Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

If I have a $C^\infty$ function $f: [0,1]^n \to \mathbb{R}$ then its Bernstein polynomials $$ B_m(x) = \sum_{k_1,\dots,k_n=0}^m f\left(\frac{k_1}{m}, \dots, \frac{k_m}{m}\right) \prod_{i=1}^n \binom{m}{k_i} x^{k_i} (1-x_i)^{m-k_i} $$ converge uniformly to $f$, and all of the partial derivatives of $B_m$ converge uniformly to the corresponding partial derivatives of $f$.

My question is whether anyone knows of a reference for this. The univariate statement can be found practically everywhere, while I found the multivariate statement for $f$ (but not its derivatives) here.

share|improve this question
    
(there are some typos in the formula: the binomials coefficients are lacking and the product should have $x_i$) –  Pietro Majer Nov 2 '12 at 8:27
    
Thanks, I believe it's fixed now. –  Joe Neeman Nov 3 '12 at 1:26
add comment

2 Answers 2

up vote 4 down vote accepted

There are several references indeed; it's in any case a consequence of the univariate case. Let me say the way I see it (for which I do not have a reference).

For the univariate case, consider the difference operator $D_n:C^0([0,1])\to C^0([0,1])$ defined as $$D_n f(x):= \frac{f\left(\big(1-\frac{1}{n}\big)x + \frac{1}{n} \right) - f\left(\big(1-\frac{1}{n}\big)x \right)}{\frac{1}{n}}\, . $$ So this is just the usual discrete difference of $f$, but first we apply the affine contraction $[0,1]\ni x\mapsto \big(1-\frac{1}{n}\big)x\in \big[0, 1- \frac{1}{n}\big]$, in order that the translation of $\frac{1}{n}$ be well defined on $C^0([0,1])$. By the mean value theorem, for any $f\in C^1([0,1])$ and $x$ we have $D_nf(x)=f'(\xi)$ for some $\xi$ with $|x-\xi| < \frac{1}{n}$, so $D_nf\to f'$ uniformly (percisely, in $\omega$ is a modulus of continuity of $f'$, $\|D _ nf -f'\| _\infty\le\omega(\frac{1}{n})$. The reason why this approximation of the derivative $Df:=f'$ is relevant in connection to the Bernstein operators, is that, as it is easy to check, $$D B_n = B_{n-1}D_n\, ,$$ which implies that for any any $f\in C^1([0,1])$ the Bernstein polynomial of $f$ converges to $f$ in $C^1$ (just because $ \| (B _ n f)'- f' \| _ \infty = \|B _ {n-1} (D _ n f-f') + B _ {n-1}f' - f'\| _ \infty \le$ $ \|D_ n f-f' \| _ \infty +\|B _{n-1}f' - f'\| _ \infty=o(1)$), and more generally, that $B_n$ converges strongly to the identity on $C^r([0,1])$.

The analogous statement for partial derivatives of functions on $[0,1]^n$ follows plainly on the same lines. Only, it is convenient to consider, more generally than the polynomial you wrote, the multivariate Bernstein polynomials $$B_m f:= \sum _ {{0\le k_i \le m_i}\atop 0\le i\le n}f\Big(\frac{k _ 1}{m _ 1},\dots,\frac{k_n}{m _ n}\Big) \prod_{1\le i\le n}\Big({m_ i\atop k_ i}\Big) x_i ^ {k _ i}(1-x _ i)^{m _ i - k _ i} $$ where now $m:=(m_1,\dots,m_n)$ is a multi-index (so the one you wrote corresponds to a constant multi-index $m_1=\dots=m_n$; the following computation is a good motivation to consider also these Bernstein polynomials with different discretizations for each variable). As observed in the link, this may be thought as a (commuting) composition of "univariate Bernstein polynomial operators" $B_{m_ i}$ each acting on $f$ as a function of the $i$-th variable. Correspondingly, for any multi-index $\alpha\in \mathbb{N}^n$ and $m\ge \alpha$ $$\partial ^\alpha B _ m = B _ { m - \alpha } \partial^\alpha _m \, ,$$ where $\partial^\alpha _m $ is the partial difference operator analogously defined by (commuting) composition of the previously defined $D ^ {\alpha _ i }_ {m _ i }$, each acting on $f$ as a function of the $i$-th variable. As a consequence, for any $f\in C^r\big([0,1]^n\big)$ and $|\alpha|\le r$ we have $\partial^\alpha B_m f\to \partial^\alpha f$ uniformly as $\min_{1\le i \le n} m _ i \to\infty$.

share|improve this answer
    
I've marked your answer as accepted because it's a very nice explanation. I would still like to have a reference, though: to put it in context, I'm writing a paper which needs some polynomial approximation, but the topic of the paper is very much unrelated to polynomial approximation and the intended audience is probably not too familiar with it. Right now I just have a proof in an appendix, but it would be nice to replace it with a reference. –  Joe Neeman Nov 15 '12 at 19:56
add comment

Kingsley (1951), "Bernstein Polynomials for Functions of Two Variables of Class C(k)", and Butzer (1953), "On two-dimensional Bernstein polynomials", prove approximation results for derivatives in two-dimensional case.

Regarding Pietro Majer's answer above: is there a reference that shows $\sup_x| \partial^{\alpha}_m f(x) - \partial^{\alpha}f(x)| \rightarrow 0$ as $\min m_i \rightarrow \infty$ for the particular difference operator he defines?

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.