# 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?

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