On the order of approximation of a continuous $2\pi$-periodic function by Fejer and Poisson means of its Fourier series

Let $\sigma_n(f)$ and $P_r(f)$ be, respectively, the Fejer and Poisson means of the Fourier series of the function $f$. The present work considers problems associated with the rapidity of approximation of a continuous $2\pi$-periodic function by means of Fejer and Poisson processes, and gives, in particular, an upper bound to the deviation of the Fejer and Poisson processes from the function in terms of moduli of continuity, and a lower bound to $\|\sigma_n(f)-f\|$ in terms of functionals composed of best approximations to the function $f$; in addition, some relationships among the quantities $\|P_r(f)-f\|$ and $\|\sigma_n(f)-f\|$ are established.