Lagrange interpolation in the zeros of Bessel functions by entire functions of exponential type and mean convergence (Q1924348)

Let \(J_\alpha\) be the Bessel function of the first kind of order \(\alpha>-1\); \(G_a:=z^{-\alpha}J_\alpha(z)\); \[ L_{\tau,\alpha}(f,z):=\sum^\infty_{\nu=-\infty,\nu\neq 0} {G_\alpha(\tau z)\over G_\alpha'(j_\nu)(\tau z-j_\nu)} f\Biggl({j_\nu\over \tau}\Biggr), \] where \(f:\mathbb{R}\to \mathbb{C}\), \(\tau>0\) is non-integer, \(j_\kappa\) is the \(\kappa\)th positive zero of \(J_\alpha\), \(j_{-\kappa}=-j_\kappa\), \(\kappa\in \mathbb{N}\). The authors had studied the mean convergence of \(L_{\tau,\alpha}(f,.)\) to \(f\) as \(\tau\to\infty\) for a sufficient wide subclass of Riemann integrable functions. This main result is analogous to two well-known theorems of J. Marcinkiewicz and R. Askey. For the proof of it many lemmas were needed. Some of them are of independent interest.