On the completeness of a system of Bessel functions of index $-3/2$ in weighted $L^2$-space

Abstract

In this paper, we study an integral representation of some class $E_{2,+}$ of even entire functions of exponential type $\sigma\le 1$. We also obtain an analog of the Paley-Wiener theorem related to the class $E_{2,+}$. In addition, we find necessary and sufficient conditions for the completeness of a system $\big\{s_k\sqrt{xs_k}J_{-3/2}(xs_k):k\in\mathbb N\big\}$ in the space $L^2((0;1);x^2 dx)$, where $J_{-3/2}$ be the Bessel function of the first kind of index $-3/2$, $(s_k)_{k\in\mathbb N}$ be a sequence of distinct nonzero complex numbers and $L^2((0;1);x^2 dx)$ be the weighted Lebesgue space of all measurable functions $f:(0;1)\rightarrow\mathbb C$ satisfying $\int_{0}^1 x^2 |f(x)|^2\, dx<+\infty$. Those results are formulated in terms of sequences of zeros of functions from the class $E_{2,+}$. We also obtain some other sufficient conditions for the completeness of the considered system of Bessel functions. Our results complement similar results on completeness of the systems of Bessel functions of index $\nu<-1$, $\nu\notin\mathbb Z$.