HW05.pdf - Math 104A Homework#5 Least Square Approximation and Orthogonal Polynomials Instructor Lihui Chai 1 The solution Pn(x to the Least Squares

# 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 solutionPn(x) to the Least Squares Approximation problem offby a polynomial ofdegree at mostnis given explicitly in terms of orthogonal polynomialsψ0(x),ψ1(x), ...,ψn(x), whereψjis a polynomial of degreej, byPn(x) =nXj=0ajψj(x),aj=hf, ψjihψj, ψji.(a) LetPnbe the space of polynomials of degree at mostn. Prove that the errorf-Pnisorthogonal to this space, i.e.hf-Pn, qi= 0 for anyq∈ Pn.(b) Using the analogy of vectors interpret this result geometrically (recall the concept oforthogonal 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 functionf(x) = exon [-1,1].(c) What is the polynomial least squares approximation of degree 4 forf(x) =x3on [-1,1]?Explain.3. Plot the monic Chebyshev polynomials˜T0(x),˜T1(x),˜T2(x),˜T3(x), and˜T4(x).4. The concentrationcof a radioactive material decays according to the law