De Branges’ theorem on approximation problems of Bernstein type

Abstract

AbstractThe Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted ${C}_{0} $-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 $\tau $ and bounded on the real axis ($\tau \gt 0$ fixed).We consider approximation in weighted ${C}_{0} $-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 $ \overline{F( \overline{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.