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

This page titled 10.4: Basic Properties of the Fourier Transform is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.