Bernstein polynomial

Bernstein polynomials approximating a curve

In the mathematical field of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials.

A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm.

Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the Stone–Weierstrass approximation theorem. With the advent of computer graphics, Bernstein polynomials, restricted to the interval [0, 1], became important in the form of Bézier curves.

DefinitionEdit

 
Bernstein basis polynomials for 4th degree curve blending

The n +1 Bernstein basis polynomials of degree n are defined as

 

where   is a binomial coefficient. So, for example,  

The first few Bernstein basis polynomials for blending 1, 2, 3 or 4 values together are:

 

The Bernstein basis polynomials of degree n form a basis for the vector space Πn of polynomials of degree at most n with real coefficients. A linear combination of Bernstein basis polynomials

 

is called a Bernstein polynomial or polynomial in Bernstein form of degree n.[1] The coefficients   are called Bernstein coefficients or Bézier coefficients.

The first few Bernstein basis polynomials from above in monomial form are:

 

PropertiesEdit

The Bernstein basis polynomials have the following properties:

  •  , if   or  
  •   for  
  •  
  •   and   where   is the Kronecker delta function:  
  •   has a root with multiplicity   at point   (note: if  , there is no root at 0).
  •   has a root with multiplicity   at point   (note: if  , there is no root at 1).
  • The derivative can be written as a combination of two polynomials of lower degree:
     
  • The transformation of the Bernstein polynomial to monomials is
     
and by the inverse binomial transformation, the reverse transformation is[2]
 
  • The indefinite integral is given by
     
  • The definite integral is constant for a given n:
     
  • If  , then   has a unique local maximum on the interval   at  . This maximum takes the value
     
  • The Bernstein basis polynomials of degree   form a partition of unity:
     
  • By taking the first derivative of   where  , it can be shown that
     
  • The second derivative of   where   can be used to show
     
  • A Bernstein polynomial can always be written as a linear combination of polynomials of higher degree:
     

Approximating continuous functionsEdit

Let ƒ be a continuous function on the interval [0, 1]. Consider the Bernstein polynomial

 

It can be shown that

 

uniformly on the interval [0, 1].[3]

Bernstein polynomials thus provide one way to prove the Weierstrass approximation theorem that every real-valued continuous function on a real interval [ab] can be uniformly approximated by polynomial functions over  .[4]

A more general statement for a function with continuous kth derivative is

 

where additionally

 

is an eigenvalue of Bn; the corresponding eigenfunction is a polynomial of degree k.

ProofEdit

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. Then we have the expected value   and

 

By the weak law of large numbers of probability theory,

 

for every δ > 0. 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 ​1n K, equal to ​1n x(1−x), is bounded from above by ​1(4n) irrespective of x.

Because ƒ, being continuous on a closed bounded interval, must be uniformly continuous on that interval, one infers a statement of the form

 

uniformly in x. Taking into account that ƒ is bounded (on the given interval) one gets for the expectation

 

uniformly in x. To this end one splits the sum for the expectation in two parts. On one part the difference does not exceed ε; this part cannot contribute more than ε. 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.

Finally, one observes that the absolute value of the difference between expectations never exceeds the expectation of the absolute value of the difference, and

 

See for instance Koralov & Sinai (2007).[5]

See alsoEdit

NotesEdit

  1. ^ G. G. Lorentz (1953) Bernstein Polynomials, University of Toronto Press
  2. ^ Mathar, R. J. (2018). "Orthogonal basis function over the unit circle with the minimax property". Appendix B. arXiv:1802.09518.
  3. ^ Natanson (1964) p. 6
  4. ^ Natanson (1964) p. 3
  5. ^ Koralov, L.; Sinai, Y. (2007). ""Probabilistic proof of the Weierstrass theorem"". Theory of probability and random processes (2nd ed.). Springer. p. 29.

ReferencesEdit

External linksEdit