3. (10 pts) Sometimes we want to fit a given curve as closely as po...

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3. (10 pts) Sometimes we want to fit a given curve as closely as possible by a polynomial of a given degree. The criterion of "Least Squares" is often used to determine the best fit. For example, if we want to fit a function f(x) on the interval (-1, - 1] by a polynomial of degree n, Er-o Ckrk we require the coefficients Ck to minimize the integral 1 n 2 1 Then fn (x) = ET-O Chrck is called the best approximation of f(x) (at order n.) in the least squares sense. Show that if we expand f(x) in Legendre series 2 1 f(x)= [auPe(x) where ae = 1 1 f(x) Pe(x) dx, e=0 the truncation of the sum up to l=n, n CaePe(x) e=0 gives the best polynomial (of degree n) approximation of f(x) in the interval [-1, 1].

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