Integral approximation of the characteristic function of an interval and the Jackson inequality in $C(\mathbb T)$

An application of the results about integral approximation of the characteristic function of an interval by the subspace $\tau_{n-1}$ of trigonometric polynomials of order at most $n-1$, which were obtained by the authors earlier, to investigation of the Jackson inequality between the best uniform approximation of a continuous periodic function by the subspace $\tau_{n-1}$ and its modulus of continuity of the second order is presented. A respective method of uniform approximation of continuous periodic functions by trigonometric polynomials is constructed.