When we use Bernstein polynomials in application When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple finite differences operators".
One aspect of my question is to see that is there any predominance to use this method instead of those were mentioned even at very especial case? BERNSTEIN Polynomials are nice themselves and have lots of properties, but are they better to use for example in computer program or other situations too?
Thank Mr. Nikita Sidorov, Mr. Pietro Maior, Mr. Neeks, Mr. quid. but when I click to accept one, the other will non-accepted automaticly, Why?
Your answers was nice all.
 A: The only practical advantage of Bernstein polynomials is their universality. They really work for any continuous function $f$. However, it is well known that if
$$
\|f-B_n(f)\|_\infty=o(1/n),\quad n\to\infty,
$$
then $f(x)=ax+b$. In other words, one cannot hope to approximate a non-linear function by a Bernstein polynomial with an error term better than $1/n$ - which is impractical. 
A: I am not an expert on this but my understanding is that in some sense Bernstein polynomials are really used all over in graphics (Adobe Flash, PostScript, Metafont, SVG,...), via Bézier curves and related things. The De Casteljau algorithm mentioned there is for numerically evaluating Bernstein polynomials, and see at the end under 'Terminology' where Bernstein polynomials are mentioned explicitly. 
A: Very recently I am also amazed by their properties and so little applications.
An useful read is: "The Bernstein polynomial basis: a centennial retrospective" [google it].
Nikita is right but their is the another aspect of not fitting the spurious peaks in the signal. This can smooth the signal better. 
A: A classical application of the Bernstein polinomials is the solution of the Hausdorff moment problem, that I mentioned in this answer.
