Laplace transform on a convex cone

Summary

The Laplace transform on a convex cone integrates a function on a closed convex cone $\Gamma \subseteq \R^n$ against $e^{-\ip{y}{x}}$. Here $y$ runs over the interior of the dual cone. For the orthant $\Gamma = [0, \infty)^n$ it is the multidimensional Laplace transform. For a function of moderate growth, the transform extends to a holomorphic function on a tube domain over the interior of the dual cone. On the cone of positive definite matrices it gives the multivariate gamma function, and it is a basic tool of analysis on symmetric cones.