BASIS IN AN INVARIANT SPACE OF ENTIRE FUNCTIONS

The existence of a basis is studied in a space of entire functions invariant under the differentiation operator. It is proved that every such space possesses a basis consisting of linear combinations of generalized eigenvectors. These linear combinations are formed within groups of exponents of arbitrarily small relative diameter. A complete description of the way to split the exponents into groups is obtained. Also, a criterion is found for the existence of a basis constructed by groups of zero relative diameter (so-called relatively small groups). In this connection a new criterion is obtained for the finiteness of the lower indicator of an entire function of exponential type.