On some extremal problems for entire functions of exponential type

In this paper we consider a number of extremal problems for nonnegative and integrable entire functions of exponential type $\leqslant\sigma$ (the class $\mathcal{E}_{1,\sigma}^+$).
The problems under consideration have the following form. Let $\Lambda_\rho$ be a translation invariant operator with a locally integrable symbol $\rho(x)$, $x\in\mathbb{R}$, such that $\rho(x)=\overline{\rho(-x)}$, $x\in\mathbb{R}$. For a fixed $\sigma>0$, it is required to find the following constants:
\begin{equation*} \begin{split} M^{\ast}(\rho,\sigma)&=\sup\{(\Lambda_\rho f)(0):f\in\mathcal{E}_{1,\sigma}^{+},\ \|f\|_1=2\pi\},\\ m^{\ast}(\rho,\sigma)&=\inf\{(\Lambda_\rho f)(0):f\in\mathcal{E}_{1,\sigma}^{+},\ \|f\|_1=2\pi\}. \end{split} \end{equation*}
This general problem reduces to an equivalent extremal problem for positive-definite functions, the solution of which is known. As consequence, we obtained exact values of $M^{\ast}(\rho,\sigma)$ and $m^{\ast}(\rho,\sigma)$ for a number of different symbols $\rho$. In particular, we consider cases where $\Lambda_\rho$ is a differential or difference operator of a special form.