Completeness of the systems of Bessel functions of index −5/2

Abstract

Let L2((0;1);x4dx) be the weighted Lebesgue space of all measurable functions f:(0;1)→C, satisfying ∫10t4|f(t)|2dt<+∞. Let J−5/2 be the Bessel function of the first kind of index −5/2 and (ρk)k∈N be a sequence of distinct nonzero complex numbers. Necessary and sufficient conditions for the completeness of the system {ρ2k√xρkJ−5/2(xρk):k∈N} in the space L2((0;1);x4dx) are found in terms of an entire function with the set of zeros coinciding with the sequence (ρk)k∈N. In this case, we study an integral representation of some class E4,+ of even entire functions of exponential type σ≤1. This complements similar results on approximation properties of the systems of Bessel functions of negative half-integer index less than −1, due to B. Vynnyts'kyi, V. Dilnyi, O. Shavala and the author.