= Solution
{c}
The theorem on <pointwise convergence of a piecewise smooth Fourier series> gives the average of the two one-sided values of the corresponding <periodic extension>. Under the usual piecewise smooth interpretation of the paper's derivative assumption, the <even extension> is continuous at $0$ and at the joined endpoints $\pm\pi$. Thus
$$
\boxed{C_f(x)=f(x)\quad(0\leq x\leq\pi)}.
$$
For the <odd extension>, the values on either side of zero are $-f(0)$ and $f(0)$; those on either side of $\pi$ are $f(\pi)$ and $-f(\pi)$. Therefore
$$
\boxed{S_f(x)=
\begin{cases}
f(x),&0<x<\pi,\\
0,&x=0\text{ or }x=\pi.
\end{cases}}
$$
The endpoint values are also immediate because each <sine> term vanishes there. If $f(0)$ or $f(\pi)$ is nonzero, the <Fourier sine series> does not reproduce that endpoint value. Merely assuming that the derivative is locally piecewise continuous in the open interval would be weaker than the usual piecewise smooth hypothesis; the convergence assertion uses the standard endpoint regularity implicit in that hypothesis.
Back to article page
