Integral representation of one class of entire functions

Abstract

In this paper, we study an integral representation of one class of entire functions. Conditions for the existence of this representation in terms of certain solutions of some differential equations are found. We obtain asymptotic estimates of entire functions from the considered class of functions. We also give examples of entire functions from this class

Completeness of the systems of Bessel functions of index $-5/2$

“…in the space L 2 (0; 1); x 4 dx in terms of an entire function with the set of zeros coinciding with the sequence of distinct nonzero complex numbers (ρ k ) k∈N (see . This complements the results of papers [3,5,10,11,13,14,20,21].…”

“…Various approximation properties of the systems of Bessel functions has been studied in many papers (see, for example, [1][2][3][4][5][6][7][10][11][12][13][14][15][16][17][18][19][20][21][22]). In particular, it is well known that the system √ xJ ν (x ρ k ) : k ∈ N is an orthogonal basis for the space L 2 (0; 1) if ν > −1 and ( ρ k ) k∈N is a sequence of positive zeros of J ν (see [1,2,4,12,22]).…”

“…Approximation properties of the systems of Bessel functions for ν < −1, ν / ∈ Z, were investigated in [5,10,11,13,14,20,21]. In particular, at studying of one boundary value problem, in [20] (see also [21]) it was proven that the system…”

Generalized eigenvectors of linear operators and biorthogonal systems

“…Let H be an Euclidean space with inner product •; • : H × H → C, N 0 = N ∪ {0}, m ∈ N 0 , n; m = [n; m] ∩ N 0 and n; m = ∅ if n > m. Suppose that a certain linear operator A : H → H has a countable set of simple eigenvalues {λ k : k ∈ N} and a corresponding system of eigenvectors {ψ k : k ∈ N} that is complete and minimal after removing, for example, the first m ∈ N members, or the adjoint operator of A has no eigenvalues. Such operators arise naturally in the study of some boundary value problems (see, for example, [3,4,10,14,16] and the reference therein), for instance, in the study of boundary value problems for Bessel's equation (see [8,12,13,18,19,25,26]). The problem is how to find a biorthogonal system (U n : n ∈ N\1; m).…”

“…Finding such biorthogonal systems often faces certain difficulties (see [3,4,8,12,13,18,19,25,26]). Sometimes, in the case of simple eigenvalues, such vectors U n can be found by using a notion of a set of generalized eigenvectors which we propose in this paper (see Section 2).…”