Fundamental principle and a basis in invariant subspaces

Abstract. In the paper we show that each analytic solution of a homogeneous convolution equation with the characteristic function of minimal exponential type is represented by a series of exponential polynomials in its domain. This series converges absolutely and uniformly on compact subsets in this domain. It is known that if the characteristic function is of minimal exponential type, the density of its zero set is equal to zero. This is why in the work we consider the sequences of exponents having zero density. We provide a simple description of the space of the coefficients for the aforementioned series. Moreover, we provide a complete description of all possible system of functions constructed by rather small groups, for which the representation by the series of exponential polynomials holds. G. Pólya [1] proved that for the zero density¯(Λ) = 0 each sum of the series has a convex domain; this function is analytic. In work [2] G. Pólya generalized this result. Consider the homogeneous convolution equationwhere is a complex-valued compactly supported measure andis the Laplace transform of the functional generating the convolution operator called the characteristic function of the operator . Assume that ( ) is a function of the minimal exponential O.A. Krivosheeva, Invariant subspaces with zero density spectrum.

“…As above, for the convex domain˜= ( , /2), Statement 5) of Theorem 3.1 in work [12] holds. According this theorem, the function is expanded into the series…”

Section: =1mentioning

confidence: 80%

“…In particular, it converges uniformly on each compact set in . Thus, Statement 2) in Theorem 3.1 in work [12] is true. According this theorem, the identity Λ ( ) = 0 holds true.…”

Section: =1mentioning

confidence: 85%

See 1 more Smart Citation

Invariant subspaces with zero density spectrum

Krivosheeva¹

2017

“…Finally, how to describe the space of the coefficients of the series over the system ℰ(Λ, )? In the case of a bounded convex domain , the answers to these questions were obtained in works [7]- [11]. In particular, there was found the criterion of the existence of the basis in the subspace constructed by the partition into relatively small groups , namely, into the groups whose diameters and the number of the points are infinitesimal as → ∞ in comparison with the absolute values of these points.…”

Section: Introductionmentioning

confidence: 99%

Basis in invariant subspace of analytical functions

Krivosheeva¹

2018

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.

“…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: 99%

On one Leontiev-Levin theorem

Krivosheev¹,

Kuzhaev²

2017

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