Gaussian semigroups on homogeneous banach spaces

The object of this paper is a study of a Cauchy problem by Gaussian semigroups methods on the 2π-torus in the setting of homogeneous Banach spaces. The approach is similar to that in: [Butzer, P. L.-Berens, H. Semi-Groupsof Operators and Approximation. Springer-Verlag, Berlin, 1967].