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?
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, J. Approximation Theory 1, 452-463 (1968). ZBL0174.09902.
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, Bull. Am. Math. Soc. 74, 1072-1078 (1968). ZBL0181.34001.
