# Tight bound on least possible norm of $C^k$ functions that interpolate given points

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.

• Now that appears rather non-trivial. If you can, you should expand your answer to explain the special case when $f(0)=f(1)=0$, and $f(1/2) \in (0, 1)$, and $k$ is 3 or 4, since that is currently the case that interests me the most. Aug 8, 2022 at 20:08