1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Science Advisor

    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


    User Avatar
    Science Advisor

    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]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Similar Threads for Bernstein's polynomial Date
Degree-Raising Formulas for Bernstein Polynomials Sep 24, 2012
Cantor-schroder-bernstein use in proof Mar 12, 2012
Bernstein polynomials Dec 7, 2010
Bernstein set Nov 17, 2010
Prove sets are equipotent using Schroder - Bernstein Oct 16, 2008