Completeness of the systems of Bessel functions of negative half-integer index less than -1 in weighted L2-spaces

Abstract

We establish necessary and sufficient conditions for the completeness of the system {ρmk √xρk J−m−1/2(xρk): k ∈ N} in the space L2((0;1);x2m dx) in terms of sequences of zeros of functions from certain classes of entire functions, where L2((0;1);t2m dt) be the weighted Lebesgue space of all measurable functions f : (0;1) → C , satisfying ∫ 1 0 t2m| f (t)|2 dt < +∞, m ∈ N, J−m−1/2 be the Bessel function of the first kind of index −m − 1/2 and (ρk )k∈N be a sequence of distinct nonzero complex numbers. We also obtain an analog of the Paley-Wiener theorem related to this system.