2008-65.md

course	course_year	question_number	tags	title	year
Numerical Analysis	IB	65	IB 2008 Numerical Analysis	4.I.8D	2008

Show that the Chebyshev polynomials, $T_{n}(x)=\cos \left(n \cos ^{-1} x\right), n=0,1,2, \ldots$ obey the orthogonality relation

$$\int_{-1}^{1} \frac{T_{n}(x) T_{m}(x)}{\sqrt{1-x^{2}}} d x=\frac{\pi}{2} \delta_{n, m}\left(1+\delta_{n, 0}\right)$$

State briefly how an optimal choice of the parameters $a_{k}, x_{k}, k=1,2 \ldots n$ is made in the Gaussian quadrature formula

$$\int_{-1}^{1} \frac{f(x)}{\sqrt{1-x^{2}}} d x \sim \sum_{k=1}^{n} a_{k} f\left(x_{k}\right)$$

Find these parameters for the case $n=3$.