Completeness of the systems of Bessel functions of index \(- 5 / 2\) (Q6589493)

In this paper the author gives necessary and sufficient conditions for the completeness of the system \(\{\rho_k^2\sqrt{x\rho_k}J_{-\frac{5}{2}}(x\rho_k): k \in \mathbb{N}\}\) in the weighted space \(L^2((0,1), x^{4} dx)\). Here \(J_{-\frac{5}{2}}\) is the first kind Bessel function of index \(-\frac{5}{2}\), and \(\{\rho_k \}_{k \in \mathbb{N}}\) is an arbitrary sequence of distinct nonzero complex numbers. These conditions are found in terms of an entire function with the set of zeros coinciding with the sequence \((\rho_k)_{k \in \mathbb{N} }\). The results extend and complement similar results on approximation properties of Bessel functions of negative fractions less than \(-1\).