Synthesizable differentiation-invariant subspaces

Baranov, Anton; Belov, Yurii

doi:10.1007/s00039-019-00474-8

Cited by 9 publications

(6 citation statements)

References 20 publications

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“…2], by the proven lemma we conclude that Φ ∈ , . We shall need some definitions and facts from the general theory of de Branges spaces [14], and also from work [7], in which this theory was successfully employed for studying -invariant subspaces in the Schwartz space (in particular, for the proof of Theorem A).…”

Section: ]mentioning

confidence: 99%

“…( 3) for each function ∈ ℋ, the function * ( ) = (¯) belongs to ℋ and has the same norm as . By means of this axiomatic description, it was established in [15], [7,Sect. 2,Thm.…”

Section: ]mentioning

confidence: 99%

“…The latter of the three above formulated results can be interpreted as follows: given a complex sequence Λ and a relatively closed in ( ; ) segment such that | | > 2 (Λ), there exists a unique -invariant subspace ⊂ ℰ( ; ) with a discrete spectrum = −iΛ and a residual segment = and this subspace is of form (1.2). In view of this interpretation, the authors of work [7] called a sequence Λ ⊂ C with (Λ) < +∞ syntheziable if a -invariant subspace with a spectrum −(iΛ) and a residual segment [− (Λ); (Λ)] is unique; in this case it is of form (1.2). In that work a complete description of synthesizable sequences was provided.…”

Section: Introductionmentioning

confidence: 99%

“…In that work a complete description of synthesizable sequences was provided. In particular, it was shown that the if the system of exponential monomials Exp Λ constructed by the sequence −(iΛ) is complete or it has a finite defect in the space 2 (− (Λ); (Λ)), then Λ is a synthesizable sequence [7,Prop. 3.2].…”

Section: Introductionmentioning

confidence: 99%

“…If the system Exp Λ has an infinite defect in 2 (− (Λ); (Λ)), then the syntesizability of Λ is determined by the conditions of the following criterion [7,Thm. 1.3…”

Section: Introductionmentioning

confidence: 99%

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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: ]mentioning

confidence: 99%

“…( 3) for each function ∈ ℋ, the function * ( ) = (¯) belongs to ℋ and has the same norm as . By means of this axiomatic description, it was established in [15], [7,Sect. 2,Thm.…”

Section: ]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…If the system Exp Λ has an infinite defect in 2 (− (Λ); (Λ)), then the syntesizability of Λ is determined by the conditions of the following criterion [7,Thm. 1.3…”

Section: Introductionmentioning

confidence: 99%

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Synthesizable sequence and principle submodules in Schwartz module

Abuzyarova

2020

Ufimsk. Mat. Zh.

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Complementing Nonuniqueness Sets in Model Spaces

Baranov

2024

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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Synthesizable differentiation-invariant subspaces

Cited by 9 publications

References 20 publications

Synthesizable sequence and principle submodules in Schwartz module

Synthesizable sequence and principle submodules in Schwartz module

Complementing Nonuniqueness Sets in Model Spaces

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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