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

We consider two types of convolutions ($\ast$ and $\star$) of functions on spaces of finite configurations (finite subsets of a phase space), and some their properties are studied. A connection of the $\ast$-convolution with the convolution of measures on spaces of finite configurations is shown. Properties of multiplication and derivative operators with respect to the $\ast$-convolution are discovered. We present also conditions when the $\ast$-convolution will be positive definite with respect to the $\star$-convolution.