Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: *[[Specializations]]* Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* Examples:: [[Fourier series of a square wave]], [[Fourier series of a triangle wave]] ---- > [!definition] Definition. ([[Fourier series]]) > Let $f$ be a [[Riemann integral|Riemann Integrable]] function on $[-\pi, \pi]$. >1. The **$n^{th}$ Fourier Coefficient** of $f$ is $\hat{f}(n):= \frac{1}{2\pi} \int _{-\pi} ^{\pi} f(x)e ^{-in x} \ d x, \ \ n \in \zz .$ >2. The **Fourier Series** of $f$ is $f(\theta) \sim \sum_{n=-\infty}^{\infty} \hat{f}(n) e ^{in \theta} \, .$ >3. The $N ^{th}$ [[partial sum]] of the **Fourier Series** of $f$ is $S _N := \sum_{n=-N}^{N} \hat{f} (n) e ^{in \theta}$. \ >Here, we make no claims regarding [[converge|convergence]]. ^0d1b80 > [!basicexample] > ![[CleanShot 2023-03-24 at [email protected]]] > [!basicproperties] > **Linearity.** $\widehat{(cf+g)}(n)$. > [!proof] Proof of Basic Properties. > **[[linear map|Linearity]].** Compute $\widehat{(cf+g)}(n)=\int _{0} ^{1} (cf(t)+g(t)) e ^{-2 \pi i nt} \, dt = c\int _{0} ^{1} f(t) e ^{-2\pi in t} \, dt + \int _{0} ^{1} g(t) e ^{-2\pi i n t} \, dt = c \hat{f}(n) + \hat{g}(n).$ **(Time) Shift.** Denote by $f_{a}$ the rightward shift of $f$ by $a$. We have $\begin{align} \widehat{f_{a}}(n) = & \int _{t=0} ^{t=1} f(t-a) e ^{- 2 \pi i n t} \, dt \\ \overbrace{=}^{s:=t-a} & \int _{-a} ^{1-a} f(s) e ^{-2\pi i n(s+a)} \, ds \\ = & e ^{- 2 \pi i n a} \int _{-a} ^{1-a} f(s)e ^{-2 \pi i n s} \, ds \\ = & e ^{- 2\pi i n a} \int _{0} ^{1} f(s) e ^{-2\pi i n s} \, ds \text{ ($f$ is 1-Periodic)} \\ =& e ^{- 2\pi i n a} \hat{f}(n). \end{align}$ **Modulation.** For $a \in \mathbb{Z}$, define $w_{a}f$ to be given by $w_{a}f(t):= e ^{2\pi i a t} f(t)$. Then $\begin{align} \big(\widehat{w_{a}f}\big)(n) = & \int _{0}^{1} e ^{2\pi ia t}f(t) e ^{-2\pi i n t}\, dt \\ = & \int _{0} ^{1} f(t) e ^{-2\pi i (n-a)t} \, dt \\ = & \hat{f}(n-a). \end{align}$ **[[Convolution]].** $\begin{align} (\widehat{\textcolor{Skyblue}{f * g}})(n)= & \int _{t=0} ^{t=1} \left[\textcolor{Skyblue}{\int_{s=0}^{s=1} f(s)g(t-s)\, \, ds \,} \right] \textcolor{Apricot}{e ^{-2\pi i n t}} \, dt \\ = & \textcolor{Skyblue}{\int_{s=0} ^{s=1}} \int_{t=0} ^{t=1} \textcolor{Skyblue}{f(s)}\textcolor{Apricot}{e ^{-2\pi i n s}}\textcolor{Skyblue}{g(t-s)} \textcolor{Apricot}{e ^{-2\pi i n (t-s)}} \, \, dt \ \textcolor{Skyblue}{ds} \text{ (Fubini)}\\ = & \int _{s=0} ^{s=1} f(s) e ^{-2\pi i n s} \underbrace{\left[\int_{t=0}^{t=1} g(t-s) e ^{-2\pi i n (t-s)}\, dt \right]}_{{}=\int _{0} ^{1} g(t) e ^{-2\pi i n t} \, dt, \text{ by periodicity} } \, ds \\ = & \left( \int_{s=0}^{s=1} f(s) e ^{-2\pi i n s}\,ds \right)\left( \int_{t=0}^{t=1}g(t) e ^{-2\pi i n t}\, dt \right) \\ =& \hat{f}(n) \hat{g}(n). \end{align}$ ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```