A basis in an invariant subspace of analytic functions

The existence problem for a basis in a differentiation-invariant subspace of analytic functions defined in a bounded convex domain in the complex plane is investigated. Conditions are found for the solvability of a certain special interpolation problem in the space of entire functions of exponential type with conjugate diagrams lying in a fixed convex domain. These underlie sufficient conditions for the existence of a basis in the invariant subspace. This basis consists of linear combinations of eigenfunctions and associated functions of the differentiation operator, whose exponents are combined into relatively small clusters. Necessary conditions for the existence of a basis are also found. Under a natural constraint on the number of points in the groups, these coincide with the sufficient conditions. That is, a criterion is found under this constraint that a basis constructed from relatively small clusters exists in an invariant subspace of analytic functions in a bounded convex domain in the complex plane.
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