Chebyshev Approximation Formula
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19-11-2021
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Chebyshev Approximation Formula
Using a Chebyshev polynomial of the first kind 
, define
Then
 |
(3)
|
It is exact for the 
zeros of 
. This type of approximation is important because, when truncated, the error is spread smoothly over ![[-1,1]](https://mathworld.wolfram.com/images/equations/ChebyshevApproximationFormula/Inline10.gif)
. The Chebyshev approximation formula is very close to the minimax polynomial.
REFERENCES:
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Chebyshev Approximation," "Derivatives or Integrals of a Chebyshev-Approximated Function," and "Polynomial Approximation from Chebyshev Coefficients." §5.8, 5.9, and 5.10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 184-188, 189-190, and 191-192, 1992.
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