10.4: Basic Properties of the Fourier Transform

The Fourier transform has several important properties. These can all be derived from the definition of the Fourier transform; the proofs are left as exercises.

1. The Fourier transform is linear: if we have two functions $f(x)$ and $g(x)$, whose Fourier transforms are $F(k)$ and $G(k)$ respectively, then for any constants $a, b \in \mathbb{C}$, $$a f(x) + b g(x) \;\;\; \overset{\mathrm{FT}}{\longrightarrow} \;\;\; a F(k) + b G(k).$$

2. Performing a coordinate translation on a function causes its Fourier transform to be multiplied by a phase factor: $$f(x+b) \;\;\; \overset{\mathrm{FT}}{\longrightarrow} \;\;\; e^{ikb} \, F(k).$$ As a consequence, translations leave the Fourier spectrum $|F(k)|^2$ unchanged.

3. If the Fourier transform of $f(x)$ is $F(k)$, then $$f^*(x) \quad \overset{\mathrm{FT}}{\longrightarrow} \;\; F^*(-k).$$ As a consequence, the Fourier transform of a real function must satisfy the symmetry relation $F(k) = F^*(-k)$, meaning that the Fourier spectrum is symmetric about the origin in k-space: $\big|\,F(k)\,\big|^2 = \big|\,F(-k)\,\big|^2.$

4. When you take the derivative of a function, that is equivalent to multiplying its Fourier transform by a factor of $ik$: $$\frac{d}{dx} f(x) \,\;\; \overset{\mathrm{FT}}{\longrightarrow} \;\;\; ik F(k).$$

   For functions of time, because of the difference in sign convention discussed in Section 10.3, there is an extra minus sign:

   $$\frac{d}{dt} f(t) \;\;\;\; \overset{\mathrm{FT}}{\longrightarrow} \;\;\; -i\omega F(\omega).$$