Hi,

I would greatly appreciate any hint for proving the following.

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

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

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