Sharp estimates of best approximations by deviations of Weierstrass-type integrals

We establish the estimates
$$ A_\sigma(f)_P\le KP(f-f*W), $$
where $W$ is a kernel of special type summable on $\mathbb R$ and $A_\sigma(f)_P$ is the best approximation (with respect to a seminorm $P$) of a function $f$ by entire functions of exponential type not greater than $\sigma$. For the uniform and the integral norm we find the least possible constant $K$. The estimates are obtained by linear methods of approximation.