# Math 224a Homework 6 ###### tags: `224a 2020` ### Due 12/21/2020 Starred questions require MATLAB/chebfun. 1. Suppose $\mu$ is a Borel measure supported on a finite interval $[a,b]$ with moments $m_k:=\int s^k d\mu(s)$, and define the $n+1\times n+1$ Hankel matrix $D_n(i,j):=m_{i+j-2}$. Let $D_n(x)$ be the determinant of $D_n$ with the bottom row replaced by $[1, x, \ldots,x^{n}]$. Show that $D_n>0$ and that the monic orthogonal polynomials $\{p_n\}_{n=0}^\infty$ with respect to $\mu$ are given by the determinantal formula $$ p_n=\frac{1}{\sqrt{\det(D_{n-1})\det(D_n)}} D_n(x).$$ 4. Trefethen 4.7 5. Trefethen 8.7 6. Trefethen 8.10 8. Trefethen 11.3 7. Trefethen 8.9* 9. Trefethen 13.1* 10. Trefethen 15.4* 11. Trefethen 15.10* 12. What are your thoughts on how this course can be improved? What were some topics you liked/didn't like/wish were included, and why?