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- Apr 14, 2013

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Could you help me at the following exercise?

$k, n \in \mathbb{N}$

$f(x)=cos(k \pi x), x \in [0,1]$

$x_i=ih, i=0,1,2,...,n, h=\frac{1}{n}$

Let $p \in \mathbb{P}_n$ the Lagrange interpolating polynomials of $f$ at the points $x_i$.

Calculate an upper bound of the maximum error $\varepsilon_p =max_{0 \leq x \leq 1}{|f(x)-p(x)|}$ as a function of $k$ and $n$, and find a relation that $k$ and $n$ should satisfy so that $\varepsilon_p \rightarrow 0$ while $k \rightarrow \infty$ and $ n \rightarrow \infty$.

I have found that an upper bound of the maximum error is $ \frac{(k \pi h)^{n+1}}{n+1}$.

How can I find the relation that $k$ and $n$ should satisfy so that $\varepsilon_p \rightarrow 0$ while $k \rightarrow \infty$ and $ n \rightarrow \infty$ ??