On Conditions of Invertibility in the Sense of Ehrenpreis in the Schwartz Algebra

Abuzyarova, Natal'ya Fairbakhovna

doi:10.1134/s1995080221060032

Cited by 8 publications

(3 citation statements)

References 9 publications

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“…Suppose the converse is true: let φ be a slowly decreasing function of exponential type σ < π such that φ(Λ) = 0. By Remark 1 in § 3.2 above and Lemma 1 in [19] we can assume that the set of zeros Z φ of φ lies on the real axis. We introduce some notation: M = Z φ \ Λ, n Zφ (t) is the number of points in Z φ lying in the interval (0, t] if t > 0, and (−n Zφ (t)) is the number of points in Z φ lying in [t, 0) if t < 0.…”

Section: 2mentioning

confidence: 99%

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Representation of invariant subspaces of the Schwartz space

Abuzyarova

2022

Sb. Math.

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A subspace $W$ of the Schwartz space $C^{\infty} (a,b)$ such that the restriction of the operator of differentiation to $W$ has a discrete spectrum is considered. Conditions for the representation of $W$ as a direct algebraic and topological sum of two subspaces, namely, the residual subspace and the subspace spanned by the exponential monomials from $W$, are investigated. One condition ensuring this representation turns out to be the existence of a functional annihilating $W$ such that the Fourier-Laplace transform of this functional is a slowly decreasing entire function. A new characteristic of complex sequences is introduced and investigated. Using this characteristic, the condition that an invariant subspace is equal to the direct sum of its residual and exponential subspaces can be put into a form that is similar to the previously discovered conditions for the possibility of weak spectral synthesis. Bibliography: 19 titles.

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Section: 2mentioning

confidence: 99%

“…By Theorem 1 in [19] must hold, where γ = σ/π ∈ (M −1 0 , 1). Setting t j = k j M 0 and t j = k j M 0 + [ln 3 (k j M 0 )], j = 0, 1, .…”

Section: 2mentioning

confidence: 99%

Representation of invariant subspaces of the Schwartz space

Abuzyarova

2022

Sb. Math.

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“…where the symbol * denotes the convolution. We say that ϕ ∈ P is invertible in the sense of Ehrenpreis if the division theorem is valid for it (see [1], [2]).…”

Section: Introductionmentioning

confidence: 99%

One property of zero set of function invertible in the sense of Ehrenpreis in the Schwartz algebra

Abuzyarova,

Idrisova,

Khasanova

2021

Preprint

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We consider those elements of the Schwartz algebra of entire functions which are Fourier-Laplace transforms of invertible distributions with compact supports on the real line. These functions are called invertible in the sense of Ehrenpreis. The presented result concerns with the properties of zero subsets of invertible in the sense of Ehrenpreis function f . Namely, we establish some properties of the zero subset formed by zeros of f laying not far from the real axis.

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Perturbations of an Integer Sequence as Zero Sets of Divisors in Some Spaces of Entire Functions

Abuzyarova

2023

Math Notes

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On Conditions of Invertibility in the Sense of Ehrenpreis in the Schwartz Algebra

Cited by 8 publications

References 9 publications

Representation of invariant subspaces of the Schwartz space

Representation of invariant subspaces of the Schwartz space

One property of zero set of function invertible in the sense of Ehrenpreis in the Schwartz algebra

Perturbations of an Integer Sequence as Zero Sets of Divisors in Some Spaces of Entire Functions

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