HW05.pdf - Math 104A Homework#5 Least Square Approximation...

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Math 104A Homework #5 Least Square Approximation and Orthogonal Polynomials * Instructor: Lihui Chai 1. The solution P n ( x ) to the Least Squares Approximation problem of f by a polynomial of degree at most n is given explicitly in terms of orthogonal polynomials ψ 0 ( x ), ψ 1 ( x ), ..., ψ n ( x ), where ψ j is a polynomial of degree j , by P n ( x ) = n X j =0 a j ψ j ( x ) , a j = h f, ψ j i h ψ j , ψ j i . (a) Let P n be the space of polynomials of degree at most n . Prove that the error f - P n is orthogonal to this space, i.e. h f - P n , q i = 0 for any q ∈ P n . (b) Using the analogy of vectors interpret this result geometrically (recall the concept of orthogonal projection). 2. (a) Obtain the first 4 Legendre polynomials in [ - 1 , 1]. (b) Find the least squares polynomial approximations of degrees 1, 2, and 3 for the function f ( x ) = e x on [ - 1 , 1]. (c) What is the polynomial least squares approximation of degree 4 for f ( x ) = x 3 on [ - 1 , 1]? Explain. 3. Plot the monic Chebyshev polynomials ˜ T 0 ( x ), ˜ T 1 ( x ), ˜ T 2 ( x ), ˜ T 3 ( x ), and ˜ T 4 ( x ). 4. The concentration c of a radioactive material decays according to the law
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