Exercise 30, § 1

a) For every $x_0\in \mathbf{T}$, there exists $f\in \mathscr{C}(\mathbf{T})$ such that the Fourier series of $f$ at $x_0$ does not converge symmetrically towards $f(x_0)$. (Let $s_n(f)$ be the symmetric partial sums of the Fourier series of $f$; consider the linear mappings $f\mapsto s_n(f)(x_0)$ and apply the Banach-Steinhaus theorem.)

b) There exists $f\in L^1(\mathbf{T})$ such that the Fourier series of $f$ does not converge symmetrically towards $f$ in $L^1(\mathbf{T})$.

c) The Fourier transform is not surjective from $L^1(\mathbf{T})$ into the Banach space of sequences $(c_n)_{n\in\mathbf{Z}}$ tending to 0 at infinity. (Apply Cor. 1 of EVT, I, p. 19.)

d) Suppose that $f\in \mathscr{C}^1(\mathbf{T})$. The sequence $s_n(f)$ converges towards $f$ in $\mathscr{C}(\mathbf{T})$.