Let $k\ge 2$, and let $f(x):[0,1]\to [0,1]$ have $k$ continuous derivatives. Define the $C^k$ norm of $f$ as— $$||f||_{C^k} = \max_{0\le i\le k} \max_{0\le x\le 1} |f^{(i)}(x)|.$$
Now, given a sequence of $n$ items $(p_n, v_n)$ and given that $f(p_n) = v_n$ for each $n$, what is a tight upper bound (or, if practical, even the exact value) on $\inf ||f||_{C^k}$ over all functions mapping $[0, 1]$ to $[0, 1]$ with $k$ continuous derivatives?
Obviously, $\inf ||f||_{C^k}$ is at least $\max |f|$ and at least $|\frac{v_{n+1}-v_n}{p_{n+1}-p_n}|$, but is greater in general.
Specific Cases
I would like you to answer this question for the case where:
- $k$ is 3, 4, 5, or 6.
- $f(0) = f(1) = 0$.
- Either:
- $f(1/2)$ is known.
- $f(1/4)$, $f(1/2)$, and $f(3/4)$ are known.
If your answer can be easily adapted to solve a more general case, that would be nice to have.
Example
Suppose $f(x)=2x(1-x)$. For $f$ and its first three derivatives:
- $\max |f| = 1/2.$
- $\max |f\prime| = 2.$
- $\max |f^{(2)}| = 4.$
- $\max |f^{(3)}| = 0.$
Thus, $||f||_{C^3} = 4$. Now suppose there is a $C^3$ function $g$ that is unknown except that it maps [0, 1] to [0, 1] and that $g(0) = 0$, $g(1/2) = 1/2$, and $g(1) = 0$. Then obviously the $C^3$ norm must be at least 1/2. But is 4 the lowest $C^k$ norm that $g$ can have? Or can it be even lower?
This illustrates the problem of finding a tight upper bound (or an exact value) on $\inf ||f||_{C^k}$ given a set of points that interpolate $f$. I have not been able to find existing results on this matter, not even on the subject of "derivative bounds" or "bounds on derivatives". A related result is Bernstein's inequality used to find derivative bounds on polynomials (but not necessarily the tightest possible), but I don't see how or whether it can help in the problem at hand, which involves a broader class of functions than polynomials.
Motivation
The motivation is to give evidence on a conjecture on a scheme to build polynomials that converge from above and below to a function and meet "consistency" requirements on their coefficients.
See also the following:
