The concepts of semimartingales and their characteristic triples are introduced for stochastic processes taking their values in a locally compact second countable abelian group. It is proved that the third characteristic always exists and that the first two characteristics always exist when the group is compact. Any continuous additive Gaussian process in a locally compact second countable abel...

We extend the classical Uncertainty Principle to the context of Gelfand pairs. The Gelfand pair setting includes riemannian symmetric spaces, compact topological groups, and locally compact abelian groups. If the locally compact abelian group is Rn we recover a sharp form of the classical Heisenberg uncertainty principle. Section

Is every locally compact abelian group which admits a Heisenberg central extension isomorphic to the product of a locally compact abelian group and its Pontryagin dual? An affirmative answer is obtained for all the commonly occurring types of abelian groups having Heisenberg central extensions, including Lie groups and certain finite Cartesian products of local fields and adèles. Furthermore, f...

The present note is devoted to study a functional equation on a locally compact Abelian group group which appears in characterization of probability distributions by the optimality of an estimate. Let X be a second countable locally compact Abelian group, Y = X be its character group, (x; y) be the value of a character y 2 Y at an element x 2 X. Denote by M(X) the convolution semigroup of proba...

‎let $g$ be a locally compact abelian group‎. ‎the concept of a generalized multiresolution structure (gms) in $l^2(g)$ is discussed which is a generalization of gms in $l^2(mathbb{r})$‎. ‎basically a gms in $l^2(g)$ consists of an increasing sequence of closed subspaces of $l^2(g)$ and a pseudoframe of translation type at each level‎. ‎also‎, ‎the construction of affine frames for $l^2(g)$ bas...

For each locally compact non-compact c-compact Abelian group G whose dual is an I-group and each compact Abelian group ,?J such that there is a one-to-one continous homomorphism from G onto a dense subgroup of Z, we show the existence of cocycles on Z whose associated unitary representations of G have purely singular continuous spectrum, as well as the existence of cocycles on Z whose associate...