On the Completeness and Minimality of Sets of Bessel Functions in Weighted \( L^2 \)-Spaces

Abstract

We establish a criterion for the completeness and minimality of the system \( (x^{-p-1}\sqrt{\rho_k}J_\nu(x\rho_k) : k \in \mathbb{{N}}) \) in the space \( L^2((0,1); x^{2p}dx) \), where \( J_\nu \) is the Bessel function of the first kind of index \( \nu \geq 1/2 \), \( p \in \mathbb{R} \) and \( (\rho_k : k \in \mathbb{N}) \) is a sequence of distinct nonzero complex numbers.