Bernstein polynomials
E480872
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
family of polynomials
ⓘ
mathematical concept ⓘ tool in approximation theory ⓘ |
| associatedWithTheorem | Weierstrass approximation theorem NERFINISHED ⓘ |
| basisOf |
Bézier curve representation
ⓘ
space of polynomials of degree at most n on [0,1] ⓘ |
| belongsTo |
functional analysis
ⓘ
real analysis ⓘ |
| constructionMethod | probabilistic interpretation via binomial distribution ⓘ |
| convergenceType | uniform convergence ⓘ |
| convergesTo | given continuous function uniformly on [0,1] ⓘ |
| definedOn | closed interval [0,1] ⓘ |
| degree | n ⓘ |
| domain | continuous functions on a closed interval ⓘ |
| field | approximation theory ⓘ |
| formsBasis | polynomial space P_n ⓘ |
| generalizedTo | closed interval [a,b] ⓘ |
| hasEndpointValue |
B_{n,0}(0) = 1
ⓘ
B_{n,k}(0) = 0 for k>0 ⓘ B_{n,k}(1) = 0 for k<n ⓘ B_{n,n}(1) = 1 ⓘ |
| hasGeneralForm | B_{n,k}(x) = C(n,k) x^k (1-x)^{n-k} ⓘ |
| hasParameter |
degree n
ⓘ
index k ⓘ |
| hasProbabilisticInterpretation | B_{n,k}(x) as probability of k successes in n Bernoulli trials with parameter x ⓘ |
| hasSymmetryProperty | B_{n,k}(x) = B_{n,n-k}(1-x) ⓘ |
| indexRange | k = 0,1,...,n ⓘ |
| introducedBy | Sergei Natanovich Bernstein NERFINISHED ⓘ |
| introducedIn | early 20th century ⓘ |
| namedAfter | Sergei Natanovich Bernstein NERFINISHED ⓘ |
| nonNegativeOn | [0,1] ⓘ |
| partitionOfUnityOn | [0,1] ⓘ |
| preserves |
convexity under suitable conditions
ⓘ
monotonicity under suitable conditions ⓘ positivity of functions ⓘ |
| relatedTo |
Bernstein operator
NERFINISHED
ⓘ
Bézier curves ⓘ |
| satisfiesProperty |
B_{n,k}(x) ≥ 0 for x in [0,1]
ⓘ
sum_{k=0}^n B_{n,k}(x) = 1 for all x in [0,1] ⓘ |
| stableUnder | shape-preserving approximation ⓘ |
| typicalDomain | continuous functions on [0,1] ⓘ |
| usedFor |
approximating continuous functions
ⓘ
constructive proof of the Weierstrass approximation theorem ⓘ uniform approximation on closed intervals ⓘ |
| usedIn |
computer-aided geometric design
ⓘ
finite element methods ⓘ geometric modeling ⓘ numerical analysis ⓘ |
| usedToDefine |
Bernstein approximation
NERFINISHED
ⓘ
Bernstein basis ⓘ |
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Bézier curve