Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V resit / Question 4 / a

The Fourier cosine series and Fourier sine series use the respective formsTheir Fourier coefficients, obtained from Fourier orthogonality, areIn particular the constant term is $a_0/2={\pi }^{-1}{\int }_0^{\pi }f(t)dt$. The Fourier cosine series is the Fourier series of the even extension $f(|x|)$, made $2\pi$-periodic. The Fourier sine series is the Fourier series of the odd extension, equal to $f(x)$ for $0<x<\pi$ and $-f(-x)$ for $-\pi <x<0$, again made $2\pi$-periodic.

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 $\pm \pi$. ThusFor 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 $-f(\pi )$. ThereforeThe 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.