It is wellknown that there is a convergence in norm for Fourier series in $L\_p$, if $1<p<\infty$, but are there some examples for pointwise divergence if $p=1,\infty$ in books, or somewhere? I have only found Kolmogorov example, but it is too complicated, and i don't need divergence almost everywhere 



Are there examples of functions in $L_1$ and $L_\infty$ whose Fourier series divergent ("weakly")?
Any function that is in $L^1$ but not in $L^\infty$ will have some point in whose neighbourhood it is unbounded, and the Fourier series is likely to diverge at such a point. A simple example is
$$ f(x) = \ln(1-\cos(x)) = -\ln(2) + \sum\_{n=1}^\infty \dfrac{2 \cos(nx)}{n} $$
where the series diverges at multiples of $2\pi$. 


EDIT: For Fejér's example of a continuous function whose series diverges at a point, see e.g. Edwards, "[Fourier series](https://books.google.ca/books?id=LsDkBwAAQBAJ)", sec. 10.3.1.


