Sampling inequalities in Sobolev spaces

Sampling inequalities in the Sobolev space  Wr,p(Ω)Wr,p(Ω), where ΩΩ is a domain of RnRn, are defined as relations like|u|l,q,Ω≤C(dr−l−n(1/p−1/q)|u|r,p,Ω+dn/q−l(∑a∈A|u(a)|p)1/p),l≤ℓ, for suitable values of rr, pp, qq and ℓℓ. In this statement, uu denotes a function in Wr,p(Ω)Wr,p(Ω), AA is a discrete   set in Ω¯ and d=supx∈Ωinfa∈A|x−a|d=supx∈Ωinfa∈A|x−a|.The structure of the sampling inequalities in spaces Wr,p(Ω)Wr,p(Ω) is analysed, their role in the study of the interpolation error by spline functions is recalled, and the analogy between these inequalities and those relative to intermediate semi-norms in spaces Wr,p(Ω)Wr,p(Ω) (cf., for example, Adams and Fournier (2003) [1, Theorem 5.2–(1)]) is shown.