When we use Bernstein polynomials in application - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:02:17Z http://mathoverflow.net/feeds/question/115509 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115509/when-we-use-bernstein-polynomials-in-application When we use Bernstein polynomials in application AmirHosein SadeghiManesh 2012-12-05T15:54:36Z 2012-12-09T03:28:03Z <p>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?</p> <p>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?</p> <p>Your answers was nice all.</p> http://mathoverflow.net/questions/115509/when-we-use-bernstein-polynomials-in-application/115510#115510 Answer by Nikita Sidorov for When we use Bernstein polynomials in application Nikita Sidorov 2012-12-05T16:17:48Z 2012-12-05T16:17:48Z <p>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. </p> http://mathoverflow.net/questions/115509/when-we-use-bernstein-polynomials-in-application/115511#115511 Answer by quid for When we use Bernstein polynomials in application quid 2012-12-05T16:25:48Z 2012-12-05T16:25:48Z <p>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 <a href="http://en.wikipedia.org/wiki/B%C3%A9zier_curve" rel="nofollow">Bézier curves</a> 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. </p> http://mathoverflow.net/questions/115509/when-we-use-bernstein-polynomials-in-application/115522#115522 Answer by Neeks for When we use Bernstein polynomials in application Neeks 2012-12-05T17:35:25Z 2012-12-05T17:35:25Z <p>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. </p> http://mathoverflow.net/questions/115509/when-we-use-bernstein-polynomials-in-application/115550#115550 Answer by Pietro Majer for When we use Bernstein polynomials in application Pietro Majer 2012-12-05T22:02:35Z 2012-12-05T22:02:35Z <p>A classical application of the Bernstein polinomials is the solution of the <a href="http://en.wikipedia.org/wiki/Hausdorff_moment_problem" rel="nofollow">Hausdorff moment problem</a>, that I mentioned in <a href="http://mathoverflow.net/questions/30220/abstract-thought-vs-calculation/30281#30281" rel="nofollow">this answer</a>. </p>