In the mathematical subfield 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. Euclid, detail from The School of Athens by Raphael. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Sergei Natanovich Bernstein (Russian: Сергей Натанович Бернштейн, sometimes Romanized as Bernshtein) (March 5, 1880 - October 26, 1968) was a Ukrainian mathematician who was born in Odessa, Ukraine and died in Moscow, USSR. His doctoral dissertation, submitted in 1904 to the Sorbonne, solved Hilberts nineteenth problem on the analytic solution of elliptic differential... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm. In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. ... In the mathematical subfield of numerical analysis the de Casteljaus algorithm, named after its inventor Paul de Casteljau, is a recursive method to evaluate polynomials in Bernstein form or Bézier curves. ...

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. In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. ... In the mathematical subfield of numerical analysis a Bézier curve is a parametric curve important in computer graphics. ...

Definition

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

The Bernstein basis polynomials of degree n form a basis for the vector space Π_n of polynomials of degree n. In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

A linear combination of Bernstein basis polynomials

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

Notes

The Bernstein basis polynomials have the following properties:

• b_ν,n(x) has a root with multiplicity ν at point x = 0
• b_ν,n(x) has a root with multiplicity n − ν at point x = 1
• b_ν,n(x) ≥ 0 if x ∈ [0,1]
• b_ν,n(x) has a maximum at x = ν/n on the interval [0,1]
• b’_ν,n(x) = n [b_{ν − 1, n − 1}(x) − b_ν,n − 1(x)]
• b_ν,n(x) = 0, if ν < 0 or ν > n
• b_ν,n(0) = δ_ν0 and b_ν,n(1) = δ_νn where δ is the Kronecker delta function
• b_n−ν,n(x) = b_ν,n(1 − x)

The Bernstein basis polynomials of degree n form a partition of unity: In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... In mathematics, a partition of unity of a topological space X is a set of continuous functions {ρi} from X to the unit interval [0,1] such that every point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all...

Example

The first few Bernstein basis polynomials are

Approximating continuous functions

Let f(x) be a continuous function on the interval [0, 1]. Consider the Bernstein polynomial In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

It can be shown that

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 In mathematical analysis, a sequence { fn } of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. ... Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ... In mathematical analysis, a sequence { fn } of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. ...

Bernstein polynomials thus afford one way to prove the Stone-Weierstrass approximation theorem that every real-valued continuous function on a real interval [a,b] can be uniformly approximated by polynomial functions over R. In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. ...

Proof

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 E(K/n) = x. A random variable is a term used in mathematics and statistics. ... In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, called success and failure. ... See binomial (disambiguation) for a list of other topics using that name. ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical...

Then the weak law of large numbers of probability theory tells us that In a statistical context, laws of large numbers imply that the average of a random sample from a large population is likely to be close to the mean of the whole population. ... Probability theory is the mathematical study of probability. ...

Because f, being continuous on a closed bounded interval, must be uniformly continuous on that interval, we can infer a statement of the form In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x affect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but...

Consequently

And so the second probability above approaches 0 as n grows. But the second probability is either 0 or 1, since the only thing that is random is K, and that appears within the scope of the expectation operator E. Finally, observe that E(f(K/n)) is just the Bernstein polynomial B_n(f,x).

See also

• Bézier curve
• Polynomial interpolation
• Newton form
• Lagrange form