On one Leontiev-Levin theorem

Abstract

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

On the necessary and sufficient conditions for the measurability of a positive sequence

“…Let δ ∈ (0; 1); then, according to Lemma 1 from [4] and Lemma 2.1 from [1], the series of inequalities is valid:…”