Chebyshev Approximation Formula

Using a Chebyshev Polynomial of the First Kind $T$, define

$$\begin{eqnarray*} c_j&\equiv& {2\over N}\sum_{k=1}^N f(x_k)T_j(x_k)\\ &=& {2\... ...ght]\cos\left\{{\pi j(k-{\textstyle{1\over 2}})\over N}\right\}. \end{eqnarray*}$$

Then

$$f(x) \approx \sum_{k=0}^{N-1} c_kT_k(x)-{\textstyle{1\over 2}}c_0.$$

It is exact for the $N$ zeros of $T_N(x)$. This type of approximation is important because, when truncated, the error is spread smoothly over $[-1,1]$. 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.