Explicit Cutoff Regularization in Coordinate Representation

In this paper, we study a special type of cutoff regularization in the coordinate representation. We show how this approach unites such concepts and properties as an explicit cut, a spectral representation, a homogenization, and a covariance. Besides that, we present new formulae to work with the regularization and give additional calculations of the infrared asymptotics for some regularized Green's functions appearing in the pure four-dimensional Yang-Mills theory and in the standard two-dimensional Sigma-model.