Hi,

I would greatly appreciate any hint for proving the following.

Question : Let f:[0, 1] ---- R. It can be proved that if [0, 1/(N+m), ... , (N+m)/(N+m) ; f ]=0, for all m=1,2, 3, ...., .... then f necessarily is a polynomial of degree less or equal to N ?

(here [0, 1/(N+m), ..., (N+m)/(N+m) ; f] denotes the divided difference of f on the knots 0, 1/(N+m), ...., 1).

Remark. In order to obtain easier a proof, it can be supposed that f satisfies some smoothness properties, as for example to be infinitely differentiable or analytic on [0, 1].

Thank you,

G