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.
-
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
\(a, b \in \mathbb{C}\)
, \[a f(x) + b g(x) \;\;\; \overset{\mathrm{FT}}{\longrightarrow} \;\;\; a F(k) + b G(k).\]
-
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.
-
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:
\(\big|\,F(k)\,\big|^2 = \big|\,F(-k)\,\big|^2.\)
-
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: \[\frac{d}{dt} f(t) \;\;\;\; \overset{\mathrm{FT}}{\longrightarrow} \;\;\; -i\omega F(\omega).\]