Polynomial of Bernstein

The polynomials of Bernstein , named thus in the honor of the Russian mathematician S. Bernstein, make it possible to give a constructive demonstration of the Théorème of Stone-Weierstrass. They are also used in the general formulation of the curved of Bézier.

Description

For a degree m , there is m+1 polynomials of Bernstein

B^m_0, \ dowries, B^m_m

defined, on the interval, by

B_i^m (U) = \ begin {pmatrix} m \ \ I \ end {pmatrix} u^i \ left (1-u \ right) ^ {semi}

where the

\ begin {pmatrix} m \ \ I \ end {pmatrix}

are the binomial coefficients.

These polynomials present four important properties:

Partition of the unit: $\ qquad \ sum_ {i=0} ^m B_i^m (U) = 1, \ qquad \ forall U \ in$
Positivity: $B_i^m (U) \ geq 0, \ qquad \ forall U \ in, \ forall I \ in 0 \ dowries m$
Symmetry: $B_i^m (U) = B_ {semi} ^m (1-u), \ qquad \ forall U \ in, \ forall I \ in 0 \ dowries m$
Formula of recurrence:

B_i^m (U) = \ begin {boxes} (1-u) B_i^ {M-1} (U), & I = 0 \ \ (1-u) B_i^ {M-1} (U) + U B_ {i-1} ^ {M-1} (U), & \ forall I \ in 1 \ dowries M-1 \ \ uB_ {i-1} ^ {M-1} (U), & I = m \ end {boxes} , \ qquad \ forall U \ in .

One will note the great resemblance of the polynomials to the Binomial distribution.

See too

Algorithm of Casteljau, makes it possible to calculate effectively the polynomials of Bernstein
Approximation of Bernstein, makes it possible to uniformly approach the continuous functions

Random links: