## 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** \(L > 0\) which satisfies this property is called the period of the function.

$L$-periodicity is preserved across…

### 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