  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.
