I am looking for an answer/reference to the following question: consider points $(x_i, y_i)$, $i=1,\ldots,n$ with $x_i, y_i \in \mathbb{R}$ and let $p \geq n$. Let $f \in C^p([x_0, x_n])$ be an interpolating function, that is $f(x_i) = y_i$ for $i=1, \ldots, n$.
What is a smallest possible bound on the derivatives $\left\| f^{(r)} \right\|_{\infty}$ for $r=1, \ldots, p$? That is, if I am free in choosing any $f$ as long as it is sufficiently often differentiable, how small can I make the derivatives?
If all that is needed for $f$ is continuity, you can obviously just use a piecewise linear function and get \begin{equation} \left\| f' \right\|_{\infty} = \max_{i=1, \ldots, n-1} \left| \frac{y_{i+1} - y_i}{x_{i+1} - x_i} \right|. \end{equation} Now I wonder if I can ''smooth out" the corners in a way to make the function $p$-times differentiable but without creating very large higher derivatives. Note that I do not need to know how $f$ looks like or how to construct it, I just need a bound on the derivatives that is as sharp as possible.
I messed around with fitting B-splines through the points and using bounds on the derivatives, but this does not seem to give a very accurate estimate.
I have also browsed various interpolation-related questions but they do not seem to give me what I need. The key difference is that I don't need $f$ to have a specific form (piecewise polynomial or whatever) as I only need the theoretical bound, not a way to actually compute $f$.
Addendum: Instead of the question above, it would probably suffice to answer the following. Given two points $(x_0, y_0)$ and $(x_1, y_1)$ and a function $f : [x_0, x_1] \mapsto \mathbb{R}$ for which $f(x_0) = y_0$ and $f(x_1) = y_1$ as well as $f^{(n)}(x_0) = f^{(r)}(x_1) = 0$ for all derivatives $r = 1, \ldots, n$ for some $n \in \mathbb{N}$. What bounds $\left\| f^{(r)} \right\|_{\infty}$?