Problem set 5, Harmonic Analysis

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Due March 8

Problem 1

1. If $latex \sum a_n$ converges to $latex s$, then it is Abel-summable to $latex s$.
2. If $latex \sum a_n$ is Cesàro-summable to $latex s$, then is it Abel-summable to $latex s$.

Problem 2 (Tauber's theorem)

If $latex \sum a_n$ is Abel-summable to $latex s$ and $latex na_n \to 0$, then $latex \sum a_n$ coverges to $latex s$.

Problem 3

Suppose $latex f$ has left and right limits at $latex \theta_0$, say

$latex \displaystyle \lim_{\theta\to\theta_0^-}f(\theta) = f(\theta_0-) \qquad \text{and}\qquad \lim_{\theta\to\theta_0^+}f(\theta) = f(\theta_0+),$

then 

$latex \displaystyle \sigma_N(\theta_0) \to \frac{f(\theta_0-) + f(\theta_0+)}{2},$

where $latex \sigma_N(\theta)$ are the Cesàro sums of its Fourier series.

Problem 4 (Bernstein's theorem)

If $latex f\in C^\alpha(\mathbb S)$, for some $latex \alpha > 1/2$, then $latex \sum |\hat{f}(n)| < \infty$.

$latex f\in C^\alpha(\mathbb S)$ means that $latex f$ is Holder continuous with exponent $latex \alpha$, that it, there exists $latex M>0$ such that

$latex |f(x) - f(y)| \le M|x-y|$.

Follow the next steps.

1. For $latex h>0$, $latex \displaystyle \frac{1}{2\pi}\int_0^{2\pi} |f(x+h) - f(x-h)|^2 dx = \sum_{n=-\infty}^\infty 4 |\sin nh|^2 |\hat f (n)|^2$.
2. For $latex h>0$, $latex \displaystyle \sum_{n=-\infty}^\infty 4 |\sin nh|^2 |\hat f (n)|^2 \le 2^{2\alpha}M^2 h^{2\alpha}$.
3. For $latex p\in\mathbb Z_+$, $latex \displaystyle \sum_{2^{p-1} < |n| \le 2^p} |\hat f (n)|^2 \le \frac{M^2 \pi^{2\alpha}}{2^{2\alpha p}}$.
4. For $latex p\in\mathbb Z_+$, $latex \displaystyle \sum_{2^{p-1} < |n| \le 2^p} |\hat f (n)| \le \frac{M \pi^\alpha}{2^{(\alpha - 1/2) p}}$.
5. Conclude $latex \sum_{n=-\infty}^\infty |\hat{f}(n)| < \infty$.

Problem 5

The Dirichlet kernel $latex \displaystyle D_N(\theta) = \sum_{n=-N}^N e^{in\theta}$ is given explicitly by

$latex \displaystyle D_N(\theta) = \frac{\sin(N+1/2)\theta}{\sin \theta/2}.$