De Branges’ Theorem on Approximation Problems of Bernstein Type

The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted C0-space on the real line. A theorem of de Branges characterizes non-density by existence of an entire function of Krein class being related with the weight in a certain way. An analogous result holds true for weighted sup-norm approximation by entire functions of exponential type at most τ and bounded on the real axis (τ > 0 fixed). We consider approximation in weighted C0-spaces by functions belonging to a prescribed subspace of entire functions which is solely assumed to be invariant under division of zeros and passing from F(z) to F(z), and establish the precise analogue of de Branges’ theorem. For the proof we follow the lines of de Branges’ original proof, and employ some results of Pitt.