(a) Let $w(x)$ be a positive weight function on the interval $[a, b]$. Show that

$$\langle f, g\rangle=\int_{a}^{b} f(x) g(x) w(x) d x$$

defines an inner product on $C[a, b]$.

(b) Consider the sequence of polynomials $p_{n}(x)$ defined by the three-term recurrence relation

$$p_{n+1}(x)=\left(x-\alpha_{n}\right) p_{n}(x)-\beta_{n} p_{n-1}(x), \quad n=1,2, \ldots$$

where

$$p_{0}(x)=1, \quad p_{1}(x)=x-\alpha_{0},$$

and the coefficients $\alpha_{n}$ (for $\left.n \geqslant 0\right)$ and $\beta_{n}$ (for $\left.n \geqslant 1\right)$ are given by

$$\alpha_{n}=\frac{\left\langle p_{n}, x p_{n}\right\rangle}{\left\langle p_{n}, p_{n}\right\rangle}, \quad \beta_{n}=\frac{\left\langle p_{n}, p_{n}\right\rangle}{\left\langle p_{n-1}, p_{n-1}\right\rangle}$$

Prove that this defines a sequence of monic orthogonal polynomials on $[a, b]$.

(c) The Hermite polynomials $H e_{n}(x)$ are orthogonal on the interval $(-\infty, \infty)$ with weight function $e^{-x^{2} / 2}$. Given that

$$H e_{n}(x)=(-1)^{n} e^{x^{2} / 2} \frac{d^{n}}{d x^{n}}\left(e^{-x^{2} / 2}\right)$$

deduce that the Hermite polynomials satisfy a relation of the form $(*)$ with $\alpha_{n}=0$ and $\beta_{n}=n$. Show that $\left\langle H e_{n}, H e_{n}\right\rangle=n ! \sqrt{2 \pi}$.

(d) State, without proof, how the properties of the Hermite polynomial $\operatorname{He}_{N}(x)$, for some positive integer $N$, can be used to estimate the integral

$$\int_{-\infty}^{\infty} f(x) e^{-x^{2} / 2} d x$$

where $f(x)$ is a given function, by the method of Gaussian quadrature. For which polynomials is the quadrature formula exact?