Basis in an invariant space of entire functions

Krivosheev, A. S.; Krivosheeva, O. A.

doi:10.1090/spmj/1387

Cited by 10 publications

(7 citation statements)

References 14 publications

(14 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The next two statements were proved in work [9], see Theorems 2.1 and 5.1 in the cited work. In the second statement the main idea of introducing quantity Λ ( ) is clarified.…”

Section: )mentioning

confidence: 73%

See 1 more Smart Citation

Invariant subspaces in half-plane

Krivosheev¹,

Krivosheeva

Rafikov

2020

Ufimsk. Mat. Zh.

View full text Add to dashboard Cite

We study subspaces of functions analytic in a half-plane and invariant with respect to the differentiation operator. A particular case of an invariant subspace is a space of solutions a linear homogeneous differential equation with constant coefficients. It is known that each solution of such equations is a linear combination of primitive solutions, which are exponential monomials with exponents being possibly multiple zeroes of a characteristic polynomial. The existence of such representation is called Euler fundamental principle. Other particular cases of invariant subspaces are spaces of solutions of linear homogeneous differential, difference and differential-difference equations with constant coefficients of both finite and infinite orders as well as of more general convolution equations and the systems of them. In the work we study the issue on fundamental principle for arbitrary invariant subspaces for arbitrary invariant subspaces of analytic functions in a half-plane. In other words, we study representation of all functions in an invariant subspace by the series of exponential monomials. These exponential monomials are eigenfunctions and adjoint functions for the differentiation operator in an invariant subspace. In the work we obtain a decomposition of an arbitrary invariant subspace of analytic functions into a sum of two invariant subspaces. We prove that the invariant subspace in an unbounded domain can be represented as a sum of two invariant subspaces. Their spectra correspond to a bounded and unbounded parts of a convex domain. On the base of this result we obtain a simple geometric criterion of the fundamental principle for an invariant subspace of analytic functions in a half-plane. It is formulated just in terms of the Krisvosheev condensation index for the sequence of exponents of the mentioned exponential monomials.

show abstract

“…The next two statements were proved in work [9], see Theorems 2.1 and 5.1 in the cited work. In the second statement the main idea of introducing quantity Λ ( ) is clarified.…”

Section: )mentioning

confidence: 73%

“…A complete solution of the fundamental principle problem for non-trivial invariant subspaces of entire functions was obtained in work [9]. It was proved that the validity of the fundamental principle in each such subspace is equivalent to the finiteness of the condensation index Λ .…”

Section: Introductionmentioning

confidence: 99%

Invariant subspaces in half-plane

Krivosheev¹,

Krivosheeva

Rafikov

2020

Ufimsk. Mat. Zh.

View full text Add to dashboard Cite

show abstract

“…By Theorem 4.1 in work [12], there exists a sequence Λ ′ = { , 1} ∞ =1 having no common points with Λ and such that 1)Λ = Λ∪Λ ′ is the zero set (counting multiplicities) of an entire function˜of an exponential type, that is,˜∈ C ;…”

Section: =1mentioning

confidence: 99%

“…Then Theorem 5.1 in work [12] implies that there exist positive numbers We let ( ) =˜( ) and = , 1. By property 1) of the sequenceΛ we obtain Statement 1) of Lemma 2.1 for each ∈ C. Statement 2) of this lemma is implied by the definition of the sets and the fact that they are mutually disjoint.…”

Section: =1mentioning

confidence: 99%

“…Concerning unbounded domains, here only two particular cases were studied when is a plane or a half-plane. The complete solution of the representation problem for invariant subspaces of entire functions was obtained in work [12]. The invariant subspaces in the half-plane we mostly studied in the case of a simple positive spectrum (see [6], [13]) and almost real spectrum [14].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Basis in invariant subspace of analytical functions

Krivosheeva¹

2018

Ufimskii Matematicheskii Zhurnal

View full text Add to dashboard Cite

Abstract. In this work we study the problem on representing the functions in an invariant subspace of analytic functions on a convex domain in the complex plane. We obtain a sufficient condition for the existence of a basis in the invariant subspace consisting of linear combinations of eigenfunctions and associated functions of differentiation operator in this subspace. The linear combinations are constructed by the system of exponential monomials, whose exponents are partitioned into relatively small groups. We apply the method employing the Leontiev interpolating function. At that, we provide a complete description of the space of the coefficients of the series representing the functions in the invariant subspace. We also find necessary conditions for representing functions in an arbitrary invariant subspace admitting the spectral synthesis in an arbitrary convex domain. We employ the method of constructing special series of exponential polynomials developed by the author.

show abstract

The distribution of singular points of the sum of a series of exponential monomials on the boundary of its domain of convergence

Krivosheev¹,

Krivosheeva²

2020

Sb. Math.

View full text Add to dashboard Cite

The problem of the distribution of the singular points of the sum of a series of exponential monomials on the boundary of the domain of convergence of the series is considered. Sufficient conditions are found for a singular point to exist on a prescribed arc on the boundary; these are stated in purely geometric terms. The singular point exists due to simple relations between the maximum density of the exponents of the series in an angle and the length of the arc on the boundary of the domain of convergence that corresponds to this angle. Necessary conditions for a singular point to exist on a prescribed arc on the boundary are also obtained. They are stated in terms of the minimum density of the exponents in an angle and the length of the arc. On this basis, for sequences with density, criteria are established for the existence of a singular point on a prescribed arc on the boundary of the domain of convergence. Bibliography: 27 titles.

show abstract

Basis in an invariant space of entire functions

Cited by 10 publications

References 14 publications

Invariant subspaces in half-plane

Invariant subspaces in half-plane

Basis in invariant subspace of analytical functions

The distribution of singular points of the sum of a series of exponential monomials on the boundary of its domain of convergence

Contact Info

Product

Resources

About