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Talk:Chebyshev coefficients: Difference between revisions
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→What is this task about?: typo, confusion with markdown :)
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Or alternatively, Clenshaw's recurrence might be added as a separate task, with the two tasks exchanging data and results. —[[User:Sonia|Sonia]] ([[User talk:Sonia|talk]]) 15:56, 21 August 2015 (UTC)
== Wikipedia doesn't seem to agree with Numerical Recipes ==
See https://en.wikipedia.org/wiki/Chebyshev_polynomials#Example_1. WP has that extra Kronecker delta that effectively divides the first term by 2, which, if I understand, gets the absolute value of that first coefficient <= 1. Am I reading that right? NR may have a pair of functions that generate and then evaluate an approximating polynomial, but if it's a variant of Chebychev, I'd prefer that we show the more mathematically correct functions. —[[User:Sonia|Sonia]] ([[User talk:Sonia|talk]]) 16:15, 21 August 2015 (UTC)
== What is this task about? ==
It's absolutely not clear what '''Chebyshev coefficients''' actually are. The name of Chebyshev is associated to many things in approximation theory
* Coefficients of Chebyshev polynomials of first and second kind
* Compute the projection of a function on the Chebyshev basis, with scalar product defined by <math>(f|g)=\int_{-1}^{1}\dfrac{f(x)g(x)}{\sqrt{1-x^2}}\;\mathrm{d}x</math>.
* Given a polynomial in the basis <math>\{x^n,n\in\Bbb{N}\}</math>, rewrite it in the Chebyshev basis, leading to ''Chebyshev economization''.
* Find the optimal approximation of a function on a given interval. This problem is related to the ''Chebyshev equioscillation theorem''.
* Find the interpolating polynomial of a function at ''Chebyshev nodes'' (roots of Chebyshev polynomials), which often leads to a better approximation than equispaced nodes.
The task has to be clarified.
[[User:Arbautjc|Arbautjc]] ([[User talk:Arbautjc|talk]]) 15:14, 9 October 2016 (UTC)
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