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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 \(d\).

MSC:

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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