| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
IB |
58 |
IB |
2016 |
Numerical Analysis |
|
Paper 1, Section I, D |
2016 |
(a) What are real orthogonal polynomials defined with respect to an inner product $\langle\cdot, \cdot\rangle ?$ What does it mean for such polynomials to be monic?
(b) Real monic orthogonal polynomials, $p_{n}(x)$, of degree $n=0,1,2, \ldots$, are defined with respect to the inner product,
$$\langle p, q\rangle=\int_{-1}^{1} w(x) p(x) q(x) d x$$
where $w(x)$ is a positive weight function. Show that such polynomials obey the three-term recurrence relation,
$$p_{n+1}(x)=\left(x-\alpha_{n}\right) p_{n}(x)-\beta_{n} p_{n-1}(x),$$
for appropriate $\alpha_{n}$ and $\beta_{n}$ which should be given in terms of inner products.