Approximation of functions of bounded $p$-variation by polynomials in terms of the Faber–Schauder system

In this paper the best polynomial approximation in terms of the system of Faber–Schauder functions in the space $C_p[0,1]$ is studied. The constant in the estimate of Jackson's inequality for the best approximation in the metric of $C_p[0,1]$ and the estimate of the modulus of continuity $\omega_{1-1/p}$ are refined.