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:
 * $\displaystyle \mathcal F f(\xi) := \int_{\R^N} f(x)\, e^{-2 \pi i x \xi} \,\mathrm{d}x \quad \text{ for } \xi \in \R^N$

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:
 * $\displaystyle \mathcal F f (\xi) := (2 \pi)^{-\frac{N}{2}} \int_{\R^N} f(x)\, e^{-i x \xi} \,\mathrm{d}x \quad \text{ for } \xi \in \R^N$

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.

Also see
One can also define the Fourier transform of a function in $L^p(\R^N)$ for $1 \leq p \leq 2$, and more in general the Fourier Transform of a Tempered Distribution on $\R^N$.