Definition:Fourier Transform/Also defined as

Fourier Transform: Also defined as

There exist a number of slightly different definitions of the Fourier transform which are commonly used.

They differ in the choice of the constant $2 \pi$ inside the exponential and/or a multiplicative constant before the integral.

The following definition is also very common:

$\ds \map {\map \FF f} {\mathbf s} := \paren {2 \pi}^{-\frac N 2} \int_{\R^N} \map f {\mathbf x} \, e^{-i \mathbf x \cdot \mathbf s} \rd \mathbf x$

for $\mathbf s \in \R^N$.

Their properties are essentially the same.

By a simple change of variable one can always translate statements using one of the definitions into statements using another one.

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