Homework 2, Real Analysis

Due date: August 18

Problem 1

Let $latex f\in C^k(\mathbb S)$, a k-continuously differentiable periodic function, with period $latex 2\pi$, and let $latex a_n$ be its nth Fourier coefficient.

1. There exists $latex C>0$ such that $latex |a_n| \le \dfrac{C}{|n|^k}$.


2. The series $latex \sum a_n e^{inx}$ converges uniformly if $latex k\ge 2$.

Problem 2

Let $latex f\in C(\mathbb S)$ be Lipschitz continuous, ie there exists A such that

$latex |f(x) - f(y)| \le A|x-y|$

for every $latex x,y$.

1. Fix $latex h>0$ and let $latex g_h(x) = f(x+h) - f(x-h)$. Then
   $latex \displaystyle \frac{1}{2\pi}\int_0^{2\pi}|g_h(x)|^2 dx = \sum_{n\in\Z} 4|a_n|^2 |\sin nh|^2$, and thus
   $latex \displaystyle \sum_{n\in\Z} |a_n|^2|\sin nh|^2 \le A^2h^2$, where the $latex a_n$ are the Fourier coefficients of f.


2. Let $latex p\in\Z_+$ and $latex h = \dfrac{\pi}{2^{p+1}}$. Then $latex \displaystyle \sum_{2^{p-1} < |n|\le 2^p} |a_n|^2 \le \frac{A^2\pi^2}{2^{2p+1}}.$


3. $latex \displaystyle \sum_{2^{p-1}<|n|\le 2^p} |a_n| \le \frac{A\pi}{2^{(p+1)/2}}.$


4. The Fourier series of f converges absolutely and uniformly.

Problem 3

For $latex N\ge 4$ and $latex x=\dfrac{\pi}{N}$,

$latex \dfrac{\sin x + \sin Nx - \sin(N+1)x}{2(1-\cos x)} \ge \dfrac{5N}{3\pi},$

and thus $latex \displaystyle \sum_{n=1}^N \sin nx$ is not uniformly bounded in x and N.

Problem 4

Let X be a metric space and $latex \bar X$ its completion. Then, for each $latex x\in\bar X$, there exists a sequence $latex x_n$ in X such that $latex x_n\to x$ in $latex \bar X$.