In this thesis we study bandlimited approximations to various functions. Bandlimited functions have compactly supported Fourier transforms, which is a desirable feature in many applications. In particular, we address the problem of determining best approximations that minimize a weighted integral error. By utilizing the theory of Hilbert spaces of entire functions developed by L. de Branges, we are able to obtain optimal solutions for several weighted approximation problems. As an application, we determine extremal majorants and minorants that vanish at a prescribed point for a class of functions, which may be used to remove contributions from undesirable points.