Bohr--Favard Inequality for differences and constants in Jackson--Stechkin Theorem

We consider uniform approximations by trigonometric polynomials. The aim of the paper is to obtain good estimates of the Jackson--Stechkin constants $J_m$. We prove that $J_m \le C 2^{-m+5/2\log_2m}$. Our proof is based on the difference analogue of the Bohr--Favard inequality.