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Bernstein's polynomial

  1. Oct 25, 2012 #1
    Find the sequence [itex](B_nf)[/itex] of Bernstein's polynomials in

    a) f(x)=x and

    b) [itex]f(x)=x^2[/itex]

    Answers (from my textbook):

    a) [itex]B_nf(x) = x[/itex] for all n.

    b) [itex]B_nf(x) = x^2 + \frac{1}{n} x (1-x)[/itex]

    I know that the bernstein's polynomial is:

    [itex]B_nf(x) = \sum_{k=0}^n f (\frac{k}{n}) \binom{n}{k} x^k (1-x)^{n-k} [/itex]

    ...but I don't know how they got the answer from this...
     
  2. jcsd
  3. Oct 25, 2012 #2

    HallsofIvy

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    Have you used that formula to calculate, say, B0 through B5 for f(x)= x and f(x)= x2? That should give you an idea.
     
  4. Dec 18, 2012 #3
    But how can I calculate [tex]B_0[/tex]? If I say n=0, then

    [itex]B_nf(x) = \sum_{k=0}^n f (\frac{k}{0}) \binom{0}{k} x^k (1-x)^{0-k} [/itex]

    So f(k/0) is undefined?
     
  5. Dec 18, 2012 #4

    HallsofIvy

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    Sorry. Clearly "B0" is not defined so calculate B1, B2, etc.

    For example, with f(x)= x,
    [tex]B_1(x)= f(0)\begin{pmatrix}1 \\ 0\end{pmatrix}x^0(1- x)^1+ f(1)\begin{pmatrix}1 \\ 1\end{pmatrix}x^1(1- x)^0= 0(1- x)+ 1x= x[/tex]
     
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