Overdetermined interpolation problems in the theory of entire functions (Q2901614)

In the present paper some criteria for the existence of an even entire function of exponential type \(\leq \sigma\) taking given values at points belonging to a given sequence of density \(>\sigma\) are obtained. The authors show that the zeros of the Bessel function or hypergeometric functions are examples of such sequences.NEWLINENEWLINE Let \(\tau>\sigma>0\), \(\{\nu_l\}_{l=1}^{\infty}\), be the sequence of all nonnegative zeros of the Bessel function \(J_{\frac n2}(\tau z)\), \(n\in\{2,3,\ldots\}\), let \(\mu_l\) be some sequence of complex numbers, \(Z_{\sigma}\) the set of all even entire functions \(w:{\mathbb C}\rightarrow {\mathbb C}\) such that \(| w(\lambda)| \leq \gamma\left(1+| \lambda| ^m\right)e^{\sigma| \mathrm{ Im}\lambda| }\) for all \(\lambda\in {\mathbb C}\) and some constants \(\gamma>0\) and \(m\in {\mathbb Z}\). Then \(w(\nu_l)=\mu_l\) for all \(l\in {\mathbb N}\) and for some \(w\in Z_{\sigma}\) if and only if the series \(\sum\limits_{l=1}^{\infty}\frac{\nu_l^{n/2+1}\mu_l}{J_{n/2+1}^2\left( \tau\nu_l\right)}J_{n/2-1}\left(t\nu_l\right)\) converges to zero in the space of distributions \({\mathcal D^{\prime}}\left(\sigma, 2\tau-\sigma\right)\). It is proved that in this case \(\mu_l=O\left(l^{\gamma}\right)\) for some \(\gamma\).