Definition:Fourier Transform

Definition
The Fourier transform of a Lebesgue integrable function $$f:\R^N \to \C$$ is the function $$\mathcal{F}f: \R^N \to \C$$ given by

\mathcal{F}f(\xi) := \int_{\R^N} f(x)\, e^{-2 \pi i x \xi} \,dx \quad \text{ for } \xi \in \R^N. $$ Here, the product $$x \xi$$ in the exponential is the scalar product of the vectors $$x$$ and $$\xi$$.

The Fourier transform of $$f$$ is also frequently denoted by $$\hat{f}$$. When confusion can arise, one may write $$\mathcal{F}[f]$$ instead of $$\mathcal{F} f$$.

Correctness of the definition
The function under the integral in the definition is Lebesgue integrable, as $$\vert f(x)\, e^{-2 \pi i x \xi} \vert = \vert f(x) \vert$$, and $$f$$ is assumed to be integrable.

Other commonly used definitions
There exist several 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:

\mathcal{F}f(\xi) := (2 \pi)^{-\frac{N}{2}} \int_{\R^N} f(x)\, e^{-i x \xi} \,dx \quad \text{ for } \xi \in \R^N. $$ Their properties are essentially the same, and by a simple change of variable one can always translate statements using one of the definitions into statements using another one.