Estimates for functions in Sobolev spaces defined on unbounded domains

Given a function uu belonging to a suitable Beppo–Levi or Sobolev space and an unbounded domain ΩΩ in RnRn, we prove several Sobolev-type bounds involving the values of uu on an infinite discrete subset AA of ΩΩ. These results improve the previous ones obtained by Madych and Potter [W.R. Madych, E.H. Potter, An estimate for multivariate interpolation, J. Approx. Theory 43 (1985) 132–139] and Madych [W.R. Madych, An estimate for multivariate interpolation II, J. Approx. Theory 142 (2006) 116–128].