1. The Opening Question: From Taylor to Fourier
Recall the Taylor series: given a smooth function $f(x)$, we write
$$ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots $$
This is local polynomial mimicry — the approximation is excellent near the point $a$ but may diverge far away. Taylor series use information at one point (derivatives at $a$) to reconstruct the function.
| Feature | Taylor Series | Fourier Series |
|---|---|---|
| Building blocks | Powers: $1, x, x^2, x^3, \ldots$ | Waves: $\cos(nx), \sin(nx)$ |
| Information used | Derivatives at one point | Integrals over one full period |
| Scope | Local (near $x = a$) | Global (entire period) |
| Best for | Smooth, analytic functions | Periodic functions (even discontinuous ones) |
The Periodic Vocabulary
A function $f(x)$ is periodic with period $T$ if $f(x + T) = f(x)$ for all $x$. The base frequency (fundamental) is $\omega_0 = \frac{2\pi}{T}$. The sine-cosine family at integer multiples of this frequency:
$$ \cos(n\omega_0 x), \quad \sin(n\omega_0 x), \quad n = 1, 2, 3, \ldots $$
are called the harmonics. In music, $n=1$ is the fundamental tone, and $n=2, 3, \ldots$ are the overtones. A musical instrument's timbre is determined by how much of each harmonic is present — this is exactly what Fourier coefficients measure.