6. An Application to Approximation 27 polynomial p(τ) = ξ0 + ξ1τ + . . . + ξmτ m such that p(τ) = for τ = τ1,... , τm+1 but there is no polynomial p(τ) such that = p(τ) for all τ [0, 1]. 3∗. Let g : [0, 1] −→ R be any function. Prove that for any m + 2 points 0 τ1 τ2 . . . τm+2 1 there is a unique polynomial p(τ) = ξ0 + ξ1τ + . . . + ξmτ m , such that |g(τ1) p(τ1)| = |g(τ2) p(τ2)| = . . . = |g(τm+2) p(τm+2)| and the signs of the differences g(τ1) p(τ1), g(τ2) p(τ2), . . . , g(τm+2) p(τm+2) alternate. Prove that the polynomial p gives the unique best (that is, with the smallest ) uniform approximation to g on the set of m + 2 points τ1,... , τm+2. The error of this approximation can be found to be = |η|, where ξ0,... , ξm and η is the (necessarily unique) solution to the system of m + 2 linear equations g(τ1) p(τ1) = η, g(τ2) p(τ2) = −η,... , g(τm+2) p(τm+2) = (−1)m+1η in m + 2 variables (ξ0, . . . , ξm,η). T T T T 1 2 3 H H H Figure 7. A linear function p(τ) = ξ0 + ξ1τ which provides the best uniform approximation for g at some three points τ1, τ2 and τ2 and satisfies p(τ1) g(τ1) = p(τ2) g(τ2) ¡ = p(τ3) g(τ3)
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