  In this paper we consider a semigroup on trigonometric expansions that will
be called the Theta semigroup since its kernel is a multiple of the third
Jacobi theta function. We study properties of this semigroup and prove that it
is a positive diffusion semigroup. We also obtain that its subordinated
semigroup is the classical Poisson semigroup. The extensions to higher
dimensions and to periodic ultra distributions are also considered.
