I'm reading Katznelson's book "Harmonic Analysis" and there is an exercise that I can't solve :

Show that if the sequence $\left\{ N_j \right\} $ tends to infinity fast enough, then the Fourier series of the function $$ f(t)=\sum_{j>0}2^{-j }K_{N_j}$$ does not converge in $L^1(\mathbb{T})$.

$K_{N_j}$ denotes the Féjer Kernel.

In a first time I tried to compute the Fourier coefficients of $f$. If $N_l<n\leq N_{l+1}$ then $\hat{f}(n)=\sum _{j> l}2^{-j}(1-\frac{|n|}{N_j+1})=\hat{f}(-n)$. So I have tried to show that $(S_n(f))$ is not a Cauchy sequence with the $L^1$ norm but I failed to find a good enough lower bound of $\int _{\mathbb{T}}|\sum _{p}^{q}\hat{f}(n) cos(nt)| dt$