Pavlov–Korevaar–Dixon interpolation problem with majorant in convergence class

We study an interpolation problem in the class of entire functions of exponential type determined by some majorant in a convergence class (non-quasianalytic majorant). In a smaller class, when the majorant possessed a concavity property, similar problem was studied by B. Berndtsson with the nodes at some subsequence of natural numbers. He obtained a solvability criterion for this interpolation problem. At that, he applied first the Hörmander method for solving a $\overline{\partial}$-problem. In works by A.I. Pavlov, J. Korevaar and M. Dixon, interpolation sequences in the Berndtsson sense were applied successfully in a series of problems in the complex analysis. At that, there was found a relation with approximative properties of the system of powers $\{z^{p_n}\}$ and with the well known Polya and Macintyre problems.
In this paper we establish the criterion of the interpolation property in a more general sense for an arbitrary sequence of real numbers. In the proof of the main theorem we employ a modification of the Berndtsson method.