MA4229 Lectures 11, 12 Weeks 9-10 Oct 12,15, 19 2010

1. MA4229 Lectures 11, 12Weeks 9-10 Oct 12,15, 19 2010 Chapter 12 Properties of Orthogonal Polynomials Chapter 11 Least Squares Approximation

2. The Least Squared Problem Consider and is the best we say that Given weighted least squares approximation from to if it minimizes Question Show that this for an appropriate scalar product Question Show that if is a finite dimensional subspace then this problem has a solution. Is it unique?

3. Characterization be a linear subspace of a real Theorem 11.1 Let is a best and let Then Hilbert space approximation from to iff Proof For every Question Show that if then there exists such that Question Show that if then for every

4. Calculation Using Normal Equations then has a basis If holds iff and the condition we obtain the normal If we represent equations Question Represent this set of (n+1) equations using a matrix (called the Gram matrix for the set Question Show that since is linearly independent its Gram matrix is nonsingular.

5. Calculation Using Gram-Schmidt whenever Theorem 11.2 If then Question Derive this from the normal equations. Gram-Schmidt Orthogonalization : Given a basis for we construct an orthogonal basis by : For

6. Three Term Recurrence then and Theorem 11.3 If where Proof. Clearly and induction  are orthogonal. are monic. Assume Clearly GS gives Question Why does Question Derive Then

7. Classic Examples http://en.wikipedia.org/wiki/Orthogonal_polynomials Legendre Hermite generating function

8. Elementary Properties and Theorem 12.1 If is a Hilbert space containing be mutually orthogonal and then every that is orthogonal to is a multiple of Proof First show that are linearly independent. Then express and observe hence

9. Elementary Properties Theorem 12.2 If is orthogonal to over (here we could take ) then has exactly k zeros in Proof Let be the number of sign changes of such that in Then there exists therefore so WHY?

10. Gaussian Quadrature Objective : Approximate Strategy : Choose to achieve = for for as large a value of as possible. Question Show that for any there exist so that = holds for Theorem 12.3 If are the orthogonal polynomials for [a,b], w and are the zeros of and then = holds for Proof

11. Gaussian Quadrature Theorem 12.4 Each is positive. Proof Question Why does each equality hold ? http://en.wikipedia.org/wiki/Gaussian_quadrature

12. Characterization of Orthogonal Polynomials Theorem 12.5 Proof iff there exists satisfying and Proof. Page 141. Example For [-1,1], w(x) = 1 let then satisfies Question What are

13. Tutorial 6 Due Thursday 22 October Write up your solutions and prepare to present on the board to the class. Staple your solutions and include your name and matric number on the front page. Pages 133-135 Exercises 11.1, 11.3, 11.7 Pages 147-149 Exercises 12.1, 12.2, 12.4 Show that the Chebyshev polynomials are orthogonal with respect to the weight over the interval function