0000039911 00000 n
0000031276 00000 n
δ 0000029350 00000 n
= for every δ > 0. 258 139
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= x 0000030222 00000 n
x x 0000019148 00000 n
Polynômes de Bernstein, courbes de Bézier 1. 0000037128 00000 n
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) ) ( a 0000035953 00000 n
k ( and. So, for example, ( 0000008956 00000 n
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The identities (1), (2), and (3) follow easily using the substitution x b 3 0000006980 00000 n
ν 0000017395 00000 n
δ It follows that the polynomials fn tend to f uniformly. x , 0000033164 00000 n
, there is a n ���i߹��o������xYhS�/~�W�q���bMi�?۳���0>��Z�j�4��N��"m���VR��n�*�7�x.�TF�s��O�"2�>���L�I4"�9�����Q_1b�ζ�32�L1��C�MB?G���L'�!��I�iO���G���v8������ �^+s���0��#j�7�,u����I��:�(� 5煒q(�Q&ԥJ^W����J�d�
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0000022639 00000 n
∑ 0000017105 00000 n
= {\displaystyle \mathbb {P} (X=i)} {\displaystyle |a-b|<\delta } 0000015559 00000 n
k The following identities can be verified: (1) 0000010226 00000 n
= Les polynômes de Bernstein, nommés ainsi en l'honneur du mathématicien russe Sergeï Bernstein (1880-1968), permettent de donner une démonstration constructive et probabiliste [1] du théorème d'approximation de Weierstrass. k and this equation can be applied twice to 0000011744 00000 n
{\displaystyle \beta _{\nu }} ( {\displaystyle t=x/(1-x)} − n C'est d'ailleurs l'interprétation qu'en fait Bernstein dans sa démonstration du théorème d'approximation de Weierstrass. {\displaystyle {\binom {m}{i}}} ) P 0000018428 00000 n
− ∈ 0000022868 00000 n
0000009572 00000 n
Bernstein-Polynome bilden das Hauptinstrument beim Beweis des Weierstraßschen Approximationssatzes, sowie bei der Konstruktion von Beziér-Kurven und Beziér-Flächen. {\displaystyle \sum _{k}\left(x-{k \over n}\right)^{2}{n \choose k}x^{k}(1-x)^{n-k}={x(1-x) \over n}.}. {\displaystyle b_{2,5}(x)={\tbinom {5}{2}}x^{2}(1-x)^{3}=10x^{2}(1-x)^{3}.}. 0000032298 00000 n
• Kac, Mark (1938). The first few Bernstein basis polynomials from above in monomial form are: The Bernstein basis polynomials have the following properties: Let ƒ be a continuous function on the interval [0, 1]. i | 0000037328 00000 n
x n 0000029000 00000 n
− 0000016195 00000 n
5 10 0000023796 00000 n
k 0000009322 00000 n
1 0000043404 00000 n
) 0000040870 00000 n
) k | ) {\displaystyle \mathbb {R} } 0000005990 00000 n
On the other hand, by identity (3) above, and since "kS�ƕ �g�NZG�9�H�\q�r���ub����E�9U½(���e�����pO
r2���b`F�P�CBz��?L��EPk�M8�����k��". n 0000009862 00000 n
Ils sont également utilisés dans la … Die Bernstein-Polynome sind benannt nach dem russischen Mathematiker Sergei Natanowitsch Bernstein. x [9][10], Suppose K is a random variable distributed as the number of successes in n independent Bernoulli trials with probability x of success on each trial; in other words, K has a binomial distribution with parameters n and x. ( 0000021633 00000 n
. ( Consider the Bernstein polynomial, uniformly on the interval [0, 1]. k 0000019428 00000 n
( 0000043088 00000 n
sup − 1 x k | 0000014675 00000 n
1 | Moreover, by continuity, 0000014995 00000 n
Within these three identities, use the above basis polynomial notation, Since f is uniformly continuous, given 260 0 obj<>stream
0000008209 00000 n
0000023141 00000 n
{\displaystyle {\tbinom {n}{\nu }}} A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm. 0000030549 00000 n
[8] See also Feller (1966) or Koralov & Sinai (2007). − < = ( Then we have the expected value ) | 0000045425 00000 n
0000031056 00000 n
0000029950 00000 n
Un article de Wikipédia, l'encyclopédie libre. 0000020269 00000 n
{\displaystyle \operatorname {\mathcal {E}} \left[{\frac {K}{n}}\right]=x\ } such that 0000039086 00000 n
0000023321 00000 n
b Moreover, this relation holds uniformly in x, which can be seen from its proof via Chebyshev's inequality, taking into account that the variance of 1⁄n K, equal to 1⁄n x(1−x), is bounded from above by 1⁄(4n) irrespective of x. 0000032728 00000 n
0000027574 00000 n
With the advent of computer graphics, Bernstein polynomials, restricted to the interval [0, 1], became important in the form of Bézier curves. ) whenever ( Partie A : Polynômes de Bernstein Pour tout entier naturel n et tout entier naturel i tel que 0 , on note ≤i ≤n B,n i le polynôme défini pour p 6.a est un polynôme de degré et de degré , est donc un polynôme de degré; et ayant les mêmes valeurs en , admet les réels comme racines distinctes. 0000025817 00000 n
0000044325 00000 n
0000041973 00000 n
0000015884 00000 n
Les m + 1 polynômes de Bernstein forment une base de l'espace vectoriel des polynômes de degré au plus m. Ces polynômes présentent plusieurs propriétés importantes : To this end one splits the sum for the expectation in two parts. {\displaystyle \delta >0} m "Une remarque sur les polynomes de M. S. Bernstein". 0000039253 00000 n
On the other part the difference exceeds ε, but does not exceed 2M, where M is an upper bound for |ƒ(x)|; this part cannot contribute more than 2M times the small probability that the difference exceeds ε. − f k For a continuous function f on the k-fold product of the unit interval, the proof that f(x1, x2, ... , xk) can be uniformly approximated by, is a straightforward extension of Bernstein's proof in one dimension. n is called a Bernstein polynomial or polynomial in Bernstein form of degree n.[1] The coefficients . 0000000016 00000 n
0000012439 00000 n
− ( ≥ doi:10.4064/sm-7-1-49-51. 0
0000036822 00000 n
0000032972 00000 n
a x Les polynômes de Bernstein, nommés ainsi en l'honneur du mathématicien russe Sergeï Bernstein (1880-1968), permettent de donner une démonstration constructive et probabiliste[1] du théorème d'approximation de Weierstrass.