Let $p_{n} \in \mathbb{P}_{n}$ be the $n$th monic orthogonal polynomial with respect to the inner product

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

on $C[a, b]$, where $w$ is a positive weight function.

Prove that, for $n \geqslant 1, p_{n}$ has $n$ distinct zeros in the interval $(a, b)$.

Let $c_{1}, c_{2}, \ldots, c_{n} \in[a, b]$ be $n$ distinct points. Show that the quadrature formula

$$\int_{a}^{b} w(x) f(x) d x \approx \sum_{i=1}^{n} b_{i} f\left(c_{i}\right)$$

is exact for all $f \in \mathbb{P}_{n-1}$ if the weights $b_{i}$ are chosen to be

$$b_{i}=\int_{a}^{b} w(x) \prod_{\substack{j=1 \\ j \neq i}}^{n} \frac{x-c_{j}}{c_{i}-c_{j}} d x$$

Show further that the quadrature formula is exact for all $f \in \mathbb{P}_{2 n-1}$ if the nodes $c_{i}$ are chosen to be the zeros of $p_{n}$ (Gaussian quadrature). [Hint: Write $f$ as $q p_{n}+r$, where $q, r \in \mathbb{P}_{n-1}$.]

Use the Peano kernel theorem to write an integral expression for the approximation error of Gaussian quadrature for sufficiently differentiable functions. (You should give a formal expression for the Peano kernel but are not required to evaluate it.)