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

TL;DR

This paper investigates two types of convolutions on finite configuration spaces, exploring their properties, connections to measure convolutions, and conditions for positive definiteness, advancing the mathematical understanding of convolutions in configuration spaces.

Contribution

It introduces and analyzes two convolutions on finite configuration spaces, establishing their properties, connections to measure convolutions, and conditions for positive definiteness.

Findings

Connection between $ ext{-}$convolution and measure convolution.

Properties of multiplication and derivative operators w.r.t. $ ext{-}$convolution.

Conditions for $ ext{-}$convolution to be positive definite.

Abstract

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.