Fourier-Laplace transform




For ff a suitable (generalized) function on an affine space, its Fourier transform is given by f^(a)f(x)e ixadx\hat f (a) \propto \int f(x) e^{i x a} d x, while its Laplace transform is f˜(a)f(x)e axdx\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 aa is allowed to be complex; this is hence called the Fourier-Laplace transform.



(Fourier-Laplace transform of compactly supported distributions)

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

n u^ ζ u(e i,ζ) \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 uu, regarded as a linear function u:C ( n)u \colon C^\infty(\mathbb{R}^n) \to \mathbb{R}, to the exponential function applied to the canonical inner product ,\langle -,-\rangle on n\mathbb{R}^n.

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

n u^ ζ u(e i,ζ) \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 n\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.