The a question is to show that Cesaro sums converge slowly
$||f - \sigma_Nf||_{L_2} \geq ||f||_2/N$,
where $\sigma_N f$ is the Cesaro sum of $\hat{f}(n)$, that is the convolusion of Fejer kenerl and f.
It can be solved by noting $||f - \sigma_Nf||_{L_2}^2 = \sum|\widehat{f - \sigma_Nf}(n)|^2$ and with another assumption $\hat{f}(0) = 0$.
But what shall I do with the $L_1$ norm? i.e., what is the lower bound for $||f - \sigma_Nf||_1$?
Thank you for any hints!