Extension of sampling inequalities to Sobolev semi-norms of fractional order and derivative data. (English) Zbl 1259.46025

For a given function \(u\) in a suitable Sobolev space defined on a domain \(\Omega\subset\mathbb R^n\), sampling inequalities usually 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 purpose of this paper is to extend the results of the authors [ibid. 107, No. 2, 181–211 (2007; Zbl 1221.41012) and J. Approx. Theory 161, No. 1, 198–212 (2009; Zbl 1181.41051)] for both cases \(\Omega\) bounded and \(\Omega= \mathbb R^n\), in two directions: to allow fractional order semi-norms on the left-hand side of the sampling inequality, and to admit derivative pointwise values on the second term of the right-hand side of the sampling inequality, so increasing the order of the approximation of the parameter \(d\).


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
41A63 Multidimensional problems
41A05 Interpolation in approximation theory
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