---
date: 2022-03-28 12:35
modification date: Monday 28th March 2022 12:35:00
title: "Fourier Analysis"
aliases: [Fourier coefficient, Fourier series, convolution, Dirichlet kernel]
---

Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
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- Refs:
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- Links:
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---

# Fourier Analysis


## Definitions


### Fourier Series


- Fourier coefficients: $$\hat{f}(n)=a_{n}=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(\theta) e^{-i n \theta} d \theta, \quad n \in \mathbb{Z}$$
- Partial sums: $$S_N f(\theta) \sim \sum_{\abs n \leq N} a_{n} e^{i n \theta}$$
- Fourier series: $$f(\theta) \sim \sum_{n=-\infty}^{\infty} a_{n} e^{i n \theta} = \lim_{N\to \infty} S_N f(\theta), \qquad c_n = \hat{f}(n)$$
- Parseval: $$\|f\|_{L^{2}(-\pi, \pi)}^{2}=\int_{-\pi}^{\pi}|f(x)|^{2} d x=2 \pi \sum_{n=-\infty}^{\infty}\left|c_{n}\right|^{2}$$
- Compuattions of Fourier expansions: 
	- $$f(\theta) \da {1\over 4}(\pi - \theta)^2 \implies f(\theta) \sim \frac{\pi^{2}}{12}+\sum_{n=1}^{\infty} \frac{\cos n \theta}{n^{2}}$$
	- $$
f(\theta)=\frac{\pi}{\sin \pi \alpha} e^{i(\pi-\theta) \alpha}
\implies 
f(\theta) \sim \sum_{n=-\infty}^{\infty} \frac{e^{i n \theta}}{n+\alpha}
$$
	- Applying Parseval: $$f(x) = x \implies c_n ={(-1)^{n}}{}\cdot {i\over n}  \implies  \zeta(2) \da \sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}$$

## Convolutions

- Convolution: $$(f * g)(x)=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(y) g(x-y) d y$$
- Dirichlet kernel: $$D_{N}(x)=\sum_{n=-N}^{N} e^{i n x}=\frac{\sin \left(\left(N+\frac{1}{2}\right) x\right)}{\sin (x / 2)}$$
	- Fourier series as a convolution: $$S_N f(x) =\left(f * D_{N}\right)(x)$$
- Poisson kernel: $$P_{r}(\theta)=\sum_{n=-\infty}^{\infty} r^{|n|} e^{i n \theta} = \frac{1-r^{2}}{1-2 r \cos \theta+r^{2}}$$
	-  Arises in solution of steady-state temperature of heat on a plate.


## Convergence

- Big $\bigo$ notation: $$f\in \bigo(g) \iff \exists x_0 \in X, M\in \RR, x\geq x_0 \implies \abs{f(x)} \leq M\abs{g(x)}$$
	- $f\in \bigo(1) \implies f$ is bounded.
- Classes of functions:
	- $C^0$: continuous functions
		- Holder continuous: $$|f(x)-f(y)| \leq C\|x-y\|^{\alpha}$$
		- A general Holder condition: $$\sup _{\theta}|f(\theta+t)-f(\theta)| \leq A|t|^{\alpha} \quad \text { for all } t$$
		- Lipschitz cts: $$d_{Y}\left(f\left(x_{1}\right), f\left(x_{2}\right)\right) \leq C\cdot d_{X}\left(x_{1}, x_{2}\right)$$
	- $C^n$: $n$ times differentiable with $f^{(n)}$ continuous.
	- Piecwise linear functions
	- Almost everywhere variants of all of the above.
	- Riemann integrable: $f$ is bounded and there exists a partition with $\abs{U - L } < \eps$
	- Lebesgue integrable
- $L^p$ norms: $$\norm{f}_{L^p} \da \qty{\int_X \abs{f}^p \dmu}^{1\over p}, \qquad \norm{f}_{L^\infty} = \ess \sup{f} = \inf \{C \geq 0:|f(x)| \leq C \text { for almost every } x\}$$
- Types of convergence:
	- Pointwise: $$\abs{f_n(x) - F(x) } \convergesto{n\to\infty} 0.$$
	- Pointwise a.e.: $$M_n \da \mu\ts{x\in X \st f_n(x) \not\to x} \convergesto{n\to\infty} 0$$
	- Uniform: $$\sup \ts{\abs{f_n(x) - f(x)} \st x\in X} \convergesto{n\to\infty}0$$
	- In sup norm $L^\infty$: $$\norm{f_n - f}_{L^\infty} \convergesto{n\to\infty}0$$
	- $L^2$: $$\norm{f_n - f}_{L^2} \convergesto{n\to\infty}0$$
		- Explicit $L^2$, i.e. mean square:$$\frac{1}{2 \pi} \int_{-\pi}^{\pi}\left|S_{N}(f)(\theta)-f(\theta)\right|^{2} d \theta \convergesto{N\to\infty} 0$$
	- Convergence in measure: $$\lim _{n \rightarrow \infty} \mu\left(\left\{x \in X:\left|f(x)-f_{n}(x)\right| \geq \varepsilon\right\}\right)\convergesto{n\to\infty}0$$

- **Theorems**:
	- The uniform limit of continuous functions is continuous.
	- $f$ is Riemann integrable iff $f$ is bounded and has null discontinuity set. Necessary because $f(x) = \chi_\QQ(x)$ is not integral on $[0, 1]$ since it is discts at every irrational.
	- If $f$ is cts and $\ts{K_n}$ is a good kernel, $$(f\convolve K_n) \convergesto{n\to\infty} f$$
		- The Dirichlet kernel is not a good kernel!
	- If $f_n\to f$ uniformly on a compact set $A$ then $\int_A f_n\to \int_A f$. Compactness is necessary: take $f_n(x) \da \chi_{[0, n]}(x)\cdot {1\over n}$.
	- If $f: S^1\to \RR$ is integrable and $\hat{f}(n) = 0$ for all $n$, then $f(x)= 0$ on the set of continuity of $f$.
	- If $f:S^1\to \RR$ is cts and $\ts{ \hat f(n)} \in \ell^1(\ZZ)$ (so the Fourier series converges absolutely), then $\hat f \to f$ uniformly.
	- If $f\in C^2(S^1)$, then $\hat f(n) \in \bigo(\abs{n}^{-2})$ and $\hat f$ converges absolutely and $S_N f \to f$ uniformly.
- Riemann-Lebesgue 
	![](attachments/Pasted%20image%2020220404164825.png)
	- ![](attachments/Pasted%20image%2020220404164951.png)
- Fourer inversion:
  ![](attachments/Pasted%20image%2020220404164839.png)
- Properties of the Fourier transform:
	![](attachments/Pasted%20image%2020220404165010.png)

# Common counterexamples

- A discontinuous Riemann-integrable function: $\chi_{[0, 1]} - \chi_{\ts{1\over 2}}$
  
- A Riemann-integrable function with countably infinitely many discontinuities:
  ![](attachments/Pasted%20image%2020220328123856.png)
- Convergence comparisons:
  ![](attachments/Pasted%20image%2020220404164617.png)
	- ![](attachments/Pasted%20image%2020220404164630.png)
	- ![](attachments/Pasted%20image%2020220404164642.png)
	- ![](attachments/Pasted%20image%2020220404164654.png)

# Exercises

- $\mathcal{F}\left(f^{\prime}\right)(\xi)=2 \pi i \xi \cdot \mathcal{F}(f)(\xi)$
- $\widehat{f\convolve g}(n) = \hat{f}(n) \cdot \hat{g}(n)$:
	- ![](attachments/Pasted%20image%2020220328130054.png)


- Show that the partial sums of Fourier coefficients are given by convolution against the Dirichlet kernel:
![](attachments/Pasted%20image%2020220328125931.png)