Interpolation Formulas with Derivatives in De Branges Spaces Ii

We consider the problem of reconstruction of entire functions of exponential type τ that are elements of certain weighted Lp(μ)-spaces from their values and the values of their derivatives up to order ν. In this paper we extend the interpolation results of [24] in which the case ν = 1 was solved. Using the theory of de Branges spaces we find a discrete set Tτ,ν of points on the real line and a frame Gτ,ν from an associated de Branges space that allow reconstruction of the function from information at the points in Tτ,ν via an interpolation series. If p = 2 we show that the series converges in L2(μ)-norm while for p 6= 2 we prove convergence on compact subsets of C. Finally, we give an application to sampling/interpolation theory in Paley-Wiener spaces.