Exercise 53, § 1

Soit $f\in \mathscr{L}^1(\mathbf{T})$.

(a) Let $k\geqslant 1$ be an integer and suppose that $f\in \mathscr{C}^k(\mathbf{T})$. The Fourier series of $f$ converges to $f$ in $\mathscr{C}^{k-1}(\mathbf{T})$. If $k\geqslant 2$, the Fourier series of $f$ converges absolutely to $f$ in $\mathscr{C}(\mathbf{T})$.

(b) For every integer $N\geqslant 1$, denote by $f_N$ the function on $\mathbf{T}$ such that

for $t\in \mathbf{T}$. Let $X\subset \mathbf{T}$ be a set such that the restriction of $f$ to X is continuous. Exercise prove that $f_N$ converges to $f$ uniformly on X.

(c) Let $X\subset \mathbf{T}$ be an open set such that the restriction of $f$ to X is of class $C^1$. Exercise prove that the Fourier series of $f$ converges symmetrically to $f$ uniformly on X.