Chapter 5.6, Problem 29E

A sequence of orthogonal polynomials usually satisfies a three-term recurrence relation. For example, the Chebyshev polynomials are related by

$T_{n+1}\left(x\right)=2x{T}_n\left(x\right)-T_{n-1}\left(x\right)$, $n=1,2,\mathrm{...}$,

where $T_0\left(x\right)=1$ and $T_1\left(x\right)=x$. Verify that the polynomials defined by the relation (R) above are indeed orthogonal in $C[-1,1]$ with respect to the inner product in Exercise 28. Verify this as follows:

a) Make the change of variables $x=\cos \theta$, and use induction to show that $T_k\left(\cos \theta \right)=\cos k\theta$, $k=0,1,\mathrm{....}$ where $T_k\left(x\right)$ is defined by (R).

b) Using part a), show that $\langle T_i,T_j\rangle =0$ when $i\ne j$.

c) Use induction to show that $T_k\left(x\right)$ is a polynomial of degree $k$, $k=0,1,\mathrm{...}$.

d) Use (R) to calculate $T_2,T_3,T_4,$ and $T_5$.

28. An inner product on $C\left[-1,1\right]$ is

$\langle f,g\rangle =\frac{2}{\pi }{\displaystyle {\int }_{-1}^1\frac{f\left(x\right)g\left(x\right)}{\sqrt{1-x^2}}}dx$.

Starting with the set $\left\{1,x,x^2,x^3,\mathrm{...}\right\}$, use the Gram-Schmidt process to find polynomials $T_0\left(x\right),T_1\left(x\right),T_2\left(x\right)$, and $T_3\left(x\right)$ such that $\langle T_i,T_j\rangle =0$ when $i\ne j$. These polynomials are called the Chebyshev polynomials of the first kind. [Hint: Make a change of variables $x=\cos \theta$.]