Question

How can every function be expressed as a sum of sine and cosine waves?

Some Setup

Imagine the $\mathbb{R}^3$ . Any point in this space is accessable by projecting onto the basis directions. This vector space is not structured enough. We can not measure length, angle, and similarity.

A inner product is a function that maps two vectors to a number (scalar). It must be:

linearity Namely, $⟨ax+by,z⟩=a⟨x,z⟩+b⟨y,z⟩$
symmetry Namely, $⟨x,y⟩=⟨y,x⟩$
positive definiteness Namely, $⟨x,x⟩≥0$ and $⟨x,x⟩=0⟺x=0$ $\langle x,y \rangle = \sum_{i} x_i y_i$

From this you can get: length (norm)

\|x\| = \sqrt{\langle x,x \rangle}

Without positivity, vectors lengths could be imaginary length.

angle

\cos\theta = \frac{|\langle x,y\rangle|}{\|x\| \|y\|}

Without symmetry, angles become asymmetric.

Orthogonality

⟨x,y⟩=0

Projection and general decomposition would fail.

In Hilbert space of functions, all elements are functions NOT vectors. A function is a point in an infinite-dimensional space. The inner product is the dot product. It measure how align two functions are.

\langle f, g \rangle = \int f(t) g(t) \, dt

Key Idea

A Fourier transform converst a signal over time into frequencies inside the signal.

\hat{f}(\omega) = \int_{-\infty}^{\infty} f(x)e^{-i\omega x}\,dx

$f(x)$ is the original signal and $\hat{f}(\omega)$ is the frequency spectrum and $w$ is the frequency. Note that it’s the original function $f(x)$ multipled against the term $e^{-iwx}$

Where does the term come from? $e^{i\theta} = cos(\theta) + i sin(\theta)$

Requirements

Absolute Integrability ( $L^1$ condition)

The function must decay fast enough so the total area is finite.

Piecewise Continuity

The function must be piecewise continuous on finite intervals.

Dirichlet Conditions

$f(x)$ is absolutely integrable over a period.
$f(x)$ has a finite number of maxima and minima in any interval.
$f(x)$ has a finite number of discontinuities in any interval.

Relaxed Conditions

To generalize it, you can define the transform using distributions

Tempered distributions include functions that are unbounded

Why Bother?

Recasting a function or math object into another coordinate system is a common trick to solve problems that otherwise would not be easily solvable.

Fourier Transform