Translation beyond Delsarte

We introduce general translations as solutions to Cauchy or Dirichlet problems. This point of view allows us to handle for instance the heat-diffusion semigroup as a translation. With the given examples, Kolmogorov–Riesz characterization of compact sets in certain L μ p spaces is 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.