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Unformatted text preview: notice that x ( t )-p ( T,t ) is orthogonal to the entire subspace of polynomials. Diﬀerentiate with respect to T , which appears both in the upper limit of integration and in p [ T,t ]. It might be helpful to do the special cases n = 1 and n = 2 ﬁrst. 5. Luenberger, Section 3.13, problem 9. 6. Luenberger, Section 3.13, problem 6. 7. Luenberger, Section 3.13, problem 18. Here is a hint which you would quickly ﬁnd on the Internet. Let P n ( t ) be the n th orthogonal polynomial, and let t 1 , ··· t m be the points where it changes sign(this set includes simple roots, but not double roots). Deﬁne the polynomial S ( t ) = m Y j =1 ( t-t j ) . Show that the assumption m < n leads to a contradiction concerning the integral of w ( t ) P n ( t ) S t . 1...
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This document was uploaded on 08/30/2011.
- Fall '09