It is well known that the interpolation error of a cubic spline has at best order $\mathcal{O}(h^4)$ which results from polynomials of degree 3.

Can I assume that if one uses polynomials of degree p and the respective function to be interpolated $f\in C^p([a,b])$, that the interpolation error of this spline is $\mathcal{O}(h^{p+1})$ ?

Is this known in Literature ? (I couldn't seem to find it.)