Invariant Spaces of Entire Functions

• Published in: Mathematical Notes Vol. 109; no. 3-4; pp. 413 - 426
• Main Authors: Krivosheev, A. S., Krivosheeva, O. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-03-2021
• Subjects: 14/34; 639/766/189; 639/766/530; 639/766/747; Mathematics; Mathematics and Statistics; analytic continuation; entire function; exponential monomial; invariant subspace; series of exponentials
• ISSN: 0001-4346; 1573-8876
• DOI: 10.1134/S0001434621030093

Subspaces of the space of analytic functions on a convex domain in the complex plane that are invariant with respect to the differentiation operator are studied. The problem of continuing all functions in an invariant subspace to entire functions is investigated. A simple geometric criterion for the existence of such a continuation is obtained. A criterion for the functions from an invariant subspace to be represented by a series of exponential monomials is also obtained. These monomials are the eigenfunctions and generalized eigenfunctions of the differentiation operator on the invariant subspace.