Sheet2 B1 Fourier Series

(a) Prove the localisation property of Fourier series: if two continuous $2 π$-periodic functions $f$ and $g$ are equal in an open interval containing $x_0$, their Fourier series either both converge at $x_0$ or both diverge at $x_0$. Solution. The function $f-g$ is 0 in $I$, so it is $1-$Hölder continuous, by theorem 2.7 $S_N(f)(x_0)-S_N(g)(x_0) → 0$ pointwise as $N→∞$, so $S_N(f)$ and $S_N(g)$ either both converge at $x_0$ or both diverge at $x_0$. (b) In the lecture, we will prove that there is a continuous function $f_0$ whose Fourier series diverges at 0. Use (a) to construct a continuous function $f_S$ whose Fourier series diverges at every point of a given finite subset $S=\{s_1Solution. Let $f_S$ be a 2π-periodic continuous function s.t. $f_S(x)=f_0(x-s_i)$ for all $x$ in a neighborhood $U_i$ of $s_i$ such that $U_i$ are disjoint. Since $f_0(x-s_i)$ diverge at $x=s_i$, by (a) $f_S$ diverge at $x=s_i$ for each $1≤i≤n$.