Interpolation Formulas With Derivatives In De Branges Spaces II

We consider the problem of reconstruction of entire functions of exponential type $\tau$ that are elements of certain weighted $L^p(\mu)$-spaces from their values and the values of their derivatives up to a fixed order. In this paper we extend the interpolation results of [F. Gon\c{c}alves, Interpolation formulas with derivatives in de Branges spaces, To appear in Trans. Amer. Math. Soc.] which considered the first derivative. Using the theory of de Branges spaces we find a discrete set $\mathcal{T}_{\tau,\nu}$ of points on the real line and a frame $\mathcal{G}_{\tau,\nu}$ from an associated de Branges space that allow reconstruction of the function from information at the points in $\mathcal{T}_{\tau,\nu}$ via an interpolation series. If $p=2$ we show that the series converges in $L^2(\mu)$-norm while for $p\neq 2$ we prove convergence on compact subsets of $\mathbb{C}$. Finally, we give an application to sampling/interpolation theory in Paley-Wiener spaces.