Some properties of principal submodules in the module of entire functions of exponential type and polynomial growth on the real axis

Abuzyarova, Natal'ya Fairbakhovna

doi:10.13108/2016-8-1-3

Cited by 9 publications

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References 8 publications

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“…, then there exist both -invariant subspaces admitting spectral synthesis in a weak [5] and subspaces not possessing this property [3], [6]; 3) if | | > 2 (Λ), then -invariant subspace with a discrete spectrum = iΛ and a residual segment admits a weak spectral synthesis (…”

Section: Introductionmentioning

confidence: 99%

Synthesizable sequence and principle submodules in Schwartz module

Abuzyarova

2020

Ufimsk. Mat. Zh.

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We consider a module of entire functions of exponential type and polynomial growth on the real axis, that is, the Schwarz module with a non-metrizable locally convex topology. In relation with the problem of spectral synthesis for the differentiation operator in the space ∞ (;), we study principle submodules in this module. In particular, we find out what functions, apart of products of the polynomials on the generating function, are contained in a principle submodule. The main results of the work is as follows: despite the topology in the Schwarz module is non-metrizable, the principle submodule coincides with a sequential closure of the set of products of its generating function by polynomials. As a corollary of the main result we prove a weight criterion of a weak localizability of the principle submodule. Another corollary concerns a notion of "synthesizable sequence" introduced recently by A. Baranov and Yu. Belov. It follows from a criterion of the synthesizable sequence obtained by these authors that a synthesizable sequence is necessary a zero set of a weakly localizable principle submodule. In the work we give a positive answer to a natural question on the validity of the inverse statement. Namely, we prove that the weak set of a weakly localizable principle submodule is a synthesizable sequence.

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Section: Introductionmentioning

confidence: 99%

Synthesizable sequence and principle submodules in Schwartz module

Abuzyarova

2020

Ufimsk. Mat. Zh.

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Principal Submodules in the Module of Entire Functions, Which is Dual to the Schwarz Space, and Weak Spectral Synthesis in the Schwartz Space

Abuzyarova

2019

J Math Sci

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Preservation of classes of entire functions defined in terms of growth restrictions along the real axis under perturbations of their zero sets

Abuzyarova

2022

St. Petersburg Math. J.

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Four special subsets of the Schwartz algebra are defined (this algebra consists of all entire functions of exponential type and of polynomial growth on the real axis). Perturbations of the zero sets for functions belonging to each of these subsets are studied. It is shown that the boundedness of the real part of the perturbing sequence is a sufficient and, generally speaking, unimprovable condition for preservation the subset from which the function in question is taken. An application of these results to spectral synthesis problems for differentiation-invariant subspaces of the Schwartz class on an interval of the real line is considered.

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Some properties of principal submodules in the module of entire functions of exponential type and polynomial growth on the real axis

Cited by 9 publications

References 8 publications

Synthesizable sequence and principle submodules in Schwartz module

Synthesizable sequence and principle submodules in Schwartz module

Principal Submodules in the Module of Entire Functions, Which is Dual to the Schwarz Space, and Weak Spectral Synthesis in the Schwartz Space

Preservation of classes of entire functions defined in terms of growth restrictions along the real axis under perturbations of their zero sets

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