Let $r:[-1,1] \rightarrow \mathbb{R}$ be a continuous function with $r(x)>0$ for all but finitely many values of $x$.

(a) Show that

$$\langle u, v\rangle=\int_{-1}^{1} u(x) v(x) r(x) d x$$

defines an inner product on $C([-1,1])$.

(b) Show that for each $n$ there exists a polynomial $P_{n}$ of degree exactly $n$ which is orthogonal, with respect to the inner product $(*)$, to all polynomials of lower degree.

(c) Show that $P_{n}$ has $n$ simple zeros $\omega_{1}(n), \omega_{2}(n), \ldots, \omega_{n}(n)$ on $[-1,1]$.

(d) Show that for each $n$ there exist unique real numbers $A_{j}(n), 1 \leqslant j \leqslant n$, such that whenever $Q$ is a polynomial of degree at most $2 n-1$,

$$\int_{-1}^{1} Q(x) r(x) d x=\sum_{j=1}^{n} A_{j}(n) Q\left(\omega_{j}(n)\right)$$

(e) Show that

$$\sum_{j=1}^{n} A_{j}(n) f\left(\omega_{j}(n)\right) \rightarrow \int_{-1}^{1} f(x) r(x) d x$$

as $n \rightarrow \infty$ for all $f \in C([-1,1])$.

(f) If $R>1, K>0, a_{m}$ is real with $\left|a_{m}\right| \leqslant K R^{-m}$ and $f(x)=\sum_{m=1}^{\infty} a_{m} x^{m}$, show that

$$\left|\int_{-1}^{1} f(x) r(x) d x-\sum_{j=1}^{n} A_{j}(n) f\left(\omega_{j}(n)\right)\right| \leqslant \frac{2 K R^{-2 n+1}}{R-1} \int_{-1}^{1} r(x) d x$$

(g) If $r(x)=\left(1-x^{2}\right)^{1 / 2}$ and $P_{n}(0)=1$, identify $P_{n}$ (giving brief reasons) and the $\omega_{j}(n)$. [Hint: A change of variable may be useful.]