Problem set 4, Harmonic Analysis

Due March 1

Problem 1

If the sequences $a_n$ and $b_n$ are bounded, then 

$\displaystyle u(r,\theta) = \sum_{n=0}^\infty r^n (a_n \cos(n\theta) + b_n \sin(n\theta))$

is harmonic in $\mathbb D$.

Problem 2

For $m,n\in\mathbb Z$,

$\displaystyle \int_0^{2\pi} e^{im\theta}e^{-in\theta} d\theta = \begin{cases} 2\pi & m=n\\0 & m\not=n.\end{cases}$

Problem 3

The series

$\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n} \sin n\theta$

converges for each $\theta$.

Problem 4

If $f$ is Riemann integrable and periodic with period $T$, then

$\displaystyle \int_a^{a+T} f(x) dx = \int_0^T f(x) dx$

for any $a\in\mathbb R$.

Problem 5

Let $g$ be the periodic function given in $[-\pi,\pi)$ by $g(\theta) = |\theta|$.

1. For each $n\in\mathbb Z$,
   
   $\displaystyle \hat g(n) = \begin{cases}\dfrac{\pi}{2} & n=0\\0 & \text{even } n\not=0\\- \dfrac{2}{\pi n^2} & \text{odd } n.\end{cases}$
2. The Fourier series of $g$ is given by
   
   $\displaystyle \frac{\pi}{2} - \frac{4}{\pi} \sum_{k=1}^\infty \frac{\cos(2k+1)x}{(2k+1)^2}.$