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 Mat. Sb., 2013, Volume 204, Number 12, Pages 49–104 (Mi msb8172)

A basis in an invariant subspace of analytic functions

A. S. Krivosheeva, O. A. Krivosheevab

a Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa
b Bashkir State University, Ufa

Abstract: 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.
Bibliography: 25 titles.

Keywords: interpolation, exponential polynomial, invariant subspace, basis.

 Funding Agency Grant Number Russian Foundation for Basic Research 10-01-00233-à Ministry of Education and Science of the Russian Federation 14.Â37.21.0358

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8172

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English version:
Sbornik: Mathematics, 2013, 204:12, 1745–1796

Bibliographic databases:

UDC: 517.537.7
MSC: 30D15, 46E15

Citation: A. S. Krivosheev, O. A. Krivosheeva, “A basis in an invariant subspace of analytic functions”, Mat. Sb., 204:12 (2013), 49–104; Sb. Math., 204:12 (2013), 1745–1796

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8172
• https://doi.org/10.4213/sm8172
• http://mi.mathnet.ru/eng/msb/v204/i12/p49

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. S. Krivosheev, O. A. Krivosheyeva, “A basis in invariant subspace of entire functions”, St. Petersburg Math. J., 27:2 (2016), 273–316
2. A. S. Krivosheev, O. A. Krivosheeva, “Fundamental Principle and a Basis in Invariant Subspaces”, Math. Notes, 99:5 (2016), 685–696
3. O. A. Krivosheyeva, A. S. Krivosheyev, “A representation of functions from an invariant subspace with almost real spectrum”, St. Petersburg Math. J., 29:4 (2018), 603–641
4. O. A. Krivosheeva, “Invariant subspaces with zero density spectrum”, Ufa Math. J., 9:3 (2017), 100–108
5. O. A. Krivosheeva, “Basis in invariant subspace of analytical functions”, Ufa Math. J., 10:2 (2018), 58–77
6. Krivosheev A., Krivosheeva O., “Representation of Analytic Functions By Series of Exponential Monomials in Convex Domains and Its Applications”, Lobachevskii J. Math., 40:9, SI (2019), 1330–1354
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