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For the Bernstein polynomial in D-module theory, see Bernstein-Sato polynomial.
Bernstein polynomials approximating a curve

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.

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.

Definizione

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I n + 1 Polinomi di base di Bernstein di grado n sono definiti come

dove

è un coefficiente binomiale.

I Polinomi di base di Bernstein di grado n formano una base per lo [spazio vettoriale |[vector space]] dei polimomi di grado n.

Una combinazione lineare di Polinomi di base di Bernstein

è detta un polinomio di Bernstein di grado n. I coefficienti βν sono detti coefficienti di Bernstein.

Osservazioni

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I Polinomi di base di Bernstein hanno le seguenti proprietà:

  • , se ν < 0 o ν > n
  • e ove è la funzione delta di Kronecker.
  • ha una radice con molteplicità ν in x = 0 (oss: se ν è 0 non ci sono radici in 0)
  • ha una radice con moltiplicità n − ν nel punto x = 1 (oss: se ν = n non ci sono radici in 1)
  • ≥ 0 per x in [0,1]
  • SE ν ≠ 0, allora ha un solo massimo locale in [0,1] in x = ν/n. Il suo valore è
  • I Polinomi di base di Bernstein di grado n formano una [partition of unity|[partizione dell' unità]]:

Esempi

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The first few Bernstein basis polynomials are

Approssimazione di funzioni continue

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Let f(x) be a continuous function on the interval [0, 1]. Consider the Bernstein polynomial

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

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.

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 k.

Dimostrazione

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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.

Then the weak law of large numbers of probability theory tells us that

for every .

Because f, being continuous on a closed bounded interval, must be uniformly continuous on that interval, we can infer a statement of the form

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 Bn(f,x).

Vedi anche

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Riferimenti

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  • Weisstein, Eric W. "Bernstein Polynomial". MathWorld.