Bernstein polynomial

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 x ∈ [0, 1], became important in the form of Bézier curves.

Definition

: $b\_\{\backslash nu,n\}(x)\; =\; \{n\; \backslash choose\; \backslash nu\}\; x^\{\backslash nu\}\; \backslash left(\; 1\; -\; x\; \backslash right)^\{n\; -\; \backslash nu\},\; \backslash quad\; \backslash nu\; =\; 0,\; \backslash ldots,\; n.$

where $\{n\; \backslash choose\; \backslash nu\}$ is a binomial coefficient.

The Bernstein basis polynomials of degree n form a basis for the vector space Π_n of polynomials of degree n.

A linear combination of Bernstein basis polynomials

:$B(x)\; =\; \backslash sum\_\{\backslash nu=0\}^\{n\}\; \backslash beta\_\{\backslash nu\}\; b\_\{\backslash nu,n\}(x)$

is called a Bernstein polynomial or polynomial in Bernstein form of degree n. The coefficients $\backslash beta\_\backslash nu$ are called Bernstein coefficients or Bézier coefficients.

Example

:

Properties

Approximating continuous functions

: $B\_n(f)(x)\; =\; \backslash sum\_\{\backslash nu\; =\; 0\}^n\; f\backslash left(\; \backslash frac\{\backslash nu\}\{n\}\; \backslash right)\; b\_\{\backslash nu,n\}(x).$

It can be shown that

: $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\{\; B\_n(f)(x)\; \}\; =\; f(x)\; \backslash ,$

uniformly on the interval [0, 1]. This is a stronger statement than the proposition that the limit holds for each value of x separately; that would be pointwise convergence rather than uniform convergence. Specifically, the word uniformly signifies that

: $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash sup\; \backslash left\backslash \{\backslash ,\; \backslash left|\; f(x)\; -\; B\_n(f)(x)\; \backslash right|\; \backslash ,:\backslash ,\; 0\; \backslash leq\; x\; \backslash leq\; 1\; \backslash ,\backslash right\backslash \}\; =\; 0.$

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

A more general statement for a function with continuous k-th derivative is

: $\{\backslash left\backslash |\; B\_n(f)^\{(k)\}\; \backslash right\backslash |\}\_\backslash infty\; \backslash le\; \backslash frac\{\; (n)\_k\; \}\{\; n^k\; \}\; \backslash left\backslash |\; f^\{(k)\}\; \backslash right\backslash |\_\backslash infty\; \backslash text\{\; and\; \}\; \backslash left\backslash |\; f^\{(k)\}-\; B\_n(f)^\{(k)\}\; \backslash right\backslash |\_\backslash infty\; \backslash to\; 0$

where additionally

: $\backslash frac\{\; (n)\_k\; \}\{\; n^k\; \}\; =\; \backslash left(\; 1\; -\; \backslash frac\{0\}\{n\}\; \backslash right)\; \backslash left(\; 1\; -\; \backslash frac\{1\}\{n\}\; \backslash right)\; \backslash cdots\; \backslash left(\; 1\; -\; \backslash frac\{k\; -\; 1\}\{n\}\; \backslash right)$

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

Proof

By the weak law of large numbers of probability theory, : $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\{\; P\backslash left(\; \backslash left|\; \backslash frac\{K\}\{n\}\; -\; x\; \backslash right|>\backslash delta\; \backslash right)\; \}\; =\; 0$ 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 K/n, equal to x(1-x)/n, 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 : $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\{\; P\backslash left(\; \backslash left|\; f\backslash left(\; \backslash frac\{K\}\{n\}\; \backslash right)\; -\; f\backslash left(\; x\; \backslash right)\; \backslash right|\; >\; \backslash varepsilon\; \backslash right)\; \}\; =\; 0$ uniformly in x. Taking into account that ƒ is bounded (on the given interval) one gets for the expectation : $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\{\; E\backslash left(\; \backslash left|\; f\backslash left(\; \backslash frac\{K\}\{n\}\; \backslash right)\; -\; f\backslash left(\; x\; \backslash right)\; \backslash right|\; \backslash right)\; \}\; =\; 0$ 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 that E(ƒ(K/n)) is just the Bernstein polynomial B_n(ƒ, x).

See for instance.

See also

Notes