Extension of sampling inequalities to Sobolev semi-norms of fractional order and derivative data

This paper, devoted to sampling inequalities, provides some extensions of previous results by Arcang,li et al. (Numer Math 107(2):181-211, 2007; J Approx Theory 161:198-212, 2009). Given a function u in a suitable Sobolev space defined on a domain Omega in , sampling inequalities typically yield bounds of integer order Sobolev semi-norms of u in terms of a higher order Sobolev semi-norm of u, the fill distance d between and a discrete set , and the values of u on A. The extensions established in this paper allow us to bound fractional order semi-norms and to incorporate, if available, values of partial derivatives on the discrete set. Both the cases of a bounded domain Omega and are considered.