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I'm studying for a final and came across the following question:

Question

In a discussion with my professor, he said using Chebyshev polynomials would be messy and unwieldy and encouraged another route - he also said the hint provided in the question gives a nudge as to the path I could take, but I'm completely lost.

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  • $\begingroup$ I don't get why he would say that, interpolating at Chebyshev roots is proven to be the optimal (least error) way to interpolate. $\endgroup$ – setholopolus Dec 14 '16 at 3:29
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Chebyshev roots are proven to be the optimal places for interpolation (optimal meaning they give you the least amount of error in your interpolation). The roots of the $n$th Chebyshev Polynomial in the interval $(-1,1)$ are given by $$ z_k = \cos \left( \frac{2k-1}{2n}\right), k = 1,2,...n$$

Since you need them to be on the interval $[0,1]$ instead, take your $z_k$ and run them through the function $$T(z) = \frac{1}{2}z + \frac{1}{2} $$ to get them into the correct interval, but still in the correct places with respect to one another.

See where this takes you! Comment if you are stuck again.

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