Abstract
Inverting the Laplace transform is a paradigm for exponentially ill-posed problems. For a
class of operators, including the Laplace transform, we give forward and inverse formulae
that have fast implementations using the fast Fourier transform. These formulae lead easily
to regularized inverses whose effects on noise and filtered data can be precisely described.
Our results give cogent reasons for the general sense of dread most mathematicians feel
about inverting the Laplace transform.
Introduction: 1. Introduction. Inversion of the Laplace transform is the paradigmatic exponen-
tially ill-posed problem. In many inverse scattering problems, the Laplace transform
is, at least implicitly, a part of the forward model, and so the solution of the inverse
scattering problem entails inverting the Laplace transform; see 12, 13, 9, 6. While
it is well understood that this inversion is problematic, to the best of our knowledge,
no one has yet spelled out the details of why, where, and how things go wrong. In
this note we introduce the harmonic analysis appropriate to this problem. On one
hand, this leads to fast numerical forward and inverse algorithms for data which is
log-uniformly sampled. On the other hand, we apply it to study regularized inverses
of the Laplace transform. We analyze the consequences of passing noisy, filtered mea-
surements through the approximate inverse. The picture that emerges is probably
much worse than most people imagine.
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