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

Abstract

This paper, devoted to sampling inequalities, provides some extensions of previous results by Arcangeli 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 Ω in $${{\mathbb{R}}^n}$$, 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 $${\overline\Omega}$$ and a discrete set $${A\subset\overline\Omega}$$, 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 Ω and $${\Omega={\mathbb{R}}^n}$$ are considered.