Advanced Calculus fi..

476 Advanced Calculus, Fifth Editionas possible, in terms of least square error, by a constant go. The error is nowwhere A and B are constants. Thus E(go) is a quadratic function of go, having aminimum when dE/dgo = 0:Hence the error is minimized whenThus the constant function y = l a is the best constant approximation, in the sense2: Oof least square error, to the function f (x).This last point of view holds for the coefficients of the general partial sum:THEOREM 2 Let f (x) be piecewise continuous for -x 5 x 5 n. The coefficientsof the partial sum .of the Fourier series off (x) are precisely those among all coefficients of the functiongn(x) = po+plcosx +ql sinx +.--+pncosnx +qnsinnxthat render the square errora minimum. Furthermore, the minimum square error En satisfies the equation:."r?r)? 74..The proof is left to Problem 7.E. = [:[f(x)l2dx - n + 2 I(a: + b:) . (7.12)2 k=lCOROLLARY If f (x) is piecewise continuous for -n 5 x 5 n and ao, al, . . . ,bl, bz, . . . are the Fourier coefficients of f (x), thenso that the series CEl(ai + b:)converges. Furthermore,lim an = 0, lim bn = 0.n-boon-b w, . --