  The 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 L. 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
($\tau>0$ 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 $\bar{F(\bar z)}$, 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 L. Pitt.
