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.
* ~~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.~~


Motivation
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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
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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—


	* $f$ 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—


	* $f'$ 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
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* 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).



Explicit bounds on the difference between Bernstein polynomials
Some comments.
The estimate $\|B\_{2n} f-B\_n f\|\_\infty \leq c \omega (f, n^{-1/2})$, ($\omega$ is the modulus of continuity of $f$) follows from $\|f-B\_n f\|\_\infty \leq c \omega (f, n^{-1/2})$. Both estimates can be improved to $\frac C n$ if $f \in C^1([0,1]$. However, since
$$ n(f-B\_nf) \to -\frac{x(1-x)}{2}f''$$
an estimate $\|f-B\_n f\|\_\infty =o(1/n)$ gives $f(x)=ax+b$ (see Chapter 1 of the book of Lorentz, Bernstein polynomials).


If $\|B\_{2n} f-B\_n f\|\_\infty \leq Cn^{-\alpha}$, then $\|B\_{2^{k+1}n} f-B\_{2^k n} f\|\_\infty \leq C2^{-\alpha k}n^{-\alpha}$ and summinng over $k$, $\|f-B\_nf \|\_\infty \leq C n^{-\alpha}$ and $\alpha \leq 1$ unless $f$ is linear.


