A closedness of set of Dirichlet series sum

Abstract. In the space of holomorphic functions in a convex domain, we study a problem on interpolation by sums of the series of exponentials converging uniformly on compact subsets of the domain. The discrete set of multiple interpolation nodes is located on the real axis in the domain and has the unique finite accumulation point. We obtain a solvability criterion in terms of distribution of limit directions at infinity for the exponents of exponentials.Keywords: holomorphic function, convex domain, interpolation with multiplicities, series of exponentials, closed ideal, closed submodule, strong dual space, duality Mathematics Subject Classification: 30E05 Formulation of the problem and preliminariesLet be a convex domain in C. We denote by ( ) the space of holomorphic functions in with the topology of uniform convergence on compact sets in . We consider an arbitrary discrete set of complex numbers Λ = { } ∈N in C.We denoteThe convergence of the series of exponentials is supposed to be absolute for each point ∈ , then ([1]) such series converges in the topology of space ( ). For a multi-dimensional situations it was shown, for instance, in work [2].Suppose that set ∩ R is non-empty. We assume that in we are given an infinite discrete set real interpolation nodes ℳ = { } ∞ =1 , ℳ ⊂ ∩ R. We also assume that to each node ∈ ℳ a multiplicity ∈ N is associated. If , ∈ ( ), we shall write ∼ = on ℳ, if ( ) ( ) = ( ) ( ) for each ∈ N and = 0, 1, . . . , − 1. In ( ) we consider the following problem on interpolation by means of series of exponentials with real nodes: For an arbitrary set of nodes ℳ ⊂ ∩ R and for each function ∈ ( ) there exists a function ∈ Σ(Λ, ), such that ∼ = on ℳ.By the classical result of interpolation by holomorphic functions [3, Corol. 1.5.4], this problem can be formulated in terms of traditional notations: For each interpolation data ∈ C, ∈ N, = 0, 1, . . . , − 1, there exists a function ∈ Σ(Λ, ), such that ( ) ( ) = , for each and .S.G. Merzlyakov, S.V. Popenov, Interpolation by series of exponentials in ( ) with real nodes.

Section: Proposition B Kernel Kersupporting

confidence: 61%

“…The most general formulation of this problem for a complex plane was considered in [19]. In work [20] series with real exponents Λ were studied in details.…”

Section: Proposition B Kernel Kermentioning

confidence: 99%

Interpolation by series of exponentials in $H(D)$ with real nodes

Merzlyakov¹,

Popenov²

2015

“…Let us prove the implication 1 =⇒ 2. Since¯0(Λ) = , by Lemma 2.1 in work [11] (see also [12,Lm. 5]) there exists a measurable sequence…”

Section: By Lemma 1 and Formula (2) This Implies The Identity¯(λ) =¯(λ)mentioning

confidence: 85%

On one Leontiev-Levin theorem

Krivosheev¹,

Kuzhaev²

2017

Self Cite

Abstract. In this work we study the relations between different densities of a positive sequence and related quantities. More precisely, in the work we consider the upper density, the maximal density introduced by G. Polya, the logarithmic block density, which seems to be introduced first by L.A. Rubel. In particular, there were obtained relations between the maximal density and a quantity being very close to the logarithmic block density. The results of these studies are applied for generalizing the classical statement obtained independently by A.F. Leont'ev B.Ya. Levin on the completeness in a convex domain of a system of exponential monomials with positive exponents; we generalized this statement for the exponents with no density. We find out that for the aforementioned result, one can weaken the condition of the measurability of the sequence (that is, the existence of a density) and replace it by the identity of upper and maximal densities. Namely, we obtain a condition under which there holds the criterion of the completeness of the system of exponential monomials in convex domains. It should be noted that this criterion holds in a rather wide class of convex domains, for instance, having vertical and horizontal symmetry axes. The main role in solving this issues was played by the results of the studies by L.A. Rubel and P. Malliavin on relation between the growth of an entire function of exponential type along the imaginary axis and the logarithmic block density of its positive zeroes. These results were applied by these authors for studying the completeness of the system of exponentials in a horizontal strip.Keywords: density of sequence, entire function, completeness, convex domain. Mathematics Subject Classification: 30D10Let Λ = { , } ∞ =1 be a multiple sequence of positive numbers. Here { } ∞ =1 is an unbounded strictly increasing sequence and is the natural number called the multiplicity of an element , 1. We recall that the upper density and lower density of the sequence Λ are respectively the quantitiesis the counting function of the sequence Λ, that is, the number of its elements counting the multiplicities located in the semi-interval (0, ]. If¯(Λ) = (Λ), the sequence Λ is called measurable and the quantityis well-defined and called the density of the sequence Λ. We let ℰ(Λ) = { } ∞, −1 =1, =0 . We shall say that a system of functions ℰ(Λ) is complete in a convex domain ⊆ C if it is complete in the space ( ) of functions analytic in the domain with the topology of uniform convergence on compact subsets in .A.S. Krivosheyev, A.F. Kuzhaev, On one Leontiev-Levin theorem. c ○ Krivosheyev A.S., Kuzhaev A

“…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%

Basis in invariant subspace of analytical functions

Krivosheeva¹

2018

Self 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.