6. An Application to Approximation 27

polynomial p(τ) = ξ0 + ξ1τ + . . . + ξmτ

m

such that p(τ) =

eτ

for τ = τ1,... , τm+1

but there is no polynomial p(τ) such that

eτ

= 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)