Intuition

Any “reasonable” function can be written as a superposition of pure sinusoids. The Fourier transform is the recipe — it tells you the amplitude and phase of each frequency component needed to reconstruct $f$.

Key properties

Property	Formula
Linearity	$\widehat{af+bg} = a\hat{f} + b\hat{g}$
Shift	$\widehat{f(x-a)}(\xi) = e^{-2\pi ia\xi}\hat{f}(\xi)$
Convolution	$\widehat{f * g} = \hat{f}\cdot\hat{g}$
Parseval	$\int	f	^2 = \int	\hat{f}	^2$
Uncertainty	$\Delta x \cdot \Delta\xi \geq \frac{1}{4\pi}$

Uncertainty principle

Time–frequency localisation is fundamentally limited: a signal that is narrow in time must be broad in frequency, and vice versa. This connects to Heisenberg’s uncertainty principle in quantum mechanics (where the Fourier pair is position/momentum).