Day 4 (September 25, 2024)

[mpham26uchicago]

Before we can prove further results about $e^{-2t}$ convergence rate, we need to show that $v(t) \to e^\nu$ at $e^{-t}$ rate. For some reason, Theorem 11.3.3 falls short. An idea to use Theorem 11.3.4. We need to prove the present theorem and also the case where $f \in BC_\ell$. Then what we can do is to multiply both sides of the integral equations for $x(t)$ by $e^t$ and to show that this approaches a constant. Ultimately, what we want to do is to show that $v(t)$ and $v\prime(t)$ has the desired asymptotic expansion.

[mpham26uchicago]

1. Restudy Sections 11.2 and 11.3
2. Prove theorem 11.3.4
3. Prove Theorem 11.3.4 but now also assume that $f \in BC_\ell$.
4. Check if this holds for $x(t)$.

[mpham26uchicago]

Look in the Exercises of Chapter 11. For example, Theorems 11.7.1 and 11.7.2, might be of interest.

[mpham26uchicago]

Theorem 7.2 looks promising. The only thing is it doesn't seem to match experimental results.