Using a Chebyshev polynomial of the first kind T(x), define c_j | congruent | 2/N sum_(k = 1)^N f(x_k) T_j(x_k) | = | 2/N sum_(k = 1)^N f[cos{(π(k - 1/2))/N}] cos{(π j(k - 1/2))/N}. Then f(x)≈ sum_(k = 0)^(N - 1) c_k T_k(x) - 1/2 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.
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