Complete biorthogonal systems of Bessel functions

Abstract

Let ν ≥ −1/2 and (ρ k ) k∈N be a sequence of nonzero complex numbers such that ρof Bessel functions of the first kind of index ν ≥ −1/2 is exact (i.e. complete and minimal) in the space L 2 (0; 1), then its biorthogonal system is also exact in L 2 (0; 1).

Generalized eigenvectors of linear operators and biorthogonal systems

“…x s k J ν (x s k ) in the space L 2 ((0; 1); x 2p dx), where ν ≥ 1/2, p ∈ R and ( s k ) k∈N is a sequence of distinct nonzero complex numbers, have been studied in [6,7,20,21,22,23,24]. [1,27]), and…”

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

“…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]).…”

“…Basis properties (completeness, minimality, basicity) of the above systems of Bessel functions and more general systems x −p−1 x ρ k J ν (x ρ k ) : k ∈ N in the space L 2 (0; 1); x 2p dx , where ν ≥ 1/2, p ∈ R and ( ρ k ) k∈N is a sequence of distinct nonzero complex numbers, have been studied in [6,7,[15][16][17][18][19]. Those results are formulated in terms of sequences of zeros of functions from certain classes of entire functions.…”

“…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…”

“…In this direction, in [13] the authors obtained a criterion for the completeness and minimality of the system ρ k √ xρ k J −3/2 (xρ k ) : k ∈ N in L 2 (0; 1); x 2 dx with an arbitrary sequence of nonzero complex numbers (ρ k ) k∈N . In addition, in [14] it was proven that the system…”