On the Theta semigroup

TL;DR

This paper introduces the Theta semigroup, a positive diffusion semigroup based on the third Jacobi theta function, and explores its properties, extensions, and relation to the classical Poisson semigroup.

Contribution

It defines the Theta semigroup, analyzes its properties, and establishes its connection to the Poisson semigroup, extending the framework to higher dimensions and ultra distributions.

Findings

The Theta semigroup is a positive diffusion semigroup.

Its subordinated semigroup is the classical Poisson semigroup.

Extensions to higher dimensions and ultra distributions are achieved.

Abstract

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.