is the coefficient next to
in the polynomial which interpolates at points .
Suppose interpolates at , while interpolates at . Then
interpolates at .
Hence the recurrence formula holds:
The Newton formula
interpolates at .
Proof
By induction:
Suppose interpolates at .
Then has degree (at most)
and roots at for (by induction hypothesis).
So it has to be .
Errors
Define the error functions .
It has roots in , hence has at least one root .
Then or for some in the interval.
Furthermore,
where interpolates at .
So certainly:
On interval , is minimized for (scaled) Chebyshev polynomials
(proof by contradiction, using that roots alternate in sign), so
and we have