Properties of the Fourier Transform

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Properties of the Fourier Transform

Let us name a function, and its Fourier transform. We use here the simple, non-unitary convention

- Definition:
- Linearity: , then
- Translation: , then
- Shifting: , then
- Scaling: , then (the so-called time reversal property is obtained with
- Conjugation: if , then
- Integration: With in the definition
- Convolution: then .
  The Fourier Transform of a product of two functions is the convolution of the Fourier Transforms of the functions.
- Uncertainty principle: the more concentrated is, the more spread out its Fourier transform must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in , its Fourier transform stretches out in . It is not possible to arbitrarily concentrate both a function and its Fourier transform.

The definition, illustrated here with scalars, also holds for being 2-D vectors in Euclidean space, the product in the definition being a standard dot product of these vectors.