Set of exponents for interpolation of exponential series by sums in all convex domains

We study the problem of multiple finite-sum interpolation in all convex domains of the complex plane of absolutely converging exponential series with exponents from a given set $\Lambda$. We obtain the following solvability criterion for this problem: each direction at infinity must be a limit direction for the set $\Lambda$. We prove that this problem is equivalent to certain particular problems of simple interpolation and to pointwise approximation of exponential series by sums in some specific domains. The same description is also obtained for problems of simple interpolation and pointwise approximation in all convex domains by functions that belong to subspaces invariant with respect to the differentiation operator and admit spectral synthesis in spaces of holomorphic functions on these domains.