Exercise 62, § 1

Identifying $\mathbf{T}$ with the interval $]-1/2,1/2]$. Let $f$ be the function on $\mathbf{R}/\mathbf{Z}$ defined by $f(x) =$ s($x$) (sign function). Let $s_n$ be the symmetric partial sum

of the Fourier series of $f$.

a) For every $x\not= 0$ in $\mathbf{T}$, one has $s_n(x)\rightarrow f(x)$ when $n\rightarrow +\infty$.

b) Let $n\geqslant 1$ and $0< x\leqslant 1/2$. One has

$s_{2n}(x) =$ sin(22 sin($\pi nx\pi x$)).

c) One has

lim$_{n\rightarrow+\infty}s_{2n}(\frac{1}{4n})= 2\int_0^{1/2}$ sin(2$\pi x\pi x$)$dx$,

and this limit belongs to the interval $]1,2[$ (“Gibbs phenomenon”).