Category:Fourier Transforms

This category contains results about Fourier Transforms.
Definitions specific to this category can be found in Definitions/Fourier Transforms.

The Fourier transform of a Lebesgue integrable function $f: \R^N \to \C$ is the function $\map \FF f: \R^N \to \C$ given by:

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

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

Here, the product $\mathbf x \cdot \mathbf s$ in the exponential is the dot product of the vectors $\mathbf x$ and $\mathbf s$.

In this context $\map \FF f$ is to be considered the operator.