On convolutions on configuration spaces. I. Spaces of finite configurations

We consider two types of convolutions (* and a similar to...) of functions on spaces of finite configurations (finite subsets of a phase space) and study some of their properties. A relationship between the *-convolution and the convolution of measures on spaces of finite configurations is described. Properties of the operators of multiplication and derivation with respect to the *-convolution are investigated. We also present conditions under which the *-convolution is positive-definite with respect to the a similar to...-convolution.