Math 128A, Spring 2007Homework 13 Solution1. For [-1,1] andw(x) = 1 find the best mean-square approximation tox3from among polynomials of degree no bigger than 2.Solution:We know that the first three Legendre polynomials, 1,x, and(3x2-1)/2, form an orthonormal basis forP2with respect to the innerproductf, g=1-1f(x)g(x)dxSo we need only computeq(x) =x3,1 1 +x3, x x+x3,(3x2-1)/2 (3x2-1)/2 =25x2. For data (0,-2), (1,-1), (2,-1), (4,3), (6,4) and weight functionw=(1,2,3,2,1), find the best mean-square linear approximation, by findingan orthonormal basis for
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