It is well-known that the interpolation error of a cubic spline has at best order $O(h^4)$, 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 $O(h^{p+1})$? Is something like this present in literature?
> 
> 
> 



Spline Interpolation error of higher degree
The following paper by de Boor suggests that this is the case, although he develops the proof only up to degree 6 splines.


*de Boor, C.*, [**On the convergence of odd-degree spline interpolation**](http://dx.doi.org/10.1016/0021-9045(68)90033-6), J. Approximation Theory 1, 452-463 (1968). [ZBL0174.09902](https://zbmath.org/?q=an:0174.09902).


The paper below by Swartz, and the references in it, also claim that this can be proved, at least under some conditions.


*Swartz, B.*, [**(O(h^{2n+2-1})) bounds on some spline interpolation errors**](http://dx.doi.org/10.1090/S0002-9904-1968-12052-X), Bull. Am. Math. Soc. 74, 1072-1078 (1968). [ZBL0181.34001](https://zbmath.org/?q=an:0181.34001).


