nLab Fourier-Laplace transform

Idea

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.

Definition

Proposition/Definition

(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)

Related concepts

• Paley-Wiener-Schwartz theorem

• wave front set

References

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