Can you show an example of a function that does not satisfy pointwise convergence theorem hypotheses for Fourier series but that is still expressible as Fourier series?
[Added after comment]
In particular, I want to know if there is a function whose Fourier series converges but which does not satisfy the hypotheses of pointwise convergence theorem:
Let $c \in \mathbb{R}$ and suppose that $f: \mathbb{R} \to \mathbb{R}$ has the following properties:
$f$ has period $2 \pi$;
$f$ is piecewise continuous on $[-\pi,\pi]$;
$D^{+}f$ and $D^{-}f$ exist.
If f is continuous at $c$, then its Fourier series is equal to $f(c)$,
whereas if $f$ has a jump discontinuity at $c$, then its Fourier series is$$\frac{1}{2} \left[f(c^{+}) + f(c^{-}) \right].$$
Best Answer
The Fourier series $\sum_n {e^{inx}\over (1+|n|)^{{1\over 2}+\epsilon}}$, with any positive epsilon, converges in $L^2$, but certainly does not converge pointwise at $0$ when $0<\epsilon<{1\over 2}$.
The series does converge at other points, but certainly not absolutely, and not uniformly pointwise.