Fourier-Laplace transform

For $f$ a suitable (generalized) function on an affine space, its Fourier transform is given by $\hat f (a) \propto \int f(x) e^{i x a} d x$, while its Laplace transform is $\tilde f(a) \propto \int f(x) e^{-a x} d x$, when defined. Clearly these are two special cases of a single “transform” where $a$ is allowed to be complex; this is hence called the *Fourier-Laplace transform*.

**(Fourier-Laplace transform of compactly supported distributions)**

For $n \in \mathbb{N}$, let $u \in \mathcal{E}'(\mathbb{R}^n)$ be a compactly supported distribution on Cartesian space $\mathbb{R}^n$. Then its *Fourier transform of distributions* is the function

$\array{
\mathbb{R}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R}
\\
\zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right)
}$

where on the right we have the application of $u$, regarded as a linear function $u \colon C^\infty(\mathbb{R}^n) \to \mathbb{R}$, to the exponential function applied to the canonical inner product $\langle -,-\rangle$ on $\mathbb{R}^n$.

This same formula makes sense more generally for complex numbers $\zeta \in \mathbb{C}^n$. This is then called the *Fourier-Laplace transform* of $u$, still denoted by the same symbol:

$\array{
\mathbb{C}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R}
\\
\zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right)
}$

This is an entire analytic function on $\mathbb{C}^n$.

(Hörmander 90, theorem 7.1.14)

- Lars Hörmander, section 2.3 of
*The analysis of linear partial differential operators*, vol. I, Springer 1983, 1990

Last revised on April 3, 2020 at 11:26:48. See the history of this page for a list of all contributions to it.