Math 641-600 Suggested Problems

Extra Problems - These are not to be handed in.

1. Let $L$ be in $\mathcal B (\mathcal H)$.
1. Show that $\|L^k\| \le \|L\|^k$, $k=2,3,\ldots$.
2. (We did this in class.) Let $|\lambda| \|L\|<1$. Show that $\big\|(I - \lambda L)^{-1} - \sum_{k=0}^{n-1}\lambda^k L^k\big\| \le \frac{|\lambda|^k \|L\|^k}{1 - |\lambda| \|L\|}.$
3. Let $L$ be as in problem 6, HW8. Estimate how many terms it would require to approximate $(I - \lambda L)^{-1}$ to within $10^{-8}$, if $|\lambda|\le 0.1$.
2. Use Newton's method (see text, problem 3.6.3) to approximate the cube root of 2. Show that the method converges.
3. Section 4.1: 6
4. Section 4.2: 1, 4, 8
5. Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
1. Show that the Green's function for this problem is $G(x,y)=\left\{ \begin{array}{rl} -(2y-1)x, & 0 \le x < y \le 1\\ -(2x-1)y, & 0 \le y< x \le 1. \end{array} \right.$
2. Verify that $0$ is not an eigenvalue for $Kf(x) := \int_0^1G(x,y)f(y)dy$.
3. Show the orthonormal set of eigenfunctions for $L$ form a complete set in $L^2[0,1]$. (Hint: use tthe results from problem 4, HW10.

Updated 12/8/2015.