Let n0phinxinfin be a family of orthonomal polynomials with

Question

Let _n=0{Φ_n(x)}^∞ be a family of orthonomal polynomials with respect to the weight function w(x). Show that the n^th degree polynomial that minimizes

_-1∫¹ w(x) (f(x) - p_n(x))² dx

is given by

P_n(x) = _k=0Σⁿ c_k Φ_k(x)

where c_k = _-1∫¹ w(x) f(x) f(x) Φ_k(x).

Hint: Expand the expression you want to minimize.