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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 Majer, Mr. Neeks, Mr. quid. but when I click to accept one, the other will non-accepted automatically, Why?

Your answers was nice all.

Daniele Tampieri
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AmirHosein Sadeghimanesh
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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.

Nikita Sidorov
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  • Lojasiewicz proves Weierstrass approximation theorem using Bernstein and Tonelli polynomials in his textbook Łojasiewicz, Stanisław An introduction to the theory of real functions. With contributions by M. Kosiek, W. Mlak and Z. Opial. Third edition. Translated from the Polish by G. H. Lawden. Translation edited by A. V. Ferreira. A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1988. x+230 pp. ISBN: 0-471-91414-2 – Margaret Friedland Dec 05 '12 at 21:28
  • Thanks! What's a Tonelli polynomial? I have to admit I've never heard of them and Google is no help either... – Nikita Sidorov Dec 05 '12 at 23:55
  • I must be missing something. Surely the approximation error is $o(1/n)$ (in particular, identically zero for all but a finite number of $n$) if $f$ is any polynomial? – Fredrik Johansson Dec 05 '12 at 23:56
  • Fredrik: not really. Take, for instance, $f(x)=x^2$; then $B_n(f;x)=x^2+x(1-x)/n$, so $B_n(f;x)-f(x)\asymp 1/n$. – Nikita Sidorov Dec 06 '12 at 00:16
  • I see. Still, any polynomial can be written in the Bernstein basis, so if you are working with the Bernstein basis for computational purposes, you always have the option to choose an exact interpolant instead of "the" Bernstein polynomial $B_n(f)$. – Fredrik Johansson Dec 06 '12 at 01:39
  • Re Tonelli polynomials: I do not remember the definition well myself and do not have the text at hand, but here is the entry from Polish Wikipedia, citing Lojasiewicz: Let $a,b\in\mathbb{R}, 0<b-a<1, f\colon [a,b]\to \mathbb{R}$ be a continuous function. If we extend it continuously to the interval $[\alpha,\beta]$ such that $[a,b]\subset [\alpha,\beta], \beta-\alpha<1$, then $T_n(x)=\int\limits_{\alpha}^\beta f(u)t_n(u-x)\ du$, where $t_n(z)=\frac{(1-z^2)^n}{\int\limits_{-1}^1(1-u^2)^n\ du},\quad n\in \mathbb N$ is called the n-th Tonelli polynomial for the function f. – Margaret Friedland Dec 06 '12 at 02:43
  • In the text, Lojasiewicz uses an argument by S. N. Bernstein himself- apparently, a proof of Weierstrass theorem was the reason for introducing the eponymous polynomials in 1912. – Margaret Friedland Dec 06 '12 at 02:54
  • Margaret: Thanks. Yes, this was the reason, and Bernstein saw it mainly as a result in Probability. – Nikita Sidorov Dec 06 '12 at 10:16
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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.

Martin Sleziak
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  • quid: Can you name some refrence? to I observe these facts by my own logic, I will be happy if you do. Thanks. – AmirHosein Sadeghimanesh Dec 09 '12 at 03:22
  • As said I am not an expert but the following notes seem nice http://www.tsplines.com/resources/class_notes/Bezier_curves.pdf (in particular not the part on the fonts; so in some sense we look at Bernstein polynomials all the time). Also the wiki page I link to gives various additional information. –  Dec 09 '12 at 11:26
  • where 'not' is 'note' of course. –  Dec 09 '12 at 11:27
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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.

Neeks
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  • Neeks: I couldn't reach this site "google.it", please say another way to see your refrence, because I think it should be important. This application you told here is a good thing. – AmirHosein Sadeghimanesh Dec 09 '12 at 03:19
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    In the suggestion 'google' was used as a verb. So Neeks recommended to use Google (or another search-engine) to find 'it' (the paper). But tangentially, I am surprised why google.it did not give you the Italian Google site, which it is. In any case, I used Google to find it for you see http://mae.engr.ucdavis.edu/~farouki/bernstein.pdf ; I even made the extra effort to use google.it :) –  Dec 09 '12 at 13:20
  • This «google it» incident is quite funny :-) – Mariano Suárez-Álvarez Dec 10 '12 at 21:04
  • @quid , the address you wrote here was useful. thank you. – AmirHosein Sadeghimanesh Jan 03 '13 at 07:09
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A classical application of the Bernstein polynomials is the solution of the Hausdorff moment problem, that I mentioned in this answer.

Martin Sleziak
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Pietro Majer
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