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However, I believe the most straightforward way to attain them in our context is in constructions of cubic splines that begin by integrating the second derivative constraints. …

There are no generic boundary conditions that guarantee that the interpolating spline reproduces a sampled polynomial. On the other hand, any $p-1$ conditions (where $p$ is the spline polynomial degre …

The $\int_a^b f(\log x) dx$ expression actually has a nice analytical form, which enables to evaluate it in an elegant manner. First, we perform a change of variables $y=\log x$, $dx = x dy = e^y dy$ …

The B-Spline basis functions as defined by the Cox-DeBoor formula cannot, in general, be constructed with convolution. The convolution construction, as I'll explain below, only works for the special c …

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 …

First, if there is a rotational symmetry in the curve, for example a line segment or an S-shape, then you cannot achieve your goal since the 180-degree rotated curve is exactly identical to the origin …

This is an interesting question and as @user100927 has correctly commented, a necessary condition for this to be possible is: $$\int_{x_i}^{x_{i+1}}f(x)dx ≤ y_{i+1}−y_i$$ for all $i$. To prove this c …

B-splines are a basis-function representation for piecwise polynomial functions. …

The final bound, which I will develop using B-Splines, is $n-1$. I'll also show that his bound is tight. … However, as I said above, we can do better using the theory of B-Splines. …

These polynomials are what the inner splines in the example converge to. The last thing that is left to do is to show that we can build a similar construction with a $C^4$ function. …

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 $(N-2) \times (N-2)$ syste …

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. …