An almost exponential sequence of exponential polynomials

A special sequence of exponential polynomials, whose exponentials are divided into relatively small groups, is studied in the article. It is proved that this sequence is almost an exponential sequence for each convex domain of a complex plane. By means of this result necessary and sufficient conditions for the considered sequence to be a basis in a closed and invariant under differentiation subspace of the space of analytic functions in a convex domain are obtained. Two methods of description of the whole class of bases in an invariant subspace, whose elements are exponential polynomials, are given.