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

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