Fourier Series

A period function $f(x)$ with period $T$ i.e. $f(x+T) = f(x)$ can be represented as a weighted sum of infinitely many sinusoids, called a Fourier series. There are two common ways of writing this series, the first with real coefficients:

\[f(x) = \sum_{n=0}^{\infty} a_n \cos(nx) + b_n \sin(nx)\]

and the second with complex coefficients:

\[f(x) = \sum_{-\infty}^{\infty} c_n e^{inx}\]

Hilbert Space Interpretation

For me, Fourier series clicked when they were explained as an orthogonal basis for the space of square-integrable functions on $[-\pi, \pi]$, denotes $L^2([-\pi, \pi])$. To see this, let’s consider the inner product between $e^{inx}$ and $e^{ikx}$ with $n, k \in \mathbf{Z}$:

\[\langle e^{inx}, e^{ikx} \rangle = \int_{-\pi}^{\pi}e^{inx} \overline{e^{ikx}} dx = \int_{-\pi}^{\pi}e^{inx} e^{-ikx} dx\]

If \(n=k\), the inner product evaluates straightforwardly to

\[\langle e^{inx}, e^{ikx} \rangle = \int_{-\pi}^{\pi} dx = \pi - (-\pi) = 2 \pi\]

If \(n \neq k\), the inner product requires a bit more work:

\[\begin{align*} \langle e^{inx}, e^{ikx} \rangle &= \int_{-\pi}^{\pi} e^{i(n-k)x} dx\\ &= \frac{1}{i(n-k)} e^{i(n-k)x} \Big\lvert_{x=-\pi}^{x=\pi}\\ &= \frac{1}{i(n-k)} (e^{i(n-k)\pi} - e^{-i(n-k)\pi})\\ &= \frac{2 \sin((n-k)\pi)}{m}\\ &= 0 \end{align*}\]

where we used three properties: (1) \(\cos(x) = \cos(-x)\), (2) \(\sin(-x) = -\sin(x)\), and (3) for any integer \(m \neq 0, \sin(m \pi) = 0\).

Fourier Series for Key Functions

Delta Function

\(\delta(x)\) is an even function i.e. \(\delta(x) = \delta(-x)\), so we expect that all the sin terms (the asymmetric terms) will vanish. Let’s practice using the complex exponential representation of the Fourier series. Define

\[\delta(x) = \sum_{n=-\infty}^{\infty} c_n e^{i n x}\]

The complex coefficient for the \(k\)th term, \(c_k\), is found by solving:

\[\begin{align*} \langle \delta(x), e^{ikx} \rangle &= \langle \sum_{n=-\infty}^{\infty} c_n e^{i n x}, e^{i k x} \rangle\\ \int_{-\pi}^{\pi} \delta(x) e^{-ikx} dx &= \sum_{n=-\infty}^{\infty} c_n \langle e^{i n x}, e^{i k x} \rangle\\ e^{-ik(0)} &= 2 \pi c_k\\ \frac{1}{2\pi} &= c_k \end{align*}\]

This tells us a remarkable fact: the series never decays! Every term has the same coefficient. This tells us that the delta function has all frequencies represented with equal strength.