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
T1
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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