Bernstein polynomial approximating a function - Maple Help

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bernstein - Bernstein polynomial approximating a function

Calling Sequence

bernstein(n, f, x)

Parameters

n

-

integer

f

-

function (specified as a procedure or operator)

x

-

algebraic expression

Description

• 

This procedure returns the nth degree Bernstein polynomial  in x approximating the function f(x) on the interval 0,1.  Note that f must be a function of one variable specified as a procedure or operator.

• 

Bernstein polynomials arise in the Stone-Weierstrass approximation theorem of analysis that says any continuous function (R->R) can be uniformly approximated on a closed interval by a sequence of polynomials.  The Bernstein polynomials are one such set for doing this.

• 

Given p:=n,i,xbinomialn,ixi1xni Bernstein is defined to be

Bernsteinn,f,x=i=0npn,i,xfin

Examples

bernstein3,x→1x+1,z

120z3+310z234z+1

(1)

f := proc(t) if t < 1/2 then 4*t^2 else 2 - 4*t^2 end if end proc:

bernstein2&comma;f&comma;x

4x2&plus;2x

(2)

See Also

binomial, polynomials


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