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Math128_hw13 - Math 128A Spring 2007 Homework 13 Solution 1...

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Math 128A, Spring 2007 Homework 13 Solution 1. For [ - 1 , 1] and w ( x ) = 1 find the best mean-square approximation to x 3 from among polynomials of degree no bigger than 2. Solution: We know that the first three Legendre polynomials, 1, x , and (3 x 2 - 1) / 2, form an orthonormal basis for P 2 with respect to the inner product f, g = 1 - 1 f ( x ) g ( x ) dx So we need only compute q ( x ) = x 3 , 1 1 + x 3 , x x + x 3 , (3 x 2 - 1) / 2 (3 x 2 - 1) / 2 = 2 5 x 2. For data (0 , - 2), (1 , - 1), (2 , - 1), (4 , 3), (6 , 4) and weight function w = (1 , 2 , 3 , 2 , 1), find the best mean-square linear approximation, by finding an orthonormal basis for
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