Math 641-600 Suggested Problems
Extra Problems - These are not to be handed
in.
- Let $L$ be in $\mathcal B (\mathcal H)$.
- Show that $\|L^k\| \le \|L\|^k$, $k=2,3,\ldots$.
- (We did this in class.) Let $|\lambda| \|L\|<1$. Show that
\[
\big\|(I - \lambda L)^{-1} - \sum_{k=0}^{n-1}\lambda^k L^k\big\| \le
\frac{|\lambda|^k \|L\|^k}{1 - |\lambda| \|L\|}.
\]
- 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
$|\lambda|\le 0.1$.
- Use Newton's method (see text, problem 3.6.3) to approximate the
cube root of 2. Show that the method converges.
- Section 4.1: 6
- Section 4.2: 1, 4, 8
- Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
- Show that the Green's function
for this problem is
\[
G(x,y)=\left\{
\begin{array}{rl}
-(2y-1)x, & 0 \le x < y \le 1\\
-(2x-1)y, & 0 \le y< x \le 1.
\end{array} \right.
\]
- Verify that $0$ is not an eigenvalue for $Kf(x) :=
\int_0^1G(x,y)f(y)dy$.
- 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.