Direct and strong converse inequalities for approximation with Fejér means

We present upper and lower estimates of the error of approximation of periodic functions by Fejér means in the Lebesgue spaces {L}_{2{\pi }}^{p}. The estimates are given in terms of a K-functional for 1\le p\le \infty and in terms of the first modulus of continuity in the case 1\lt p\lt \infty . We pay attention to the involved constants.