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Basis in an invariant space of entire functions

Authors: A. S. Krivosheev and O. A. Krivosheeva
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 27 (2015), nomer 2.
Journal: St. Petersburg Math. J. 27 (2016), 273-316
MSC (2010): Primary 30D10
Published electronically: January 29, 2016
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Abstract: 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.

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Additional Information

A. S. Krivosheev
Affiliation: Institute of Mathematics With Computer Centrum, Russian Academy of Sciences, ul. Chernyshevskogo 112, 450048 Ufa, Russia

O. A. Krivosheeva
Affiliation: Baskhir State University, ul. Zaki Validi 32, 450076 Ufa, Russia

Keywords: Entire function, basis, invariant subspace, interpolation
Received by editor(s): February 5, 2014
Published electronically: January 29, 2016
Article copyright: © Copyright 2016 American Mathematical Society