The Fourier Transform
A Fourier series describes a periodic function with a discrete set of frequencies
n\pi/L. But what about a one-off pulse that never repeats? Let the
period grow — push L \to \infty — and the spacing between allowed
frequencies shrinks to zero. The sum over discrete n melts
into an integral over a continuous frequency k. That limit is
the Fourier transform.
\hat{f}(k) = \int_{-\infty}^{\infty} f(x)\, e^{-ikx}\,dx, \qquad f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(k)\, e^{ikx}\,dk.
The first integral analyses f into its frequency content
\hat{f}(k); the second synthesises f back
from it. The discrete coefficients c_n have become a continuous
spectrum.
The hallmark: narrow in, wide out
The Fourier transform trades localisation between the two domains. A sharply concentrated pulse
needs many frequencies to build, so its transform is spread out; a broad, gentle pulse needs
only a few low frequencies, so its transform is narrow. This reciprocity — the foundation of the
Heisenberg uncertainty principle — is cleanest for the Gaussian, which is its
own transform shape:
f(x) = e^{-a x^2} \;\Longrightarrow\; \hat{f}(k) = \sqrt{\tfrac{\pi}{a}}\; e^{-k^2 / (4a)}.
A wide bump in x (small a) becomes a narrow
bump in k, and vice versa.
Two Gaussians, reciprocal widths
The pulse is on the left, its transform on the right. Slide the width parameter
a: as the pulse narrows, its transform broadens —
the two widths move in opposite directions. You can never make both narrow at once.