  We establish new estimates for the constant $J_a(k,\alpha)$ in the
Brudnyi-Jackson inequality for approximation of $f \in C[-1,1]$ by algebraic
polynomials:
  $$ E_{n}^a (f) \le J_a(k, \alpha) \ \omega_k (f, \alpha \pi /n ), \quad
\alpha >0 $$
  The main result of the paper implies the following inequalities
  $$ 1/2< J_a (2k, \alpha) < 10, \quad n \ge 2k(2k-1), \quad
  \alpha \ge 2 $$
