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Interpolating polynomial for sin([itex]\pi{x}[/itex])

  1. Dec 7, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider the function sin([itex]\pi[/itex][itex]x[/itex]) on [-1,1] and its approximations by interpolating polynomials. For integer [itex]n[/itex][itex]\geq[/itex]1, let [itex]x_{n,j}=-1+\frac{2j}{n}[/itex] for [itex]j=0,1,...,n[/itex], and let [itex]p_{n}(x)[/itex] be the [itex]n[/itex]th-degree polynomial interpolating sin([itex]\pi[/itex][itex]x[/itex]) at the nodes [itex]x_{n,0},...,x_{n,n}[/itex]. Prove that

    [itex]\max_{x\in[-1,1]}\left | {sin}{(\pi{x})-p_{n}{(x)}} \right | \to 0[/itex] as [itex]n \to \infty[/itex]

    2. Relevant equations

    3. The attempt at a solution
    I have no idea how to start!!!
  2. jcsd
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