On the Sharpness of Jackson's Inequality in the Spaces $L_p$ on the Half-Line with Power Weight

In the space $L_p$, $1\leq p<2$, on the half-line with power weight, Jackson's inequality between the value of the best approximation of a function by even entire functions of exponential type and its modulus of continuity defined by means of a generalized shift operator is well known. The question of the sharpness of the inequality remained open. For the constant in Jackson's inequality, we obtain a lower bound, which proves its sharpness.