Harmonic Analysis on Conelike Bodies and Holomorphic Functions on Tube Domains

In this paper, we establish Bochner–Weil type theorems and integral formulas of Lévy–Khinchin type in the setting of locally compact commutative semigroups with involution. These results are used to prove some new holomorphic extension results. For a conelike involution semigroup S in a finite dimensional real space V, we establish integral characterizations for continuous positive definite and definitizable functions on S. In particular, we obtain holomorphic and Cauchy–Riemann extensions of these functions by means of an integral representation. We prove that ifz0is a point in the complexified vector space W :=V+iV and if a function is defined onz0+S then either positive definiteness or definitizability offextends holomorphy from a domainΩcontainingz0to a tube domain in W containing the sum of 0 and the convex cone spanned by S. We also obtain the holomorphic extension as a generalized Fourier–Laplace transform of some measure or a Lévy–Khinchin type integral representation.