In the
mathematicalMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
field of
numerical analysisNumerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square.Being able to compute the sides...
, a
Bernstein polynomial, named after
Sergei Natanovich BernsteinSergei Natanovich Bernstein was a mathematician who was born in Odessa, Imperial Russia and died in Moscow, USSR....
, is a
polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
in the
Bernstein form, that is a
linear combinationIn mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.- Definition:Suppose that K is a...
of
Bernstein basis polynomials.
A
numerically stableIn the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....
way to evaluate polynomials in Bernstein form is
de Casteljau's algorithmIn the mathematical subfield of numerical analysis the de Casteljau's 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 theoremIn 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...
. With the advent of computer graphics, Bernstein polynomials, restricted to the interval
x ∈ [0, 1], became important in the form of
Bézier curveIn the mathematical field of numerical analysis, a Bézier curve is a parametric curve important in computer graphics and related fields.Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.Bézier curves were widely...
s.
The
Bernstein basis polynomials of degree are defined as
where is a
binomial coefficientIn mathematics, the binomial coefficient is the coefficient of the x k term in the polynomial expansion of the binomial power n....
.
The Bernstein basis polynomials of degree form a
basisIn linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others...
for the
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
of polynomials of degree .
A linear combination of Bernstein basis polynomials
is called a
Bernstein polynomial or
polynomial in Bernstein form of degree .
In the
mathematicalMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
field of
numerical analysisNumerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square.Being able to compute the sides...
, a
Bernstein polynomial, named after
Sergei Natanovich BernsteinSergei Natanovich Bernstein was a mathematician who was born in Odessa, Imperial Russia and died in Moscow, USSR....
, is a
polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
in the
Bernstein form, that is a
linear combinationIn mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.- Definition:Suppose that K is a...
of
Bernstein basis polynomials.
A
numerically stableIn the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....
way to evaluate polynomials in Bernstein form is
de Casteljau's algorithmIn the mathematical subfield of numerical analysis the de Casteljau's 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 theoremIn 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...
. With the advent of computer graphics, Bernstein polynomials, restricted to the interval
x ∈ [0, 1], became important in the form of
Bézier curveIn the mathematical field of numerical analysis, a Bézier curve is a parametric curve important in computer graphics and related fields.Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.Bézier curves were widely...
s.
Definition
The
Bernstein basis polynomials of degree are defined as
where is a
binomial coefficientIn mathematics, the binomial coefficient is the coefficient of the x k term in the polynomial expansion of the binomial power n....
.
The Bernstein basis polynomials of degree form a
basisIn linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others...
for the
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
of polynomials of degree .
A linear combination of Bernstein basis polynomials
is called a
Bernstein polynomial or
polynomial in Bernstein form of degree . The coefficients are called
Bernstein coefficients or
Bézier coefficients.
Example
The first few Bernstein basis polynomials are:
| |; |
|
| |; |
|; |
|
| |; |
|; |
|; |
|
| |; |
|; |
|; |
|; |
|
| |; |
|; |
|; |
|; |
|... |
Properties
The Bernstein basis polynomials have the following properties:
- , if or .
- and where is the Kronecker delta
In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise...
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).
- for .
- .
- The derivative
In calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
can be written as a combination of two polynomials of lower degree:
- 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
In mathematics, a partition of unity of a topological space X is a set of continuous functions, , from X to the unit interval [0,1] such that for every point, ,...
:
- A Bernstein polynomial can always be written as a linear combination of polynomials of higher degree:
Approximating continuous functions
Let be a
continuous functionIn mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous...
on the interval . Consider the Bernstein polynomial
It can be shown that
uniformlyIn the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence { f
n } of functions converges uniformly to a limiting function f if the speed of convergence of f
n to f does not depend on x.The concept is...
on the interval . This is a stronger statement than the proposition that the limit holds for each value of separately; that would be
pointwise convergenceIn mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...
rather than
uniform convergenceIn the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence { f
n } of functions converges uniformly to a limiting function f if the speed of convergence of f
n to f does not depend on x.The concept is...
. Specifically, the word
uniformly signifies that
Bernstein polynomials thus afford one way to prove the
Stone-Weierstrass approximation theoremIn 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...
that every real-valued continuous function on a real interval can be uniformly approximated by polynomial functions over .
A more general statement for a function with continuous k-th derivative is
- and
where additionally
is an eigenvalue of ; the corresponding eigenfunction is a polynomial of degree .
Proof
Suppose is a
random variableIn mathematics, random variables are used in the study of probability. They were developed to assist in the analysis of games of chance, stochastic events, and the results of scientific experiments by capturing only the mathematical properties necessary to answer probabilistic questions...
distributed as the number of successes in independent
Bernoulli trialIn the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure".In practice it refers to a single experiment which can have one of two possible outcomes...
s with probability of success on each trial; in other words, has a
binomial distributionIn probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli...
with parameters and . Then we have the
expected valueIn probability theory and statistics, the expected value of a random variable is the integral of the random variable with respect to its probability measure....
.
Then the
weak law of large numbersIn probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
of
probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
tells us that
for every .
Because , being continuous on a closed bounded interval, must be
uniformly continuousIn mathematics, a function ƒ is uniformly continuous if, roughly speaking, it is possible to guarantee that ƒ and ƒ be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between ƒ and ƒ cannot...
on that interval, we can infer a statement of the form
Consequently
And so the second probability above approaches 0 as grows. But the second probability is either 0 or 1, since the only thing that is random is , and that appears
within the scope of the expectation operator . Finally, observe that is just the Bernstein polynomial .
See also
- Bézier curve
In the mathematical field of numerical analysis, a Bézier curve is a parametric curve important in computer graphics and related fields.Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case.Bézier curves were widely...
- Polynomial interpolation
In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. In other words, given some data points , the aim is to find a polynomial which goes exactly through these points.- Applications :Polynomials can be used to...
- Newton form
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form...
- Lagrange form
In numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolating polynomial for a given set of data points in the Lagrange form...