Suggest edit — SU-MATH53 FEB252024

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---
title: "SU-MATH53 FEB252024"
source: https://www.jemoka.com/posts/kbhsu_math53_feb252024/
date: 2024-02-25
---

Fourier Decomposition Main idea, any induction $f(x)$ on an interval $[0, L]$ can be written as a sum:
\begin{equation} f(x) = a_0 + \sum_{k=1}^{\infty} a_{k} \cos \qty( \frac{2\pi k}{L} x) + \sum_{k=1}^{\infty} b_{k} \sin \qty( \frac{2\pi k}{L} x) \end{equation}
L-periodicity A function is $L$-periodic if $f(x+L) = f(x)$ for nonzero $L$ for all $x$. The smallest $L &gt; 0$ which satisfies this property is called the period of the function.
$L$-periodicity is preserved across&hellip;
translation we are just moving it to the right/left
dilation Suppose $f(x)$ is $L$ periodic and let $g(x) = f(kx)$, then, $g$ is also $L$ periodic.
Proof:
$g(x+L) = f(k(x+L)) = f(kx + kL) = f(kx) = g(x)$. So $g$ would also be $L$ periodic. However, importantly, $g$ would also be $\frac{L}{k}$ periodic (verified by using the same sketch as before)
linear combinations Suppose $f,g$ are $L$ periodic and $h(x) = af(x) + bg(x)$, then $h$ is also $L$ periodic.
Proof:
\begin{equation} h(x+L) = af(x+L) + bg(x+L) = af(x) + bg(x) = h(x) \end{equation}
Fourier Series see Fourier Series