MATH36022 Numerical Analysis 2 Approximation of Functions – Week 2 Exercises

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Attempt these exercises in advance of the tutorial in Week 3

1. Find the best $L_{\infty }$ approximation to $f(x)=x^{n+1}+{\sum }_{k=0}^n{a}_k{x}^k$  
   on $[-1,1]$ by polynomials of degree at most $n$.
   ANS: Suppose $p_n(x)$ is the best $L_{\infty }$ approximation to $f(x)$. Then, by Chebyshev Equioscillation Theorem, $f(x)-p_n(x)$ with leading coefficient $1$ should exists at least $n+2$ oscillated maximum modulus values. Recall that the Chebyshev polynomial $2^{-n}\ast T_{n+1}(x)$ has leading coefficient $1$. And achieves its maximum modulus on $[-1,1]$ at $n+2$ distinct points in [ − 1 , 1 ]

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