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Due date: August 18

Problem 1

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

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

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

Problem 2

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

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

for every $x,y$.

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

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

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

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

Problem 3

For $N\ge 4$ and $x=\dfrac{\pi}{N}$,

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

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

Problem 4

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