We all know the following formula for the maximum error (evenly spaced) polynomial interpolation:
$|f(x) - p_n(x)| \leq \frac{h^{n+1}}{4(n+1)} \max_{x\in [a,b]} f^{(n+1)}(x)$
where $p_n(x)$ is the polynomial of degree $n$ coinciding at the points $x_i = a + i h$ ($i=0,\ldots,n-1, b = x_{n-1}$) with the given function $f(x)$ (i.e. $f(x_i) = p_n(x_i)$).
So far so good. But I would like to know whether there is a similar error bound if we do the matching only on $n/2$ points but also match the derivative (in case of even $n$).
The particular case I am interested in is for $n=3$, where we construct the interpolating polynomial $p_3(x)$ as
$p_3(a) = f(a)$
$p_3'(a) = f'(a)$
$p_3(b) = f(b)$
$p_3'(b) = f'(b)$
This of course uniquely defines $p_3$. The question is what can we say about
$|f(x) - p_n(x)| \leq ???$
perhaps in an analogous way as the formula above.