Sharp estimates of best approximations in terms of holomorphic functions of Weierstrass-type operators

We establish the estimates
$$ A_\sigma(f)_P\le KP\bigl(\Phi(\mathcal W)f\bigr), $$
where $W$ is a kernel of special type summable on $\mathbb R$, a function $\Phi$ is holomorphic in the neighborhood of the spectrum of $W$, $A_\sigma(f)_P$ is the best approximation of a function $f$ by entire functions of exponential type not greater than $\sigma$, with respect to seminorm $P$. In some cases for the uniform and the integral norm we find the least possible constant $K$. The estimates are obtained by linear methods of approximation.