10.4: Basic Properties of the Fourier Transform
( \newcommand{\kernel}{\mathrm{null}\,}\)
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.
- 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∈C, af(x)+bg(x)FT⟶aF(k)+bG(k).
- Performing a coordinate translation on a function causes its Fourier transform to be multiplied by a phase factor: f(x+b)FT⟶eikbF(k).As a consequence, translations leave the Fourier spectrum |F(k)|2 unchanged.
- If the Fourier transform of f(x) is F(k), then f∗(x)FT⟶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: |F(k)|2=|F(−k)|2.
- When you take the derivative of a function, that is equivalent to multiplying its Fourier transform by a factor of ik: ddxf(x)FT⟶ikF(k).
For functions of time, because of the difference in sign convention discussed in Section 10.3, there is an extra minus sign: ddtf(t)FT⟶−iωF(ω).