Let $f:[0,1]\to [0,1]$ be continuous.  Let—

$$B_n(f)(x)=\sum_{k=0}^n f(k/n) {n \choose k} x^k (1-x)^{n-k},$$ 

be the _Bernstein polynomial_ of $f$ of degree $n$.

This question relates to the difference between two Bernstein polynomials of the same function $f(x)$, namely the question of finding a simple and explicit function $\phi(f, n)$ such that— $$|B_{2n}(f)(x) - B_{n}(f)(x)| \le \phi(f, n) \text{ whenever } 0 \le x \le 1. \tag{1}$$

I have not been able to find results answering this question.  However:

- Butzer (1953) showed a very similar result on a family of operators, one of which is $2 B_{2n}(f) - B_n(f)$, but without explicit error bounds.
- <s>For the polynomials $2x(1-x)$ and $2x^2(1-x)$, the left-hand side of $(1)$ appears to converge to 0 at the rate $O(1/n^2)$.  I suspect this is the case whenever $f(x)$ has a continuous second derivative.</s>

## Motivation

Answers to this question may help solve another question of mine, on finding [bounds on the expected value of a hypergeometric random variable](https://mathoverflow.net/questions/429037/bounds-on-the-expectation-of-a-function-of-a-hypergeometric-random-variable).

## Questions

Let $M(f, r)$ equal the maximum absolute value of $f$ and its derivatives up to the $r$-th derivative.

1. Given that $f$ has a continuous second derivative, can $\phi(f, n)$ equal $CM(f, 2)/n^2$ and/or $C/n^2$?  In either case, find an upper bound for $C$.
2. Given that $f$ is continuous, what is an explicit upper bound for $\phi(f, n)$, perhaps given the modulus of continuity of $f$?
3. Does the same bound for $C$ as in question 1 hold if, instead&mdash;

    - $f$ has a Lipschitz-continuous derivative, and 
    - $M(f, r)$ equals the maximum absolute value of $f$ and the Lipschitz constants of $f$ and its derivative?
    
4. Does the same bound for $C$ as in question 1 hold if, instead&mdash;

    - $f'$ is in the _Zygmund class_, that is, for some constant $D\ge 0$ it has the property $|f'(x) + f'(y) - 2f'((x+y)/2)| \le D\epsilon$ for every $\epsilon>0$, whenever $x$ and $y$ are in $[0, 1]$ and $|x-y|\le\epsilon$, and
    - $M(f, r) = \max(D, \max |f|, \max |f'|)$?

It would be nice if the answers to questions 1 and 4 can easily carry over to the problem of finding an upper bound for $|B_{n+1}(f)(x) - B_{n}(f)(x)|$.

## References

- Butzer, Paul (1953) "[Linear Combinations of Bernstein Polynomials](https://doi.org/10.4153/CJM-1953-063-7)". Canadian Journal of Mathematics, 5, 559-567. doi:10.4153/CJM-1953-063-7, [MR0058023](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0058023), [Zbl 0051.05002](https://zbmath.org/?q=an%3A0051.05002).