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Actually, we can learn this from the uniqueness of the spline interpolation (for a given set of $(p+1)n$ constraints). …

Not only is the answer "no", but for any number $N$ you can construct a monotone function and sample it such that the natural spline approximation will have $N$ extremum points. See the figure (and i …

Solving the B-Spline interpolation problem is standard (see for example, Chapter 9 of The NURBS Book, or here). … This can be proved by explicitly solving the interpolation equations but a simpler proof comes from the mean value theorem. …

From the properties of integrals it is not hard to derive an upper bound on the $L^2$ norm from the upper bound on the $L^\infty$ norm. Since you have (from Hall & Meyer) a bound $\left| f(x)-s(x) \ri …

There is an analytical solution to the problem in the following sense: Given a number $N$, the optimal interpolation points $x_0=0, x_1, ..., x_{N-1}=1$ are the roots of an $(N-2) \times (N-2)$ system … x_i$ to the right and reduce the maximal interpolation error. …

Using the recursive derivative formula (see for example here): $$N'_{i,p}(t) = \frac{p}{t_{i+p}-t_i} N_{i,p-1}(t) - \frac{p}{t_{i+p+1}-t_{i+1}} N_{i+1,p-1}(t)$$ We get that the maximum is achieved whe …

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. … Swartz, B., (O(h^{2n+2-1})) bounds on some spline interpolation errors, Bull. Am. Math. Soc. 74, 1072-1078 (1968). ZBL0181.34001. …