Let \begin{equation*} (A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k \cos(kt) + b_k \sin(kt), \\ a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} x(t) \cos(kt) dt, \\ b_k = \frac{1}{\pi} \int_{-\pi}^{\pi} x(t) \sin(kt) dt. \end{equation*} I proved that for $L_p[-\pi;\pi], \;p\not=1,\infty$ we have that $A_n \xrightarrow{s} A\;$("strongly"="in norm") using Minkowski inequality and Hausdorff–Young inequality, and that we do not have uniform convergence (if look at $f_n(t)=\sin((n+1)t)$ ).
How do I prove that there is no even "weak" convergence in $L_1[−π;π]$, i.e An⇀̸Id in $L_1[−π;π]$ ?
The answer by Coudy show that there exists an example of function in $\mathbb L^1$ whose Fourier series does not convergence in measure, hence it cannot be convergent in $\mathbb L^1$.