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Physics LibreTexts

10.4: Basic Properties of the Fourier Transform

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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,bC, af(x)+bg(x)FTaF(k)+bG(k).

  2. Performing a coordinate translation on a function causes its Fourier transform to be multiplied by a phase factor: f(x+b)FTeikbF(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)FTF(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.

  4. When you take the derivative of a function, that is equivalent to multiplying its Fourier transform by a factor of ik: ddxf(x)FTikF(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)FTiωF(ω).


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.

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