Quaternion-valued positive definite functions on locally compact Abelian groups and nuclear spaces

In this talk we introduce and study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space of countably infinite tuples of real numbers endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner–Minlos theorem. We will see that in all these various settings the integral representation is with respect to a quaternion-valued measure which has certain symmetry properties. This talk is based on joint work with D. Alpay, F. Colombo and I. Sabadini

On a Class of Quaternionic Positive Definite Functions and Their Derivatives

In this paper, we start the study of stochastic processes over the skew field of quaternions. We discuss the relation between positive definite functions and the covariance of centered Gaussian processes and the construction of stochastic processes and their derivatives. The use of perfect spaces and strong algebras and the notion of Fock space are crucial in this framework

Quaternion-valued positive definite functions on locally compact Abelian groups and nuclear spaces

In this paper we study quaternion-valued positive definite functions on locally compact Abelian groups, real countably Hilbertian nuclear spaces and on the space RN=(x1,x2,...):xdϵR endowed with the Tychonoff topology. In particular, we prove a quaternionic version of the Bochner-Minlos theorem. A tool for proving this result is a classical matricial analogue of the Bochner-Minlos theorem, which we believe is new. We will see that in all these various settings the integral representation is with respect to a quaternion-valued measure which has certain symmetry properties