For a suitable (generalized) function on an affine space, its Fourier transform is given by , while its Laplace transform is , when defined. Clearly these are two special cases of a single “transform” where is allowed to be complex; this is hence called the Fourier-Laplace transform.
(Fourier-Laplace transform of compactly supported distributions)
For , let be a compactly supported distribution on Cartesian space . Then its Fourier transform of distributions is the function
where on the right we have the application of , regarded as a linear function , to the exponential function applied to the canonical inner product on .
This same formula makes sense more generally for complex numbers . This is then called the Fourier-Laplace transform of , still denoted by the same symbol:
This is an entire analytic function on .
(Hörmander 90, theorem 7.1.14)
Last revised on June 19, 2024 at 22:41:13. See the history of this page for a list of all contributions to it.