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Polynomial interpolation

  1. Apr 10, 2010 #1
    Let [tex]x_{0}, x_{1}, \cdots , x_{n}[/tex] be distinct points in the interval [a,b] and [tex]f \in C^{1}[a,b][/tex].

    We show that for any given [tex]\epsilon >0[/tex] there exists a polynomial p such that

    [tex]\left\| f-p \right\|_{\infty} < \epsilon[/tex] and [tex]p(x_{i}) = f(x_{i})[/tex] for all [tex]i=1,2, \cdots , n [/tex]

    I know [tex]\left\| f\right\| _{\infty}= max_{x \in [a,b]}|f(x)| [/tex] and I wonder if the polynomial they are asking for is the Lagrangian polynomial interpolating f at the nodes [tex]x_{0}, x_{1}, \cdots , x_{n}[/tex]. If yes, I am not sure how to prove the problem.
  2. jcsd
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