Suppose that the functions 1, ... , n satisfy the

Chapter 11, Problem 5

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Suppose that the functions 1, ... , n satisfy the orthonormality relation (1) and that agiven function f is to be approximated by Sn(x) = c11(x) ++ cnn(x), where the coef-ficients ci are not necessarily those of Eq. (9). Show that the mean square error Rn givenby Eq. (6) may be written in the formRn =10r(x)f 2(x) dx ni=1a2i +ni=1(ci ai)2,where the ai are the Fourier coefficients given by Eq. (9). Show that Rn is minimized ifci = ai for each i.