  We introduce general translations as solutions to Cauchy or Dirichlet
problems. This point of view allows us to handle the heat-diffusion semigroup
as a translation. With the given examples Kolmogorov-Riesz characterization of
compact sets in certain $L^p_\mu$ spaces are given. Pego-type characterizations
are also derived. Finally for some examples the equivalence of the
corresponding modulus of smoothness and K-functional is pointed out.
