment through $2j\pi$, and a positive plateau. The jump size is $\pi$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-v-fourier-extensions.png]
{title=Periodic limits and 60-term Fourier partial sums}
The blue curves are the <partial sums> through $n=60$. The open endpoint circles and filled zero dots in the lower panel distinguish the one-sided limits from the value of the <Fourier sine series> at each jump. Near these jumps the <partial sums> exhibit the <Gibbs phenomenon>: the oscillations become concentrated in narrower regions as the number of terms grows, while their limiting overshoot remains about $0.08949$ times the jump, here about $0.281$.
Back to article page
