2.II.18F

(a) Let $\left\{Q_{n}\right\}_{n \geqslant 0}$ be a set of polynomials orthogonal with respect to some inner product $(\cdot, \cdot)$ in the interval $[a, b]$. Write explicitly the least-squares approximation to $f \in C[a, b]$ by an $n$ th-degree polynomial in terms of the polynomials $\left\{Q_{n}\right\}_{n \geqslant 0}$.

(b) Let an inner product be defined by the formula

$(g, h)=\int_{-1}^{1}\left(1-x^{2}\right)^{-\frac{1}{2}} g(x) h(x) d x$

Determine the $n$th degree polynomial approximation of $f(x)=\left(1-x^{2}\right)^{\frac{1}{2}}$ with respect to this inner product as a linear combination of the underlying orthogonal polynomials.

*Typos? Please submit corrections to this page on GitHub.*